Geografic-Information-System/lib/cdhc/doc/goodness.tex

\documentclass[draft]{article}
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%\title{\(\rightarrow\)DRAFT\(\leftarrow\)\\
%Goodness of Fit Techniques}

\title{Goodness of Fit Tests\\ 
{\large Documentation on {\tt libcdhc.a}}\\
{\large and}\\
{\large A GRASS Tutorial on {\tt s.normal}}}

\author{James Darrell McCauley\thanks{USDA National Needs Fellow,
Department of Agricultural Enginering, Purdue University. Email:
{\tt mccauley@ecn.purdue.edu}}}

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\def\libname{{\tt cdhc}}

\def\returns#1{\sffamily\slshape Returns \(\mathsf{#1}\).}

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\begin{document}
\bibliographystyle{plain}
\maketitle

\begin{abstract}
The methods used by the GRASS program {\tt s.normal}
are presented. These are various goodness of fit statistics for testing
the null hypothesis of normality. Other additional tests found in
\libname\, a C programming library, 
are also documented (this document serves two puposes:
a tutorial for the GRASS geographic information system
and documentation for the library).
\end{abstract}

\section{Introduction}

This document is a programmer's 
manual for \libname, a C programming library useful
for testing whether a sample is normally, lognormally,
or exponentially distributed.
Prototypes for library functions\footnote{%
Each function in the library returns a pointer to static double.
The \libname\ library was inspired by Johnson's
STATLIB collection of FORTRAN routines for testing 
distribution assumptions~\protect\cite{johnson94}.  
Some functions in \libname\
are loosely based on Johnson's work (they have been completely
rewritten, reducing memory requirements and number
of computations and fixing a few bugs). Others are based on
algorithms found in \emph{Applied Statistics}, \emph{Technometrics},
and other related journals.} 
are given in the margins near
corresponding mathematical explanations. Hence, it is also
a user's guide for programs using \libname.
Readers should be equipped with at least one graduate course
in probability and statistics. Much of the background
and derivation/justification of each test has been
omitted. A good text for more background information
is {\em Goodness-of-Fit Techniques\/} by
D'Agostino and Stephens~\cite{dagostino86b} (see also references in text). 

\subsection{Hypothesis Testing}

Before beginning the description of the tests, a few definitions
should be given. The general framework for mosts tests is that
the {\em null\/} hypothesis \(H_0\) is that a random variable \(x\)
follows a particular distribution \(F\left(x\right)\). 
Generally, the {\em alternative\/} hypothesis is that
\(x\) does not follow \(F\left(x\right)\) (with no additional
usuable information; the Kotz Separate Families test in \S\ref{sec:kotz}
is one exception). 
This may differ from the way that some have learned hypothesis testing
in that some tests are set up to reject the null hypothesis in
favor of the alternative.

A {\em simple\/} hypothesis implies that \(F\left(x\right)\)
is completely specified, e.g., \(x\sim N\left(0,1\right)\).
A {\em composite\/} hypothesis means that 
one (or more) of the parameters of \(F\left(x\right)\)
is not completely specified, e.g.,  \(x\sim N\left(\mu,\sigma\right)\).
That is, the composite hypothesis may be:
\begin{displaymath}
H_0 : F\left(x\right) = F_0\left(x; \theta\right)
\end{displaymath}
where \(\theta=\left[\theta_1, \ldots,\theta_p\right]'\)
is a \(p\) vector of \emph{nuisance} parameters whose values
are unknown and must be estimated from data. 

% Less is known
% about the theory of this later
% case, which is the most commonly encountered in practice.

\subsection{Probability Plots}

In addition to these analytical techniques, graphical
methods are valuable supplements. The most important
graphical technique is probability plotting. A \emph{probability plot}
\label{pplot}
is a plot of the cumulative distribution function \(F\left(x\right)\)
on the vertical axis versus \(x\) on the horizontal axis.
The vertical axis is scaled such that, if the data fit
the assumed distribution, the resulting plot will lie on
a straight line. Special plotting paper may be purchased
to do these plots; however, most modern scientific
plotting programs have this capability (e.g., {\tt gnuplot}).
Each test presented below
should be used in conjunction with a probability plot.
 
\subsection{Shape of Distributions}

Through much of the literature are references to Johnson
curves: \(S_U\) or \(S_B\) (see \S\ref{sec:johnson-su}, 
page~\pageref{sec:johnson-su}).
These refer to a system of distributions introduced by 
Johnson~\cite{johnson49} where a standard normal random
variable \(Z\) is translated to \(\left(Z-\gamma\right)/\delta\)
and transformed using \(T\):
\begin{equation}
Y=T\left(\frac{Z-\gamma}{\delta}\right).
\end{equation}
Three families in Johnson's~\cite{johnson49} system are:
\begin{enumerate}
\item a family of bounded distributions, denoted by \(S_B\), where:
\begin{equation}
Y=T\left( \frac{e^x}{1+e^x} \right);
\end{equation}
\item a family lognormal distributions where:
\begin{equation}
Y=T\left( e^x \right);
\end{equation}
\item and a family of unbounded distributions, denoted by \(S_U\), where:
\begin{equation}
Y=\sinh\left(x\right) = T\left( e^x-e^{-x} \right).
\end{equation}
\end{enumerate}
In the \(S_B\) and \(S_U\) families, \(\gamma\) and \(\delta\)
govern the shape of the distribution. In the lognormal families,
\(\delta\) governs the shape while \(\gamma\) is only a scaling
factor~\cite{hoaglin85c}. Other approaches to exploring the 
shape of a distribution include \(g\)- and 
\(h\)-distributions~\cite{hoaglin85c} and 
Pearson curves (see Bowman~\cite{bowman86}).

\subsection{Miscellaneous}

Many tests are presented here without mention of their relative
merits. Users are advised to consult the cited literature to
determine which test is appropriate for their situation. Sometimes
a certain test will have more \emph{power} than another; that is,
a test may have a better ability to reject a model when
the model is incorrect. 

\section{Moments: \(b_2\) and \(\protect\sqrt{b_1}\)}

\function{omnibus\_moments(x,n)}
         {double* \\ 
          \hbox{omnibus\_moments(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[\sqrt{b_1},b_2\right]'}}%

Let \(x_1,  x_2, \ldots,  x_n\) be the \(n\)
observations with mean:
\begin{equation}
m_1 = \frac{1}{n}\sum_{j=1}{n} x_j.
\end{equation}
The central moments are defined as:
\begin{equation}
\label{eqn:moments}
m_i = \frac{1}{n}\sum_{j=1}{n}\left( x_j - m_i\right)^i,\: i=2,3,4.
\end{equation}
The sample skewness \(\left(\sqrt{b_1}\right)\)
and kurtosis \(\left(b_2\right)\) are defined as:
\begin{equation}
\sqrt{b_1}  =  m_3/m_2^{3/2} = \sqrt{n}
\left(\sum_{j=1}^n\left(x_i-\bar{x}\right)^3\right)/
\left( \sum_{j=1}^n\left(x_i-\bar{x}\right)^2 \right)^{3/2}
\end{equation}
and
\begin{equation}
\label{eqn:4th-sample-moment}
b_2  =  m_4/m_2^2.
\end{equation}
These are invariant under both origin and scale changes~\cite{bowman86}.
When a distribution is specified, these are denoted as 
\(\sqrt{\beta_1}\) and \(\beta_2\).

For a standard normal, \(\sqrt{\beta_1}=0\) and \(\beta_2=3\).
To use either or both of these statistics to test for
departure from normality, these are sometimes transformed
to their standardized to their normal equivalent
deviates, \(X\left(\sqrt{b_1}\right)\) and \(X\left(b_1\right)\).

For \(X\left(\sqrt{b_1}\right)\), D'Agostino and 
Pearson~\cite{dagostino73} gave coefficients \(\delta\)
and \(\lambda\) (\(n=8\) to 1000) for:
\begin{equation}
X\left(\sqrt{b_1}\right) = \delta \sinh^{-1}
\left(\sqrt{b_1}/\lambda\right)
\end{equation}
that transforms \(\sqrt{b_1}\) to a standard normal
using a Johnson \(S_U\) approximation (Table~\ref{tbl:johnson}).
\label{sec:johnson-su}
An equivalent approximation~\cite{dagostino86} 
that avoids the use of tables is given by:
\begin{enumerate}
\item Compute \(\sqrt{b_1}\) from the sample data.
\item Compute:
\begin{eqnarray}
Y &=& \sqrt{b_1} \left[\frac{\left(n+1\right)\left(n+3\right)}
                         {6\left(n-2\right)}\right]^{\frac{1}{2}}, \\
\beta_2 &=& \frac{3\left(n^2+27n-70\right)\left(n+1\right)\left(n+3\right)}
        {\left(n-2\right)\left(n+5\right)\left(n+7\right)\left(n+9\right)},\\
W^2 &=& \sqrt{2\left(\beta_2-1\right)}-1, \\
\delta &=& 1/\sqrt{\log W}, \\
\mbox{and}\\
\alpha &=& \sqrt{2/\left(W^2-1\right)}.
\end{eqnarray}
\item Compute the standard normal variable:
\begin{equation}
Z = \delta \log\left[Y/\alpha + \sqrt{\left(Y/\alpha\right)^2+1}\,\right].
\end{equation}
\end{enumerate}
This procedure is applicable for \(n\ge8\).

%D'Agostino~\cite{dagostino86} also notes 
%that the normal approximation given by
%\begin{equation}
%\sqrt{\beta_1}\left[\frac{\left(n+1\right)\left(n+3\right)}
                         %{6\left(n-2\right)}\right]^{\frac{1}{2}}
%\end{equation}
%is valid for \(n\ge150\)~\cite{dagostino86}.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(\sqrt{b_1} = 0.2373\).  Suppose that we wish to test the
hypothesis of normality:

\(H_0\): \(\sqrt{\beta_1}=0\) (normality)

\noindent versus the two-sided alternative

\(H_1\): \(\sqrt{\beta_1}\ne0\) (non-normality)

\noindent at a level of significance of 0.05.
Following the procedure given above,
\(Y =2.3454\), 
\(\beta_2 = 3.0592\),
\(W^2 = 1.0294\),
\(\delta=12.6132\),
\(\alpha=8.2522\), and
\(Z=1.5367\).
At a 0.05 significance level for a two-sided test, we reject
the null hypothesis of normality if \(\left|Z\right|\ge1.96\). In
this instance, we cannot reject \(H_0\).
\end{example}

The fourth standardized moment \(b_2\) may be used to
test the normality hypothesis by the following 
procedure~\cite{anscombe63}:
\begin{enumerate}
\item Compute \(b_2\) from the sample data.
\item Compute the mean and variance of \(b_2\):
\begin{equation}
E\left(b_2\right) = \frac{3\left(n-1\right)}{n+1}
\end{equation}
and
\begin{equation}
Var\left(b_2\right) = \frac{24n\left(n-2\right)\left(n-3\right)}
{\left(n+1\right)^2\left(n+3\right)\left(n+5\right)}.
\end{equation}
\item Compute the standardized value of \(b_2\):
\begin{equation}
y = \frac{b_2-E\left(b_2\right)}{Var\left(b_2\right)}.
\end{equation}
\item Compute the third standardized moment of \(b_2\):
\begin{equation}
\sqrt{\beta_1\left(b_2\right)} = 
\frac{6\left(n^2-5n+2\right)}{\left(n+7\right)\left(n+9\right)}
\sqrt{\frac{6\left(n+3\right)\left(n+5\right)}
           {n\left(n-2\right)\left(n-3\right)}}.
\end{equation}
\item Compute:
\begin{equation}
A=6+\frac{8}{\sqrt{\beta_1\left(b_2\right)}}\left[
\frac{2}{\sqrt{\beta_1\left(b_2\right)}} +
\sqrt{1+\frac{4}{\sqrt{\beta_1\left(b_2\right)}}}\,\right].
\end{equation}
\item Compute:
\begin{equation}
\label{eqn:z-b2}
Z = \left(\left(1-\frac{2}{9A}\right)-
\left[\frac{1-2/A}{1+y\sqrt{2/\left(A-4\right)}}\right]^{\frac{1}{3}}\right)/
\sqrt{2/\left(9A\right)}
\end{equation}
where \(Z\) is a standard normal variable with
zero mean and variance of one.
\end{enumerate}

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(b_2 =1.9148\).  Suppose that we wish to test the
hypothesis of normality:

\(H_0\): \(\beta_2=3\) (normality)

\noindent versus the one-sided alternative

\(H_1\): \(\beta_2>3\) (non-normality)

\noindent at a level of significance of 0.05. We would
reject \(H_0\) if \(Z\) (eqn.~\ref{eqn:z-b2}) is larger
than 1.645 (Table~\ref{tbl:normal}).  Following the procedure given above,
\(E\left(b_2\right)=2.9897\), 
\(Var\left(b_2\right)=0.0401\),
\(y=-26.8366\), 
\(\sqrt{\beta_1\left(b_2\right)}=0.0989\),
\(A=2163\), and
\(Z=-131.7\).
Therefore, we cannot reject \(H_0\).
\end{example}

\subsection{Omnibus Tests for Normality}

\section{Geary's Test of Normality}
\label{sec:geary}

\function{geary\_test(x,n)}
         {double*\\
          \hbox{geary\_test(x,n)}\\ 
          double *x;\\
          int n;\\ 
          \returns{\left[\sqrt{a},y\right]'}}

Let \(x_1,  x_2, \ldots,  x_n\) be the \(n\)
observations. The ratio of the mean deviation
to the standard deviation is given as:
\begin{equation}\label{eqn:geary}
a = \frac{1}{n\sqrt{m_2}}\sum_{j=1}^n \left|x_i-\bar{x}\right|
\end{equation}
where \(\bar{x}=\sum_{i=1}^n x_i\) and \(m_2\) is defined
by eqn.~\ref{eqn:moments}.
This ratio can be transformed
a standard normal~\cite{dagostino86} via
\begin{equation}\label{eqn:geary-normal}
y = \frac{\sqrt{n}\left(a-0.7979\right)}{0.2123}.
\end{equation}
This test is valid for \(n\ge41\). 

More generally, Geary~\cite{geary47} considered tests of the
form
\begin{equation}
a\left(c\right) = 
\frac{1}{nm_2^{c/2}}
\sum_{j=1}^n \left|x_i-\bar{x} \right|^c \: \mbox{for}\: c\ge1
\end{equation}
where \(a\left(1\right)=a\) of eqn.~\ref{eqn:geary}, and
\(a\left(4\right)=b_2\) of eqn.~\ref{eqn:4th-sample-moment}.

D'Agostino and Rosman~\cite{dagostino74} conclude that
Geary's \(a\) test has good power for symmetric alternatives
and skewed alternatives with \(\beta_2 < 3\) when compared to
other tests, though for symmetric alternatives, \(b_2\)
(eqn.~\ref{eqn:4th-sample-moment}) can sometimes be more powerful and
for skewed alternatives, \(W\) (eqn~\ref{eqn:w-test})
or \(W'\) (eqn~\ref{eqn:w-prime-test})
usually dominate \(a\).
The Geary test (eqns.~\ref{eqn:geary}-\ref{eqn:geary-normal})
is seldom used today---D'Agostino~\cite{dagostino86} include it 
in his summary work because it is of ``historical interest.''

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(a = 0.8823\).  Suppose that we wish to test the
hypothesis of normality:

\(H_0\): normality

\noindent versus the two-sided alternative

\(H_1\): non-normality

\noindent at a level of significance of 0.05.
From eqn.~\ref{eqn:geary-normal}, \(y=9.9607\).
\end{example}

\section{Extreme Normal Deviates}
\label{sec:extreme}

\function{extremes(x,n)}
         {double* \\
          \hbox{extremes(x,n)}\\ 
          double *x;\\
          int n;\\ 
          \returns{\left[x_n-\bar{x}, x_1-\bar{x}\right]'}}

Let \(x_1 \le x_2 \le \cdots \le x_n\) be the \(n\)
observations. Given a known normal deviation \(\sigma\),
the largest and smallest deviation from a normal population
may be computed:
\begin{equation}
u_n = \frac{x_n-\bar{x}}{\sigma}
\end{equation}
and
\begin{equation}
u_1 = -\frac{x_1-\bar{x}}{\sigma},
\end{equation}
respectively. These statistics are potentially
useful for detecting outliers for populations
with a known \(\sigma\) but an unknown mean.
Table 25 in Pearson and Hartley~\cite{pearson76}
gives percentage points for this statistic.
Pearson and Hartley~\cite{pearson76} also give examples
of the use of extreme deviates when an estimator of
\(\sigma\) (independent of the sample) is
known and when a combined ``internal''
and ``external'' estimate is used.

\section{EDF Statistics for Testing Normality}

[Note: This section follows closely the presentation of 
Stephens~\cite{stephens86}.]

Let \(x_1 \le x_2 \le \cdots \le x_n\) be the \(n\)
observations. Suppose that the continuous distribution of \(x\)
is \(F\left(x\right)\). The empirical distribution function (EDF)
is \(F_n\left(x\right)\) defined by:
\begin{equation}
F_n\left(x\right) = \frac{1}{n}\left(\mbox{number of observations} 
\le x\right); \: -\infty < x < \infty
\end{equation}
or
\begin{displaymath}
\begin{array}{rclll}
F_n\left(x\right)& = &0, & x<x_1\\
F_n\left(x\right)& = &\frac{1}{n}, & x_i\le x<x_{i+1}, & i=1,\ldots,n-1\\
F_n\left(x\right)& = &1, & x_n\le x.
\end{array}
\end{displaymath}
Thus \(F_n\left(x\right)\) is a step function calculated from
the data. As \(n\rightarrow\infty\), \\
\(\left|F_n\left(x\right)- F\left(x\right)\right|\) 
decreases to zero with probability one~\cite{stephens86}.

EDF statistics that measure the difference between
\(F_n\left(x\right)\) and \(F\left(x\right)\) are divided
into two classes: supremum and quadratic.
On a graph of 
\(F_n\left(x\right)\) and \(F\left(x\right)\) versus \(x_i\),
denote the largest vertical distance when
\(F_n\left(x\right)>F\left(x\right)\) as \(D^+\).
Also, let \(D^-\) denote the largest vertical distance when
\(F_n\left(x\right)<F\left(x\right)\). These two
measures are supremum statistics.
Quadratic statistics are given by the Cram\'er--von Mises family
\begin{equation}
\label{eqn:cramer-family}
Q = n\int_{-\infty}^{\infty} 
\left(F_n\left(x\right) - F\left(x\right)\right)^2
\psi\left(x\right) d F\left(x\right)
\end{equation}
where \(\psi\left(x\right)\) is a weighting function~\cite{stephens86}.

To compute these statistics, the Probability Integral Transformation
is used: \(z=F\left(x\right)\) where \(F\left(x\right)\) is
the Gaussian distribution. The new variable, \(z\), is uniformly
distributed between 0 and 1. Then \(z\) has distribution 
function \(F^*\left(z\right)=z\), \(0\le z\le1\).
A sample \(x_1, x_2, \ldots, x_n\) gives values \(z_i=F\left(x_i\right)\),
\(i=1, \ldots, n\), and \(F^*_n\left(z\right)\) is the EDF of
values \(z_i\). For testing normality,
\begin{equation}
z_{\left(i\right)} = \Phi\left(
\left(x_{\left(i\right)}-\hat{\mu}\right)/\hat{\sigma}
\right)
\end{equation}
where \(\hat{\mu}\) and \(\hat{\sigma}\) are estimated from
the data and \(\Phi\left(\cdot\right)\) denotes the cumulative
probability of a standard normal. For testing if the data
follows an exponential distribution \(\mbox{Exp}\left(\alpha,\beta\right)\),
where \(\alpha\) is known to be zero, \(\hat{\beta}\)
is estimated by \(\bar{x}\) (the sample mean) and
\begin{equation}
z_{\left(i\right)} = 1-\exp\left(-x_{\left(i\right)}/\bar{x}\right).
\end{equation}

Now, EDF statistics can be computed by comparing \(F^*_n\left(z\right)\)
and a uniform distribution for \(z\). These take the same values
as comparisons between \(F_n\left(x\right)\) and \(F\left(x\right)\):
\begin{equation}
F_n\left(x\right) - F\left(x\right) = 
F^*_n\left(z\right) - F^*\left(z\right) =
F^*_n\left(z\right) - z.
\end{equation}
After ordering \(z\)-values,
\(z_{\left(1\right)}\le 
z_{\left(2\right)} \le \cdots 
\le z_{\left(n\right)}\) and computing \(\bar{z}=\sum_{i=1}^n z_i/n\),
the supremum statistics are
\begin{equation}
\label{eqn:dplus}
D^+=\max_{i=1,\ldots,n}\left(i/n-z_{\left(i\right)}\right)
\end{equation}
and
\begin{equation}
\label{eqn:dminus}
D^-=\max_{i=1,\ldots,n}\left(z_{\left(i\right)}-\left(i-1\right)/n\right).
\end{equation}

\subsection{Kolmogorov \(D\)}

\function{kolmogorov\_smirnov(x,n)}
         {double* \\ 
          \hbox{kolmogorov\_smirnov(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[D^n,D\right]'}}

\function{kolmogorov\_smirnov\_exp(x,n)}
         {double* \\ 
          \hbox{kolmogorov\_smirnov\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[D^e,D\right]'}}

The most well-known EDF statistic is Kolmogorov's \(D\), computed
from supremum statistics:
\begin{equation}
D = \sup_x\left|F_n\left(x\right) - F\left(x\right)\right| =
\max\left(D^+,D^-\right).
\end{equation}
The modified form for testing a completely specified
distribution~\cite{stephens86}:
\begin{equation}
D^*=D\left(\sqrt{n}+0.12+0.11/\sqrt{n}\right).
\end{equation}
For testing a normal distribution with \(\mu\) and \(\sigma\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
D^n=D\left(\sqrt{n}-0.01+0.85/\sqrt{n}\right).
\end{equation}
For testing an exponential distribution with \(\alpha\) and \(\beta\)
% origin and scale
unknown, \(D\) does not need modified~\cite{stephens86}.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(D^n = 4.0314\) and \(y=\).  Suppose that we wish to test the
hypothesis of normality:

\(H_0\): normality

\noindent versus the two-sided alternative

\(H_1\): non-normality

\noindent at a level of significance of 0.05.
\end{example}

\subsection{Kuiper's \(V\)}
\label{sec:kuiper}

\function{kuipers\_v(x,n)}
         {double* \\ 
          \hbox{kuipers\_v(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[V^n,V\right]'}}

\function{kuipers\_v\_exp(x,n)}
         {double* \\ 
          \hbox{kuipers\_v\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[V^e,V\right]'}}

Kuiper's~\cite{kuiper60} \(V\) is another statistic computed
from supremum statistics:
\begin{equation}
\label{eqn:kuipers-v}
V = D^+ + D^-.
\end{equation}
The modified form for testing a completely specified
distribution~\cite{stephens86}:
\begin{equation}
V^*=V\left(\sqrt{n}+0.155 +0.24\sqrt{n}\right).
\end{equation}
For testing a normal distribution with \(\mu\) and \(\sigma\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
V^n=V\left(\sqrt{n}+0.05+0.82/\sqrt{n}\right).
\end{equation}
For testing an exponential distribution with \(\alpha\) and \(\beta\)
unknown, \(V\) the modified equation is~\cite{stephens86}:
\begin{equation}
V^e=\left(V-0.2/\sqrt{n}\right)
    \left(\sqrt{n}+0.24+0.35/\sqrt{n}\right).
\end{equation}

\subsection{Pyke's Statistics}
\label{sec:pyke}

For some purposes, eqns.~\ref{eqn:dplus} and~\ref{eqn:dminus}
may be modified to~\cite{pyke59}:
\begin{equation}
\label{eqn:cplus}
C^+=\max_{0\le i\le n}\left(\frac{i}{n+1}-z_{\left(i\right)}\right),\:
z_{\left(0\right)}=0,
\end{equation}
and
\begin{equation}
\label{eqn:cminus}
C^-=\max_{0\le i\le n}\left(z_{\left(i\right)}-\frac{i}{n+1}\right)
\end{equation}
(following the modification of notation by Durbin~\cite{durbin73}).  Then,
\begin{equation}
C = \max\left(C^+,C^-\right).
\end{equation}
Durbin~\cite{durbin73} notes that these modifications to 
eqns.~\ref{eqn:dplus} and~\ref{eqn:dminus} are related to
the fact that \(E\left(z_{\left(i\right)}\right)=i/\left(n+1\right)\).
Percentage points were given by Durbin~\cite{durbin69}.

\subsection{Brunk's \(B\)}
\label{sec:brunk}

As an alternative to Kuiper's \(V\) (eqn.~\ref{eqn:kuipers-v}), 
Brunk~\cite{brunk62} suggests:
\begin{equation}
\label{eqn:brunks-b}
B = C^+ + C^-
\end{equation}
where \(C^+\) and \(C^-\) are given by eqns.~\ref{eqn:cplus}
and \ref{eqn:cminus}.

\subsection{Cram\'er--von Mises \(W^2\)}
\label{sec:cramer-von-mises}

\function{cramer\_von\_mises(x,n)}
         {double* \\ 
          \hbox{cramer\_von\_mises(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[W^{2,n},W^2\right]'}}

\function{cramer\_von\_mises(x,n)}
         {double* \\ 
          \hbox{cramer\_von\_mises\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[W^{2,e},W^2\right]'}}

Quadratic statistics are computed from
the Cram\'er--von Mises family given in eqn~\ref{eqn:cramer-family}.
When \(\psi\left(x\right)=1\) in eqn~\ref{eqn:cramer-family}, the statistic is
the Cram\'er--von Mises statistic \(W^2\):
\begin{equation}
W^2=\sum_{j=1}^n\left(Z_i - \left(2j-1\right)/\left(2n\right)\right)^2
+\frac{1}{12n}
\end{equation}
(When \(\psi\left(x\right)= 
\left(F\left(x\right)\left(1 - F\left(x\right)\right)\right)^{-1}\),
this yields the Anderson--Darling statistic given below
in \S\ref{sec:anderson-darling}~\cite{stephens86}.)
The modified form for testing a completely specified
distribution~\cite{stephens86}:
\begin{equation}
W^{2,*} = \left(W^2-0.4/n +0.6/n^2\right)/\left(1 + 1/n\right).
\end{equation}
For testing a normal distribution with \(\mu\) and \(\sigma\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
W^{2,n}=W^2\left(1.0 + 0.5/n\right).
\end{equation}
For testing an exponential distribution with \(\alpha\) and \(\beta\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
W^{2,e}=W^2\left(1.0 + 2.8/n -3/n^2\right).
\end{equation}

\subsection{Watson \(U^2\)}
\label{sec:watson}

\function{watson\_u2(x,n)}
         {double* \\ 
          \hbox{watson\_u2(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[U^{2,n}, U^{2}\right]'}}

\function{watson\_u2\_exp(x,n)}
         {double* \\ 
          \hbox{watson\_u2\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[U^{2,e}, U^{2}\right]'}}

\begin{equation}
U^2=W^2-n\left(\bar{Z}-0.5\right)^2
\end{equation}
where \(W^2\) is the Cram\'er--von Mises statistic 
(\S\ref{sec:cramer-von-mises}).
The modified form for testing a completely specified
distribution~\cite{stephens86}:
\begin{equation}
U^{2,*} = \left(U^2-0.1/n +0.1/n^2\right)/\left(1 + 0.8/n\right).
\end{equation}
For testing a normal distribution with \(\mu\) and \(\sigma\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
U^{2,n}=U^2\left(1.0 + 0.5/n\right).
\end{equation}
For testing an exponential distribution with \(\alpha\) and \(\beta\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
U^{2,e}=U^2\left(1.0 + 2.3/n -3/n^2\right).
\end{equation}

\subsection{Anderson--Darling \(A^2\)}
\label{sec:anderson-darling}

\function{anderson\_darling(x,n)}
         {double* \\ 
          \hbox{anderson\_darling(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[A^{2,n}, A^{2}\right]'}}

\function{anderson\_darling\_exp(x,n)}
         {double* \\ 
          \hbox{anderson\_darling\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\
          \returns{\left[A^{2,e}, A^{2}\right]'}}
Anderson and Darling~\cite{anderson54} present
another EDF test statistic which is sensitive at the
tails of the distribution (rather than near
the median). 
When \(\psi\left(x\right)= 
\left(F\left(x\right)\left(1 - F\left(x\right)\right)\right)^{-1}\)
in eqn.(\ref{eqn:cramer-family}),
this yields the Anderson--Darling statistic~\cite{anderson54,stephens86}:
\begin{equation}
A^2 = -n - \frac{1}{n} \sum_{j=1}^n \left(2j-1\right)
\left[ \ln z_j + \ln\left(1-z_{n-j+1}\right)\right].
\end{equation}
Equivalently~\cite{stephens86},
\begin{equation}
A^2 = -n - \frac{1}{n} \sum_{j=1}^n\left[ \left(2j-1\right)
\ln z_j + \left(2n+1-2j\right) \ln\left(1-z_{j}\right)\right].
\end{equation}
Anderson and Darling~\cite{anderson54} give
the following asymptotic significance values of \(A^2\):

\begin{center}
\begin{tabular}{cc}\hline
Significance 	& Significance \\
Level 		& Point \\ \hline \hline
0.10		& 1.933\\
0.05		& 2.492\\
0.01		& 3.857\\ \hline
\end{tabular}
\end{center}
Anderson and Darling~\cite{anderson54} state that
sample size should be at least 40; however, Stephens~\cite{stephens86}
give the same asymptotic values (for more significance levels)
for a sample size \(\ge5\).

For testing a completely specified distribution, \(A^2\)
is used unmodified.
For testing a normal distribution with \(\mu\) and \(\sigma\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
A^{2,n}=A^2\left(1.0 + 0.75/n+2.25/n^2\right).
\end{equation}
For testing an exponential distribution with \(\alpha\) and \(\beta\)
unknown, the modified equation is~\cite{stephens86}:
\begin{equation}
A^{2,e}=A^2\left(1.0 + 5.4/n -11/n^2\right).
\end{equation}


\subsection{Durbin's Exact Test}
\label{sec:durbin}

\function{durbins\_exact(x,n)}
         {double* \\ 
          \hbox{durbins\_exact(x,n)}\\ 
          double *x;\\ 
          int n;\\
          \returns{\left[K_m,\sqrt{n}K_m\right]'}}

Durbin~\cite{durbin61} presented a modified
Kolmogorov test. The discussion that follows
has been adapted from Durbin's work~\cite{durbin61}.

Let \(x_1, x_2, \ldots, x_n\) be the \(n\)
i.i.d.\ observations and suppose that 
it is desired to test the hypothesis that
they come from the continuous distribution \(F\left(x\right)\).
If the null hypothesis is true, then \(u_j=F\left(x_j\right)\)
(\(j=1,\ldots,n)\) are independent \(U\left(0,1\right)\)
variables and are randomly scattered on the (0,1) interval.
Clustering may indicated a departure from the null hypothesis.
Denoting the ordered \(u\)'s by 
\(0 \le u_{\left(1\right)} \le \cdots \le u_{\left(n\right)} \le 1\), 
let \(c_1=u_{\left(1\right)}\), 
\(c_2=u_{\left(j\right)}-u_{\left(j-1\right)}\)
(\(j=2,\ldots,n\)), and \(c_{n+1}=1-u_{\left(n\right)}\).

Since the interest is in relative magnitudes of \(c\)'s, these
are ordered:
\(c_{\left(1\right)} \le c_{\left(2\right)} \cdots 
\le c_{\left(n\right)}\). Then, the following transformation
is applied:
\begin{equation}
\label{eqn:durbin:g}
g_j=\left(n+2-j\right)\left(c_{\left(j\right)}
-c_{\left(j-1\right)}\right)\:
\left(c_{\left(0\right)}=0;\: j=1,\ldots,n+1\right).
\end{equation}
Durbin~\cite{durbin61} shows that the \(g\)'s, which depend
on the {\em ordered\/} intervals, have the same
distribution as the {\em unordered\/} \(c\)'s.

Letting
\begin{equation}
\label{eqn:durbin:w}
w_r = \sum_{j=1}^r g_j
\end{equation}
it follows that \(w_1, \ldots, w_n\) have the same distribution
as the ordered \(U\left(0,1\right)\) variables
\(u_{\left(1\right)}, \ldots, u_{\left(n\right)}\). 

From eqns.~\ref{eqn:durbin:g} and \ref{eqn:durbin:w}, \(w_j\)
can be expressed as:
\begin{equation}
w_j=c_{\left(1\right)} + \cdots
+ c_{\left(j-1\right)} + \left(n+2-j\right)c_{\left(j\right)},\:
\left(j=1,\ldots,n\right),
\end{equation}
where
\(c_{\left(1\right)} \le \cdots \le c_{\left(n\right)}\) is
the ordered set of intervals.

In addition to two other test, Durbin~\cite{durbin61} introduces
the {\em modified Kolmogorov test}. The test statistic is:
\begin{equation}
K_m = \max_{r=1,\ldots,n}\left(\frac{r}{n}-w_r\right).
\end{equation}
The test procedure is to reject when \(K_m\) is greater
than the value tabulated for a one-sided Kolmogorov test.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(K_m = 0.4127\). To test the
hypothesis of normality:

\(H_0\): normality

\noindent versus the one-sided alternative

\(H_1\): non-normality

\noindent at a level of significance of 0.05, we would
reject \(H_0\) if \(K_m\) is larger
than 0.895 (critical value of \(D\) for \(\alpha=0.05\).  
Therefore, we cannot reject \(H_0\).
\end{example}

\section{Chi-Square Test}

\function{chi\_square(x,n)}
         {double* \\ 
          \hbox{chi\_square(x,n)}\\ 
          double *x;\\ 
          int n;\\
          \returns{\left[x^2,k-3\right]'}}

\function{chi\_square\_exp(x,n)}
         {double* \\ 
          \hbox{chi\_square\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\
          \returns{\left[x^2,k-2\right]'}}

According to Shapiro~\cite{shapiro90},
the chi-square goodness of fit test is the oldest
procedure for testing distributional assumptions.
It is useful for testing normality and exponentiality
when the number of observations is large (because its power
is poor for small samples when compared to other tests).
It is also useful when data are discrete~\cite{shapiro90}.

The basic idea is to divide the \(n\) data into \(k\) cells
and compare the observed number in each cell with the 
expected number in each cell. The resulting statistic
is distributed as a chi-square random variable with
\(k-1-t\) degrees of freedom, where \(t\) is the number
of parameters estimated. The number of cells is taken
as 
\begin{equation}
k=\mbox{(int)} 4\left[0.75\left(n-1\right)^2\right]^{1/5}.
\end{equation}
\marginpar{what should the notation be for rounding? For ceil, 
we use \(\lceil x\rceil\). For floor, we use \(\lfloor x\rfloor\).}
The ratio \(n/k\) should be at least 5; otherwise another
test should be used~\cite{shapiro90}. In this implementation,
\(k\) is decremented by one until \(n/k\ge5\).

Let \(x_{\left(1\right)}, 
x_{\left(2\right)},\ldots, x_{\left(k\right)}\) 
be the upper boundaries of cells. Choose \(x_{\left(i\right)}\)
so that the probability of being in any cell
is the same: 
\begin{equation}
P\left(x\le x_{\left(i\right)}\right) = \frac{i}{k},\:
i=1,2,\ldots,k
\end{equation}
In thse implmentation, only the case of raw data, as opposed
to pre-tabulated data, is considered (i.e., equal probability cells).

For testing the normality hypothesis,
let \(x_{\left(0\right)}=-\infty\) and
\(x_{\left(k\right)}=\infty\).
The values of \(x_{\left(i\right)}\) are:
\begin{equation}
x_{\left(i\right)} = \bar{x} + s\,Z_{i/k} 
\end{equation}
where \(\bar{x}\) and \(s\) are estimated
mean and variance parameters and \(Z_{i/k}\)
are percentiles of the standard normal distribution.
The test statistic is
\begin{equation}
\label{eqn:chi-square}
x^2 = \frac{k}{n}\sum_{i=1}^k f_i^2-n
\end{equation}
where \(f_i\) is the number of observations in cell \(i\).
The hypothesis of normality is rejected at an \(\alpha\)
level if \(x^2\) is greater \(x^2_{\alpha}\), a 
\(\chi^2\) random variable with \(k-3\) degrees of freedom.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(x^2 = 952.7\) with \(\nu=45\) degrees of freedom. 
Since \(\chi^2_{45,0.05}\approx30.33\) (Table~\ref{tbl:chisq}),
we reject \(H_0\) at an \(\alpha=0.05\) level.
\end{example}

For testing the exponentiality hypothesis,
let \(x_{\left(0\right)}=0\) and
\(x_{\left(k\right)}=\infty\).
The values of \(x_{\left(i\right)}\) are:
\begin{equation}
x_{\left(i\right)} = -\frac{1}{\lambda}\ln\left(1-\frac{i}{k}\right),
i=1,2,\ldots,k-1.
\end{equation}
The parameter \(\lambda\) is estimated from
\begin{equation}
\hat{\lambda} = n \left(\sum_{i=1}^n x_i\right)^{-1}
\end{equation}
where \(x_i\) is the \(i\)th observation in 
the sample.  Equation~(\ref{eqn:chi-square})
is the statistic used for testing exponentiality. The hypothesis
of exponentiality is rejected at an \(\alpha\) level if
\(x^2\), a \(\chi^2\) random variable with \(k-2\)
degrees of freedom.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(x^2 = 308.11\) with \(\nu=46\) degrees of freedom. 
Since \(\chi^2_{46,0.05}\approx31.16\) (Table~\ref{tbl:chisq}),
we reject \(H_0:\) exponentiality, at an \(\alpha=0.05\) level.
\end{example}

\section{Analysis of Variance Tests}

\subsection{Shapiro-Wilk \(W\)}
\label{sec:shapiro-wilk}

\function{shapiro\_wilk(x,n)}
         {double* \\ 
          \hbox{shapiro\_wilk(x,n)}\\ 
          double *x;\\ 
          int n;\\
          \returns{\left[W,S^2\right]'}}

\function{shapiro\_wilk\_exp(x,n)}
         {double* \\ 
          \hbox{shapiro\_wilk\_exp(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[W,S^2\right]'}}

%\marginpar{3, 4, 206, 208, 211, 252, 393, 403--406}

Recall the description of a probability plot given on page~\pageref{pplot}.
Ordered observations are plotted against expected values of
order statistics from the distribution being tested. The plot tends
to be linear if the distributional assumption is correct. If
a genearlized least squares is performed, an \(F\)-type ratio could
be used to test the fit of a linear model. This was the basis of
test introduced by Shapiro and Wilk~\cite{shapiro65}. Foregoing
many of the details in the derivation, the test procedures
for normality and exponentiality are given
below.

Let \(x_1 \le x_2 \le \cdots \le x_n\) be the \(n\)
ordered observations and let
\begin{equation}
S^2 = \sum_{i=1}^n x_i^2 - \frac{1}{n} \left(\sum_{i=1}^n x_i\right)^2.
\end{equation}
Calculate
\begin{equation}
b = \sum_{i=1}^k a_{n-i+1}\left(x_{n-i+1}- x_i\right)
\end{equation}
where \(k=n/2\) if \(n\) is even,
\(k=\left(n-1\right)/2\) if \(n\) is odd, and
\(a_{n-i+1}\) are found in Table~\ref{tbl:shapiro-wilk-a}.
Then a test of normality
for small samples (\(3\le n\le 50\)) is defined as
\begin{equation}
\label{eqn:w-test}
W = \frac{b^2}{S^2}
\end{equation}
Small values of \(W\) indicate non-normality (``lower-tail''). Hence
if the computed value of \(W\) is less than the
\(W_{\alpha}\) shown in Table~\ref{tbl:w-test}, the hypothesis
of normality is rejected.

\begin{example}
Using the first 40 observations from the sample data given in 
Table~\ref{tbl:pine},
\(W=0.0000245\). Using \(\alpha=0.05\) and Table~\ref{tbl:w-test},
\(W_{0.05}=0.940\). Since \(W<W_{0.05}\), we reject \(H_0\).
\end{example}

For testing exponentiality, no tabulated constants are needed
for calculation of \(b\):
\begin{equation}
b = \sqrt{\frac{n}{n-1}}\left(\bar{x}-x_1\right)
\end{equation}
where 
\begin{equation}
\bar{x} = \frac{1}{n} \sum_{i=1}^n x_i.
\end{equation}
This assumes that the origin parameter is unknown. It also
differs from the test of normality in that it is a two-tailed
procedure. That is, too small or too large a value of the
test statistic indicates non-exponentiality~\cite{shapiro90}.

\begin{example}
Using the first 40 observations from the sample data given in 
Table~\ref{tbl:pine},
\(W=0.0909\). Using \(\alpha=0.05\) and Table~\ref{tbl:w-test-e},
\(W_{0.025}=0.0148\) and \(W_{0.975}=0.0447\). 
Since \(W\) is not contained in the
interval \(\left[W_{0.025},W_{0.975}\right]\), 
we reject \(H_0\): exponentiality.
\end{example}

\subsection{Modified Shapiro--Francia \(W'\)}
\label{sec:shapiro-francia}
 
\function{shapiro\_francia(x,n)}
         {double* \\ 
          \hbox{shapiro\_francia(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[W',S^2\right]'}}

%\marginpar{213, 223, 399, 403--406}

The \(W\) test of normality in the previous
section for sample sizes of 50 or less.
Shapiro and Francia~\cite{shapiro72b} extended
the \(W\) test for \(n\) up to 99 by replacing
the values \(a_{n-i+1}\) in Table~\ref{tbl:shapiro-wilk-a}
\(b_{n-i+1}\) in Table~\ref{tbl:shapiro-francia-b}.
The test procedure follows.

Let \(x_1 \le x_2 \le \cdots \le x_n\) be the \(n\)
ordered observations. Then a test of normality
for large samples is defined as:
\begin{equation}
\label{eqn:w-prime-test}
W' = \frac{b'}{S^2}
\end{equation}
The numerator \(b'\) is defined as:
\begin{equation}
b' = \sum_{i=1}^k b_{n-i+1} \left(x_{n-i+1} - x_i\right)
\end{equation}
where \(k=n/2\) if \(n\) is even and \(k=\left(n-1\right)/2\)
is \(n\) is odd.
Significant values,
determined empirically by Shapiro and Francia~\cite{shapiro72b}
are given in Table~\ref{tbl:w-prime-test}.
D'Agostino~\cite{dagostino86} notes that the values given
by Shapiro and Francia~\cite{shapiro72b} in the lower
tail were ``higher than what they should be'' since too few
samples were used in determining these significance levels.

\begin{example}
Using the first 99 observations from the sample data given in 
Table~\ref{tbl:pine},
\(W'=1.0139\). Using \(\alpha=0.05\) and Table~\ref{tbl:w-prime-test},
\(W'_{0.05}=0.976\). Since \(W'>W'_{0.05}\), we cannot reject \(H_0\).
\end{example}

\subsection{Weisberg-Bingham \(\tilde{W'}\)}

\function{weisberg\_bingham(x,n)}
         {double* \\ 
          \hbox{weisberg\_bingham(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[\tilde{W'},S^2\right]'}}

An alternative way of computing \(b'\) is to note that
the vector \(\left[b_1, b_2,\ldots,b_n\right]'\)
is equivalent to \(m'/\left(m'm\right)^{1/2}\)
where \( m' = \left(m_1, m_2, \ldots, m_n\right)\) denotes
a vector of expected normal order statistics.
One approximation for normal order statistics
attributed to Blom~\cite{blom58} is:
\begin{equation}
E\left(r,n\right) = -\Phi^{-1}\left(\frac{r-\alpha}{n-2\alpha+1}\right)
\end{equation}
with a recommended ``compromise value \(\alpha=0.375\)~\cite{royston82c}.'' 
Define this new statistic as \(\tilde{W'}\).
So, instead of hardcoding constants (as done in 
\S\ref{sec:shapiro-wilk}-\ref{sec:shapiro-francia}), 
this approximation is used.  Since \(\tilde{W'}\)
is essentially the same as \(W'\), the table of 
critical values for \(W'\) (Table~\ref{tbl:w-prime-test}) may be used.

\subsection{D'Agostino's \(D\) Test of Normality}
\label{sec:dagostino-d}

\function{dagostino\_d(x,n)}
         {double* \\
          \hbox{dagostino\_d(x,n)}\\ 
          double *x;\\
          int n;\\ 
          \returns{\left[D,y\right]'}}

D'Agostino~\cite{dagostino86} presents a modified
Shapiro-Wilk \(W\) test that eliminates the need for
a table of weights. The test statistic is given as
\begin{eqnarray}
D &=& T/\left(n^2\sqrt{m_2}\right) \\ \nonumber
  &=& T/\left(n^{3/2}\sqrt{\sum_{j=1}^n\left(x_j-\bar{x}\right)^2}\right) 
\end{eqnarray}
where
\begin{equation}
T = \sum_{i=1}^n \left(i-\frac{1}{2}\left(n+1\right)\right)x_i.
\end{equation}
An approximate standard variable is
\begin{equation}
\label{eqn:xform-d}
y=\frac{\sqrt{n}\left(D-0.28209479\right)}{0.02998598}.
\end{equation}
Significant values are given in Table~\ref{tbl:d-test}.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(D = 0.2859\) and \(y=3.0667\).  Suppose that we wish to test the
hypothesis of normality:

\(H_0\): normality

\noindent versus the two-sided alternative

\(H_1\): non-normality

\noindent at a level of significance of 0.005. From Table~\ref{tbl:d-test}
(linearly interpolating),
we reject \(H_0\) if \(y<-3.006\) or \(y>2.148\). Therefore, we
cannot reject \(H_0\).
\end{example}

\subsection{Royston's Modification}

\function{royston(x,n)}
         {double* \\ 
          \hbox{royston(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[W, P\right]'}}

Royston~\cite{royston82a} also presented a modified \(W\) statistic for
\(n\) up to 2000 that did not require extensive use of tabulated
constants. 
If \( m' = \left(m_1, m_2, \ldots, m_n\right)\) denotes
a vector of expected values of standard normal order
statistics and \(V=\left(v_{ij}\right)\) denote the corresponding
\(n\times n\) covariance matrix, then \(W\) may be written as:
\begin{equation}
W=\left[\sum_{i=1}^n a_i x_{\left(i\right)}\right]^2/
\sum_{i=1}^n \left( x_{\left(i\right)} - \bar{x}\right)^2
\end{equation}
where
\begin{equation}
a'=m'V^{-1}\left[\left(m'V^{-1}\right)\left(V^{-1} m' \right)\right]^{1/2}.
\end{equation}
Let \(a^* = m'V^{-1}\); The following 
approximation for \(a^*\) is used:
\begin{equation}
\label{eqn:astar}
\hat{a}^* = \cases{
2m_i, & i=2,3,\ldots,n-1\cr\cr
\left(\frac{\hat{a}_1^2}{1-2\hat{a}_1^2}
  \sum_{i=2}^{n-1} \hat{a}_i^{*2}\right)^{1/2}, & i=1, i=n}
\end{equation}
where
\begin{equation}
\hat{a}_1^2=\hat{a}_n^2 = \cases{
g\left(n-1\right), n\le20\cr\cr
g\left(n\right), n>20}
\end{equation}
and
\begin{equation}
g\left(n\right)=\frac{\Gamma\left(\frac{1}{2}\left[n+1\right]\right)}
                     {\sqrt{2\Gamma\left(\frac{1}{2}n+1\right)}}.
\end{equation}
The function \(g\left(n\right)\) is approximated using:
\begin{equation}
\label{eqn:stirling}
g\left(n\right)=\left[\frac{6n+7}{6n+13}\right]
\left(\frac{\exp\left(1\right)}{n+2}
\left[\frac{n+1}{n_2}\right]^{n-2}\right)^{1/2}
\end{equation}
Royston~\cite{royston82a} used eqns.~\ref{eqn:astar}--\ref{eqn:stirling}
for the range \(7\le n\le2000\), but exact values of \(a_i\)
for \(n<7\). 

Royston~\cite{royston82a} used the following normalizing transformation:
\begin{equation}
y=\left(1-W\right)^\lambda
\end{equation}
so that
\begin{equation}
z=\left[\left(1-W\right)^\lambda-\mu_y\right]/\sigma_y
\end{equation}
can be compared with the upper tail of a standard normal. Large
values of \(z\) indicate non-normality of the original sample.

This implementation in \libname\
closely follows Royston's published FORTRAN code~\cite{royston82b,royston82c}.
It returns \(W\) and a corresponding \(P\) value (smallest level
at which we could have preset \(\alpha\) and still have been able
to reject \(H_0\)).
It also utilizes algorithms by Hill~\cite{hill73} and 
Wichura~\cite{wichura88}.

%\section{Modified Maximum Likelihood Ratio Test}
%
%If the third moment is less than zero:
%\begin{equation}
%\sum_i\left(x_i-\bar{x}\right) \le 0
%\end{equation}
%then the distribution is normal. Otherwise, the test
%statistic is:
%\begin{equation}
%\frac{\sqrt{\frac{1}{n}\sum_i\left(x_i-\sigma/n\right)^2}}
%    {\exp\left(\sigma/n\right) \sqrt{\frac{1}{n}\left(x_i-\bar{x}\right)^2}}
%\end{equation}
%
%\function{mod\_maxlik\_ratio(x,n)}
%         {double* \\ 
%         \hbox{mod\_maxlik\_ratio(x,n)}\\ 
%         double *x;\\ 
%         int n;\\ 
%         \returns{?}}
%\section{Coefficient of Variation Test}
%
%pages 424, 428, 435, 457
%
%\begin{equation}
%\sqrt{\exp\left(\frac{1}{n-1}\sqrt{\exp\left(\frac{1}{n-1}\sum_i
%\left(\log x_i - \frac{1}{n}\sum_j x_j\right)^2\right)-1}\right)-1}
%\end{equation}
%
%\function{coeff\_variation(x,n)}
%         {double* \\ 
%          \hbox{coeff\_variation(x,n)}\\ 
%          double *x;\\ 
%          int n;\\ 
%          \returns{?}}
%

\section{Kotz Separate Families \(T'_f\)}
\label{sec:kotz}
% move as subsection to EDF Stats?

\function{kotz\_families(x,n)}
         {double* \\ 
          \hbox{kotz\_families(x,n)}\\ 
          double *x;\\ 
          int n;\\ 
          \returns{\left[T_f', T_f\right]'}}

Kotz~\cite{kotz73} developed a test where the null hypothesis
\(H_0\) is that the sample \(x_1, x_2, \ldots, x_n\) came
from a lognormal distribution, and the alternate hypothesis
is that the parent population was normal. The test statistic,
given as:
\begin{equation}
T'_f = \frac{\log\frac{\hat{\beta}_2}{\beta_{2,\hat{\alpha}}}}
{2\sqrt{n}\{\frac{1}{4}\left(e^{4\hat{\alpha}_2}+
2e^{3\hat{\alpha}_2} -4\right) -\hat{\alpha}_2 -
\frac{\hat{\alpha}_2\left(2e^{\hat{\alpha}_2}-1\right)^2}
     {2\left(2e^{\hat{\alpha}_2}-1\right)^2}
+\frac{3}{4}e^{\hat{\alpha}_2}\}^{1/2}}
\end{equation}
is asymptotically normal~\cite{cox62}.

\begin{example}
For the sample data given in Table~\ref{tbl:pine} (\(n=584\)), 
\(T'_f = -0.6021\).  Suppose that we wish to test the hypothesis

\(H_0:\) lognormal

\noindent versus

\(H_1:\) normal

\noindent at a level of significance of 0.05. We would
reject \(H_0\) if \(T'_f\) is larger
than 1.645.  Therefore, we reject \(H_0\).
\end{example}

The discussion that follows explains in more detail how this 
statistic is calculated and how it was derived. The remainder
of this section
was taken directly from the work of Kotz~\cite{kotz73}
(pages 123,124--126).

\ldots\ A test for this special situation was considered 
by Roy~\cite{roy50}, where he bases his decision on the
statistic
\begin{equation}
R=\frac{L_l}{L_n}
\end{equation}
where \(L_l\) denotes the likelihood of the sample under the
lognormal hypothesis and \(L_n\) that under the normal
hypothesis. If \(R>1\) one accepts lognormality,
and if \(R<1\) normality is accepted. More recently Cox~\cite{cox61,cox62}
has elaborated on Roy's heuristic approach, and has derived a general
class of tests to discriminate between hypotheses that are {\em separate\/}
(in the sense that an arbitrary simple hypothesis in \(H_0\) cannot
be obtained as a limit---in the parameter space---of a simple hypothesis
in \(H_1\). We will now apply Cox's general theory to testing
lognormality against normality\ldots

Suppose \(x_1, x_2, \ldots, x_n\) is a random sample from a certain
population. The null hypothesis, \(H_f\), is that the p.d.f.\ of the
\(x\)'s is log-normal and the alternate hypothesis, \(H_g\), is
that the p.d.f.\ is normal, that is, for \(H_f\)
\begin{equation}
f\left(y,\beta\right) = \frac{1}{\sqrt{2\pi\beta}}
\exp-\left(\frac{\left(\log y-\beta\right)^2}{2\beta}\right),
\: -\infty < y< \infty.
\end{equation}
and for \(H_g\):
\begin{equation}
g\left(y,\alpha\right) = \frac{1}{y\sqrt{2\pi\alpha_2}}
\exp-\left(\frac{\left(y-\alpha_1\right)^2}{2\alpha_2}\right),
\: y>0.
\end{equation}
From the maximum likelihood equations we find that
\begin{equation}
\hat{\alpha}_1=\frac{1}{n}\sum\log x_i; \:
\hat{\alpha}_2=\frac{1}{n}\sum\left(\log x_i-\hat{\alpha}_1\right),
\end{equation}
and analogous equations for \(\hat{\beta}_1\) 
and \(\hat{\beta}_2\).

Under \(H_f\), the log-normal null hypothesis, as the sample
size \(n\) increases to infinity,
\(\hat{\alpha}_1\rightarrow\alpha_1\),
\(\hat{\alpha}_2\rightarrow\alpha_2\),
\(\hat{\beta}_{1,\alpha}\rightarrow\beta_{1,\alpha}\),
and
\(\hat{\beta}_{2,\alpha}\rightarrow\beta_{2,\alpha}\)
where 
\begin{equation}
\hat{\beta}_{1,\alpha}=\exp\left(\alpha_1+\frac{\alpha_2}{2}\right)
\end{equation}
and
\begin{equation}
\hat{\beta}_{2,\alpha}=\exp
\left(2\alpha_1+\alpha_2\right)
\left[\exp\left(\alpha_2\right) -1\right].
\end{equation}
Cox's test is based on the log likelihood ratio
\begin{equation}
L_{fg}=\sum_{i=1}^n\log
\frac{f\left(x_i,\hat{\alpha}\right)}
     {g\left(x_i,\hat{\beta}\right)}
\end{equation}
and his test statistic is given by
\begin{equation}
T_f=L_{fg}-E_{\hat{\alpha}}\left(L_{fg}\right)
\end{equation}
where \(E_{\hat{\alpha}}\left(L_{fg}\right)\) is the expected
value under \(H_f\) when \(\alpha\) takes the value
\(\hat\alpha\). Writing
\begin{equation}
F=\log f\left(x,\alpha\right), \: 
F_{\alpha_i} = \frac{\partial\log f\left(x,\alpha\right)}{\partial\alpha_i},\:
i=1,2
\end{equation}
\null\begin{equation}
F_{\alpha_i\alpha_j} = \frac{\partial^2\log f\left(x,\alpha\right)}
{\partial\alpha_i\partial\alpha_j}, \:
G = \log g\left(x,\beta\right)
\end{equation}
\null\begin{equation}
G_{\beta_i}=\frac{\partial\log g\left(x,\beta\right)}{\partial\beta_i}, \:
\mbox{etc.,}
\end{equation}
Cox shows that \(T_f\) is asymptotically normal with zero mean and
variance
\begin{equation}
V_\alpha\left(T_f\right)=
nV_\alpha\left(F-G\right) -
\sum\frac{C_\alpha^2\left(F-G, F_{\alpha_i}\right)}
         {V_\alpha\left(F_{\alpha_i}\right)}
\end{equation}
where \(V_\alpha\left(\cdot\right)\),
\(C_\alpha\left(\cdot\right)\), denote variance and covariance under \(H_f\).

In our case it can be shown that 
\begin{equation}
T_f=\frac{n}{2}\log\frac{\hat{\beta}_2}{\hat{\beta}_{2,\hat{\alpha}}}
\end{equation}
Results of the following type are used in the derivation of 
\(V_\alpha\left(T_f\right)\):
\begin{equation}
E_\alpha\left[x^2\log x\right] = 
\left(\alpha_1+2\alpha_2\right)\exp\left(2\alpha_1+2\alpha_2\right)
\end{equation}
\null\begin{equation}
E_\alpha\left[x^2\log^2x\right] = 
\left(\alpha_2+\alpha_1^2+4\alpha_1\alpha_2+4\alpha_2^2\right)
\exp\left(2\alpha_1+2\alpha_2\right)
\end{equation}
\null\begin{equation}
E_\alpha\left[\left(\log x\right)\left(\log x-\alpha\right)\right] = 
\alpha_2
\end{equation}
\null\begin{equation}
E_\alpha\left[\left(\log x\right)\left(\log x-\beta_1\right)^2\right] =
\beta_2\left(\alpha_1+2\alpha_2\right)
\end{equation}
\null\begin{equation}
E_\alpha\left[\left(\log x -\alpha_1\right)
\left(\log x-\beta_1\right)^2\right] =
2\alpha_2\beta_2.
\end{equation}
Using these results, after a considerable amount of simplification,
we get
\begin{equation}
V_\alpha\left(T_f\right)=n\left[
\frac{1}{4}\left(e^{4\alpha_2}+
2e^{3\alpha_2}+
3e^{\alpha_2}-4\right)
\alpha_2-
\frac{\alpha_2\left(2e^{\alpha_2}-1\right)^2}
     {2\left(2e^{\alpha_2}-1\right)^2}\right]
\end{equation}
Cox~\cite{cox62} has shown that
\begin{equation}
T'_f=\frac{T_f}{\sqrt{V_\alpha\left(T_f\right)}}
\end{equation}
is asymptotically standardized normal. In our case we get,
after substituting the estimators for the parameters,
\begin{equation}
T'_f = \frac{\log\frac{\hat{\beta}_2}{\beta_{2,\hat{\alpha}}}}
{2\sqrt{n}\{\frac{1}{4}\left(e^{4\hat{\alpha}_2}+
2e^{3\hat{\alpha}_2} -4\right) -\hat{\alpha}_2 -
\frac{\hat{\alpha}_2\left(2e^{\hat{\alpha}_2}-1\right)^2}
     {2\left(2e^{\hat{\alpha}_2}-1\right)^2}
+\frac{3}{4}e^{\hat{\alpha}_2}\}^{1/2}}
\end{equation}
%\begin{equation}
%T'_f = \frac{\log\frac{\hat{\beta}_2}{\beta_{2,\hat{\alpha}}}}
%{2\sqrt{n}\{\frac{1}{4}\left(\exp\left({4\hat{\alpha}_2}\right)+
%2\exp\left({3\hat{\alpha}_2}\right) -4\right) -\hat{\alpha}_2 -
%\frac{\hat{\alpha}_2\left(2\exp\left({\hat{\alpha}_2}\right)-1\right)^2}
%     {2\left(2\exp\left({\hat{\alpha}_2}\right)-1\right)^2}
%+\frac{3}{4}\exp\left({\hat{\alpha}_2}\right)\}^{1/2}}
%\end{equation}

\section{Utility Functions}

This section describes some useful functions included in
\libname\ but not necessarily described in the previous
sections, e.g., normal order statistics, normal probabilities,
inverse normals.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibliography{goodness}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\clearpage
\appendix

\begin{table}
\caption{Cumulative Standard Normal Distribution.}
\label{tbl:normal}
\centerline{Area Under the Normal Curve from}
\begin{displaymath}
-\infty\:\:\mbox{to}\:\:z=\frac{X_i-\mu}{\sigma}.
\end{displaymath}
\centerline{Computed by the author using 
algorithm 5666 for the error function, from
Hart \emph{et al.}~\cite{hart68}.}
\footnotesize
\begin{center}
\begin{tabular}{c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
                 @{\extracolsep{4pt}}c%
}\hline
\(z\)& 0.00& 0.01  & 0.02  & 0.03  & 0.04  & 0.05  & 0.06  & 0.07  & 0.08  & 0.09\\ \hline
0.0&0.50000&0.50399&0.50798&0.51197&0.51595&0.51994&0.52392&0.52790&0.53188&0.53586\\
0.1&0.53983&0.54380&0.54776&0.55172&0.55567&0.55962&0.56356&0.56749&0.57142&0.57535\\
0.2&0.57926&0.58317&0.58706&0.59095&0.59483&0.59871&0.60257&0.60642&0.61026&0.61409\\
0.3&0.61791&0.62172&0.62552&0.62930&0.63307&0.63683&0.64058&0.64431&0.64803&0.65173\\
0.4&0.65542&0.65910&0.66276&0.66640&0.67003&0.67364&0.67724&0.68082&0.68439&0.68793\\
0.5&0.69146&0.69497&0.69847&0.70194&0.70540&0.70884&0.71226&0.71566&0.71904&0.72240\\
0.6&0.72575&0.72907&0.73237&0.73565&0.73891&0.74215&0.74537&0.74857&0.75175&0.75490\\
0.7&0.75804&0.76115&0.76424&0.76730&0.77035&0.77337&0.77637&0.77935&0.78230&0.78524\\
0.8&0.78814&0.79103&0.79389&0.79673&0.79955&0.80234&0.80511&0.80785&0.81057&0.81327\\
0.9&0.81594&0.81859&0.82121&0.82381&0.82639&0.82894&0.83147&0.83398&0.83646&0.83891\\
1.0&0.84134&0.84375&0.84614&0.84849&0.85083&0.85314&0.85543&0.85769&0.85993&0.86214\\
1.1&0.86433&0.86650&0.86864&0.87076&0.87286&0.87493&0.87698&0.87900&0.88100&0.88298\\
1.2&0.88493&0.88686&0.88877&0.89065&0.89251&0.89435&0.89617&0.89796&0.89973&0.90147\\
1.3&0.90320&0.90490&0.90658&0.90824&0.90988&0.91149&0.91309&0.91466&0.91621&0.91774\\
1.4&0.91924&0.92073&0.92220&0.92364&0.92507&0.92647&0.92785&0.92922&0.93056&0.93189\\
1.5&0.93319&0.93448&0.93574&0.93699&0.93822&0.93943&0.94062&0.94179&0.94295&0.94408\\
1.6&0.94520&0.94630&0.94738&0.94845&0.94950&0.95053&0.95154&0.95254&0.95352&0.95449\\
1.7&0.95543&0.95637&0.95728&0.95818&0.95907&0.95994&0.96080&0.96164&0.96246&0.96327\\
1.8&0.96407&0.96485&0.96562&0.96638&0.96712&0.96784&0.96856&0.96926&0.96995&0.97062\\
1.9&0.97128&0.97193&0.97257&0.97320&0.97381&0.97441&0.97500&0.97558&0.97615&0.97670\\
2.0&0.97725&0.97778&0.97831&0.97882&0.97932&0.97982&0.98030&0.98077&0.98124&0.98169\\
2.1&0.98214&0.98257&0.98300&0.98341&0.98382&0.98422&0.98461&0.98500&0.98537&0.98574\\
2.2&0.98610&0.98645&0.98679&0.98713&0.98745&0.98778&0.98809&0.98840&0.98870&0.98899\\
2.3&0.98928&0.98956&0.98983&0.99010&0.99036&0.99061&0.99086&0.99111&0.99134&0.99158\\
2.4&0.99180&0.99202&0.99224&0.99245&0.99266&0.99286&0.99305&0.99324&0.99343&0.99361\\
2.5&0.99379&0.99396&0.99413&0.99430&0.99446&0.99461&0.99477&0.99492&0.99506&0.99520\\
2.6&0.99534&0.99547&0.99560&0.99573&0.99585&0.99598&0.99609&0.99621&0.99632&0.99643\\
2.7&0.99653&0.99664&0.99674&0.99683&0.99693&0.99702&0.99711&0.99720&0.99728&0.99736\\
2.8&0.99744&0.99752&0.99760&0.99767&0.99774&0.99781&0.99788&0.99795&0.99801&0.99807\\
2.9&0.99813&0.99819&0.99825&0.99831&0.99836&0.99841&0.99846&0.99851&0.99856&0.99861\\
3.0&0.99865&0.99869&0.99874&0.99878&0.99882&0.99886&0.99889&0.99893&0.99896&0.99900\\
3.1&0.99903&0.99906&0.99910&0.99913&0.99916&0.99918&0.99921&0.99924&0.99926&0.99929\\
3.2&0.99931&0.99934&0.99936&0.99938&0.99940&0.99942&0.99944&0.99946&0.99948&0.99950\\
3.3&0.99952&0.99953&0.99955&0.99957&0.99958&0.99960&0.99961&0.99962&0.99964&0.99965\\
3.4&0.99966&0.99968&0.99969&0.99970&0.99971&0.99972&0.99973&0.99974&0.99975&0.99976\\
3.5&0.99977&0.99978&0.99978&0.99979&0.99980&0.99981&0.99981&0.99982&0.99983&0.99983\\\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Cumulative Chi-Square Distribution.}
\label{tbl:chisq}
Computed by the author using CDFLIB~\cite{brown93},
with the exception of items marked with a dagger (\dag), which
were found in {\em Biometrika Tables for Statisticians} (1966),
3rd.~Ed., University College, London, as cited by Shapiro~\cite{shapiro90}.
 
\scriptsize
\begin{center}
\begin{tabular}{r%
                                       r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}r@{.}l%
                 @{\extracolsep{1.0pt}}
}
\hline
& \multicolumn{20}{c}{\(\alpha\)} \\ \cline{2-21}
\(\nu\) &
0&995 & 0&990 & 0&975 & 0&950 & 0&900 & 0&100 & 0&050 & 0&025 & 0&010 & 0&005\\
\hline
 1 & 0&\(0000393^{\dag}\) & 0&\(000157^{\dag}\) & 0&\(000982^{\dag}\) 
& 0&\(0158^{\dag}\) & 0&\(102^{\dag}\) &  2&71 &  3&84 &  5&02 &  6&63 &  7&88 \\
 2 & 0&0100 & 0&0201& 0&0506& 0&103 & 0&211 &4&61 & 5&99 & 7&38 & 9&21 & 10&6 \\
 3 & 0&0717 & 0&115 & 0&216 & 0&352 & 0&584 &6&25 & 7&81 & 9&35 &11&3  & 12&8 \\
 4 & 0&207  & 0&297 & 0&484 & 0&711 & 1&06  &7&78 & 9&49 &11&1  &13&3  & 14&9 \\
 5 & 0&412  & 0&554 & 0&831 & 1&15  & 1&61  &9&24 &11&1  &12&8  &15&1  & 16&8 \\
\\
 6 & 0&676 &0&872 & 1&24 & 1&64 & 2&20 & 10&6  & 12&6 & 14&5& 16&8& 18&5 \\
 7 & 0&989 & 1&24 & 1&69 & 2&17 & 2&83 & 12&0  & 14&1 & 16&0& 18&5& 20&3 \\
 8 & 1&34 &  1&65 &  2&18 &  2&73 &  3&49 & 13&4  & 15&5 & 17&5 & 20&1 & 22&0 \\
 9 & 1&73 &  2&09 &  2&70 &  3&33 &  4&17 & 14&7  & 16&9 & 19&0 & 21&7 & 23&6 \\
10 & 2&16 &  2&56 &  3&25 &  3&94 &  4&87 & 16&0  & 18&3 & 20&5 & 23&2 & 25&2 \\
\\
11 & 2&60 & 3&05 & 3&82 & 4&57 & 5&58 & 17&3 & 19&7 & 21&9 & 24&7 & 26&8 \\
12 & 3&07 & 3&57 & 4&40 & 5&23 & 6&30 & 18&6 & 21&0 & 23&3 & 26&2 & 28&3 \\
13 & 3&57 & 4&11 & 5&01 & 5&89 & 7&04 & 19&8 & 22&4 & 24&7 & 27&7 & 29&8 \\
14 & 4&07 & 4&66 & 5&63 & 6&57 & 7&79 & 21&1 & 23&7 & 26&1 & 29&1 & 31&3 \\
15 & 4&60 & 5&23 & 6&26 & 7&26 & 8&55 & 22&3 & 25&0 & 27&5 & 30&6 & 32&8 \\
\\
16 & 5&14 & 5&81 & 6&91 & 7&96 &  9&31 & 23&5 & 26&3 & 28&9 & 32&0 & 34&3 \\
17 & 5&70 & 6&41 & 7&56 & 8&67 & 10&1  & 24&8 & 27&6 & 30&2 & 33&4 & 35&7 \\
18 & 6&26 & 7&01 & 8&23 & 9&39 & 10&9  & 26&0 & 28&9 & 31&5 & 34&8 & 37&2 \\
19 & 6&84 & 7&63 & 8&91 & 10&1 & 11&7  & 27&2 & 30&1 & 32&9 & 36&2 & 38&6 \\
20 & 7&43 & 8&26 & 9&59 & 10&9 & 12&4  & 28&4 & 31&4 & 34&2 & 37&6 & 40&0 \\
\\
21 & 8&03 & 8&90 & 10&3 & 11&6 & 13&2 & 29&6 & 32&7 & 35&5 & 38&9 & 41&4 \\
22 & 8&64 & 9&54 & 11&0 & 12&3 & 14&0 & 30&8 & 33&9 & 36&8 & 40&3 & 42&8 \\
23 & 9&26 & 10&2 & 11&7 & 13&1 & 14&9 & 32&0 & 35&1 & 38&0 & 41&6 & 44&2 \\
24 & 9&89 & 10&9 & 12&4 & 13&9 & 15&7 & 33&2 & 36&4 & 39&4 & 43&0 & 45&6 \\
25 & 10&5 & 11&5 & 13&1 & 14&6 & 16&5 & 34&4 & 37&7 & 40&6 & 44&3 & 46&9 \\
\\
26 & 11&2 & 12&2 & 13&8 & 15&4 & 17&3 & 35&6 & 38&9 & 41&9 & 45&6 & 48&3 \\
27 & 11&8 & 12&9 & 14&6 & 16&2 & 18&1 & 36&7 & 40&1 & 43&2 & 47&0 & 49&6 \\
28 & 12&5 & 13&6 & 15&3 & 16&9 & 18&9 & 37&9 & 41&3 & 44&5 & 48&3 & 51&0 \\
29 & 13&1 & 14&3 & 16&0 & 17&7 & 19&8 & 39&1 & 42&6 & 45&7 & 49&6 & 52&3 \\
30 & 13&8 & 15&0 & 16&8 & 18&5 & 20&6 & 40&3 & 43&8 & 47&0 & 50&9 & 53&7 \\
\hline
\end{tabular}
\end{center}
\normalsize
According to Shapiro~\cite{shapiro90}, for situations with
larger than 30 degrees of freedom, \(\chi^2_{\nu,\alpha} =
0.5 \left(z_{\alpha}+\sqrt{2\nu-1}\right)^2\), where
\(z_{\alpha}\) is the 100\(\alpha\)\% point of the standard normal
distribution, e.g., \(z_{0.05}=-1.645\) from Table~\ref{tbl:normal}.
\end{table}
\clearpage

\begin{table}
\caption{Signficant Values of D'Agostino's D Test (\(y\) statistic
of eqn.~\protect\ref{eqn:xform-d}).}
\centerline{Reproduced from D'Agostino~\protect\cite{dagostino86}.}
\label{tbl:d-test}
\scriptsize
\begin{center}
\begin{tabular}{rllllllllll}\hline
& \multicolumn{10}{c}{Percentiles} \\ \cline{2-11}
n & 0.5 & 1.0 & 2.5 & 5 & 10 & 90 & 95 & 97.5 & 99 & 99.5 \\ \hline
10&-4.66&-4.06&-3.25&-2.62&-1.99&0.149&0.235&0.299&0.356&0.385\\
12&-4.63&-4.02&-3.20&-2.58&-1.94&0.237&0.329&0.381&0.440&0.479\\
14&-4.57&-3.97&-3.16&-2.53&-1.90&0.308&0.399&0.460&0.515&0.555\\
16&-4.52&-3.92&-3.12&-2.50&-1.87&0.367&0.459&0.526&0.587&0.613\\
18&-4.47&-3.87&-3.08&-2.47&-1.85&0.417&0.515&0.574&0.636&0.667\\
20&-4.41&-3.83&-3.04&-2.44&-1.82&0.460&0.565&0.628&0.690&0.720\\
\\
22&-4.36&-3.78&-3.01&-2.41&-1.81&0.497&0.609&0.677&0.744&0.775\\
24&-4.32&-3.75&-2.98&-2.39&-1.79&0.530&0.648&0.720&0.783&0.822\\
26&-4.27&-3.71&-2.96&-2.37&-1.77&0.559&0.682&0.760&0.827&0.867\\
28&-4.23&-3.68&-2.93&-2.35&-1.76&0.586&0.714&0.797&0.868&0.910\\
30&-4.19&-3.64&-.291&-2.33&-1.75&0.610&0.743&0.830&0.906&0.941\\
\\
32&-4.16&-3.61&-2.88&-2.32&-1.73&0.631&0.770&0.862&0.942&0.983\\
34&-4.12&-3.59&-2.86&-2.30&-1.72&0.651&0.794&0.891&0.975&1.02\\
36&-4.09&-3.56&-2.85&-2.29&-1.71&0.669&0.816&0.917&1.00&1.05\\
38&-4.06&-3.54&-2.83&-2.28&-1.70&0.686&0.837&0.941&1.03&1.08\\
40&-4.03&-3.51&-2.81&-2.26&-1.70&0.702&0.857&0.964&1.06&1.11\\
\\
42&-4.00&-3.49&-2.80&-2.25&-1.69&0.716&0.875&0.986&1.09&1.14\\
44&-3.98&-3.47&-2.78&-2.24&-1.68&0.730&0.892&1.01&1.11&1.17\\
46&-3.95&-3.45&-2.77&-2.23&-1.67&0.742&0.908&1.02&1.13&1.19\\
48&-3.93&-3.43&-2.75&-2.22&-1.67&0.754&0.923&1.04&1.15&1.22\\
50&-3.91&-3.41&-2.74&-2.21&-1.66&0.765&0.937&1.06&1.18&1.24\\
\\
60&-3.81&-3.34&-2.68&-2.17&-1.64&0.812&0.997&1.13&1.26&1.34\\
70&-3.73&-3.27&-2.64&-2.14&-1.61&0.849&1.05&1.19&1.33&1.42\\
80&-3.67&-3.22&-2.60&-2.11&-1.59&0.878&1.08&1.24&1.39&1.48\\
90&-3.61&-3.17&-2.57&-2.09&-1.58&0.902&1.12&1.28&1.44&1.54\\
100&-3.57&-3.14&-2.54&-2.07&-1.57&0.923&1.14&1.31&1.48&1.59\\
\\
150&-3.409&-3.009&-2.452&-2.004&-1.520&0.990&1.233&1.423&1.623&1.746\\
200&-3.302&-2.922&-2.391&-1.960&-1.491&1.032&1.290&1.496&1.715&1.853\\
250&-3.227&-2.861&-2.348&-1.926&-1.471&1.060&1.328&1.545&1.779&1.927\\
300&-3.172&-2.816&-2.316&01.906&-1.456&1.080&1.357&1.528&1.826&1.983\\
350&-3.129&-2.781&-2.291&-1.888&-1.444&1.096&1.379&1.610&1.863&2.026\\
\\
400&-3.094&-2.753&-2.270&-1.873&-1.434&1.108&1.396&1.633&1.893&2.061\\
450&-3.064&-2.729&-2.253&-1.861&-1.426&1.119&1.411&1.652&1.918&2.090\\
500&-3.040&-2.709&-2.239&-1.850&-1.419&1.127&1.423&1.668&1.938&2.114\\
550&-3.019&-2.691&-2.226&-1.841&-1.413&1.135&1.434&1.682&1.957&2.136\\
600&-3.000&-2.676&-2.215&-1.833&-1.408&1.141&1.443&1.694&1.972&2.154\\
\\
650&-2.984&-2.663&-2.206&-1.826&-1.403&1.147&1.451&1.704&1.986&2.171\\
700&-2.969&-2.651&-2.197&-1.820&-1.399&1.152&1.458&1.714&1.999&2.185\\
750&-2.956&-2.640&-2.189&-1.814&-1.395&1.157&1.465&1.722&2.010&2.199\\
800&-2.944&-2.630&-2.182&-1.809&-1.392&1.161&1.471&1.730&2.020&2.221\\
850&-2.933&-2.621&-2.176&-1.804&-1.389&1.165&1.476&1.737&2.029&2.221\\
\\
900&-2.923&-2.613&-2.710&-1.800&-1.386&1.168&1.481&1.743&2.037&2.231\\
950&-2.914&-2.605&-2.164&-1.796&-1.383&1.171&1.485&1.749&2.045&2.241\\
1000&-2.906&-2.599&-2.159&-1.792&-1.381&1.174&1.489&1.754&2.052&2.249\\
1500&-2.845&-2.549&-2.123&-1.765&-1.363&1.194&1.519&1.793&2.103&2.309\\
2000&-2.807&-2.515&-2.101&-1.750&-1.353&1.207&1.536&1.815&2.132&2.342\\
\hline
\end{tabular}
\end{center}
\end{table}
\clearpage

\begin{table}
\caption{Sample Data. Diameters at Breast Height (cm)
of 584 Longleaf Pine Trees.}
\label{tbl:pine}
Locations and Diameters at Breast Height (dbh, in centimeters)
of all 584 Longleaf Pine Trees in the 4 hectare Study Region.
The \(x\) coordinates are distances (in meters) from the tree to the 
southern boundary. The \(y\) coordinates are distances (in meters) from
the tree to the eastern boundary.
Reproduced from Table~8.1 of Cressie~\protect\cite{cressie91}.
\scriptsize
\begin{center}
\begin{tabular}{rrrrrrrrrrrr}
\hline
\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh \\
\hline
200.0&  8.8& 32.9&199.3& 10.0& 53.5&193.6& 22.4& 68.0&167.7& 35.6& 17.7\\
183.9& 45.4& 36.9&182.5& 47.2& 51.6&166.1& 48.8& 66.4&160.7& 42.4& 17.7\\
162.9& 29.0& 21.9&166.4& 33.6& 25.7&163.0& 35.8& 25.5&156.1& 38.7& 28.3\\
157.6& 42.8& 11.2&154.4& 36.2& 33.8&150.8& 45.8&  2.5&144.6& 25.4&  4.2\\
142.7& 25.4&  2.5&144.0& 28.3& 31.2&143.5& 36.9& 16.4&123.1& 14.3& 53.2\\
113.9& 13.1& 67.3&114.9&  8.1& 37.8&101.4&  9.3& 49.9&105.7&  9.1& 46.3\\
106.9& 14.7& 40.5&127.0& 29.7& 57.7&129.8& 45.8& 58.0&136.3& 44.2& 54.9\\
106.7& 49.4& 25.3&103.4& 49.6& 18.4& 89.7&  4.9& 72.0& 10.8&  0.0& 31.4\\
 26.4&  5.4& 55.1& 11.0&  5.5& 36.0&  5.1&  3.9& 28.4& 10.1&  8.5& 24.8\\
 18.9& 11.3& 44.1& 28.4& 11.0& 50.9& 41.1&  9.2& 47.5& 41.2& 12.6& 58.0\\
\\
 33.9& 21.4& 36.9& 40.8& 39.8& 65.6& 49.7& 18.2& 52.9&  6.7& 46.9& 39.5\\
 11.6& 46.9& 42.7& 17.2& 47.9& 44.4& 19.4& 50.0& 40.3& 26.9& 47.2& 53.5\\
 39.6& 47.9& 44.2& 38.0& 50.7& 53.8& 19.1& 45.2& 38.0& 32.1& 35.0& 48.3\\
 28.4& 35.5& 42.9&  3.8& 44.8& 40.6&  8.5& 43.4& 34.5& 11.2& 40.2& 45.7\\
 22.4& 34.3& 51.8& 23.8& 33.3& 52.0& 24.9& 29.8& 44.5&  9.0& 38.9& 35.6\\
 10.4& 61.2& 19.2& 30.9& 52.2& 43.5& 48.9& 67.8& 33.7& 49.5& 73.8& 43.3\\
 46.3& 80.9& 36.6& 44.1& 78.0& 46.3& 48.5& 94.8& 48.3& 45.9& 90.4& 20.4\\
 44.2& 84.0& 40.5& 37.0& 64.3& 44.0& 36.3& 67.7& 40.9& 36.7& 71.5& 51.0\\
 35.3& 78.3& 36.5& 33.5& 81.6& 42.1& 29.3& 83.8& 15.6& 22.4& 84.1& 18.5\\
 17.1& 84.7& 43.0& 27.3& 89.4& 28.9& 27.9& 90.6& 21.3& 48.4& 99.5& 30.9\\

 43.6& 98.4& 42.7& 39.0& 97.3& 37.6& 14.9& 91.2& 47.1&  6.1& 96.2& 44.6\\
 10.7& 98.6& 44.3& 22.2&100.0& 26.1\\
     &     &     &     &     &     & 32.7& 99.1& 25.9&  0.9&100.0& 41.4\\
 93.5& 96.2& 59.5& 85.1& 90.6& 26.1& 92.8& 61.5& 11.4& 91.3& 69.5& 33.4\\
 95.9& 59.7& 35.8& 93.4& 71.5& 54.4& 89.6& 86.3& 33.6& 99.5& 78.9& 35.5\\
100.6& 53.1&  7.4&103.5& 72.1& 36.6&104.7& 74.0& 19.1&104.0& 67.1& 34.9\\
104.2& 64.7& 37.3&105.0& 59.8& 16.3&111.8& 73.2& 39.1&112.4& 69.8& 36.5\\
110.0& 65.9& 25.0&120.4& 79.2& 46.8&109.4& 62.5& 18.7&109.7& 62.9& 23.2\\
113.3& 60.4& 20.4&118.0& 69.3& 42.3&126.5& 69.2& 38.1&125.1& 68.2& 17.9\\
114.2& 54.6& 39.7&110.6& 51.5& 14.5&147.3& 73.8& 33.5&146.7& 73.0& 56.0\\
148.1& 86.2& 66.1&138.2& 73.4& 26.3&135.7& 70.7& 44.8&134.9& 72.7& 24.2\\
 98.0& 27.7& 39.0& 93.5& 28.7& 15.1& 82.3& 16.8& 35.6& 79.2& 25.3& 21.6\\
 84.2& 29.0& 17.2& 88.8& 35.1& 22.3& 82.5& 36.3& 18.2& 75.6& 28.1& 55.6\\
 72.9& 36.2& 23.2& 79.1& 43.6& 27.0& 50.0& 48.8& 50.1& 59.9& 34.4& 45.5\\
 60.5& 13.0& 47.2& 60.2& 11.4& 37.8& 66.5& 15.9& 31.9& 70.4&  6.6& 38.5\\
 70.7&  2.2& 23.8& 71.7&  1.9& 46.3&179.5& 92.6&  2.8&186.1& 91.0&  3.2\\
178.3& 92.4&  5.8&178.6& 91.8&  3.5&186.2& 90.3&  2.3&185.2& 89.9&  3.8\\
185.5& 89.8&  3.2&185.8& 89.1&  4.4&186.5& 88.8&  3.9&176.7& 92.3&  7.8\\
177.7& 91.5&  4.7&184.0& 89.0&  4.8& 11.0& 34.4& 44.1& 17.5& 21.9& 51.5\\
  4.3& 31.3& 51.6&  5.9&  8.1& 33.3&  1.9& 68.5& 13.3&  1.8& 71.0&  5.7\\
  1.1& 82.5&  3.3&  2.4& 95.3& 45.9&  4.6& 94.0& 32.6&  3.1& 79.5& 11.4\\
  3.9& 72.1&  9.1&  4.1& 70.9&  5.2&  7.9& 68.7&  4.9& 14.8& 81.8& 42.0\\
  9.4& 67.7& 32.0& 15.9& 78.7& 32.8& 16.6& 78.8& 22.0& 18.2& 80.3& 20.8\\
174.1&135.6&  7.3&173.0&127.4&  3.0&174.0&125.7&  2.2&177.3&121.0&  2.2\\
177.6&120.3&  2.2&195.7&144.1& 59.4&197.0&142.5& 48.1&178.2&112.6& 51.5\\
173.8&112.7& 50.3&172.8&124.4&  2.9&162.7&114.6& 19.1&164.6&120.9& 15.1\\
 80.4& 90.7& 21.7& 71.0& 88.8& 42.4& 73.0& 85.6& 40.2& 56.7& 95.3& 37.4\\
 66.5& 86.2& 40.1& 67.0& 84.7& 39.5& 62.9& 87.9& 32.5& 61.8& 89.0& 39.5\\
 51.9& 94.5& 35.6& 60.9& 71.6& 44.1& 61.0& 69.8& 42.2& 61.7& 66.2& 39.4\\
 57.3& 68.4& 35.5& 54.2& 76.4& 39.1& 76.1& 52.9&  9.5& 67.2& 57.6& 48.4\\
 81.9& 58.5& 31.9& 90.1& 59.6& 30.7&135.3&126.6& 15.0&135.0&124.0& 24.5\\
\hline
\end{tabular}
\end{center}
\end{table}
\clearpage

\scriptsize
\begin{center}
{\normalsize Table~\thetable (continued).}
\par\vspace{\baselineskip}\par
\begin{tabular}{rrrrrrrrrrrr}
\hline
\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh \\
\hline
136.2&122.1& 15.0&129.7&127.0& 22.2&134.8&120.2& 27.5&136.9&116.8& 10.8\\
137.0&116.0& 26.2&128.9&124.2& 10.2&127.5&125.0& 18.9&127.6&121.7& 44.2\\
129.7&119.0& 13.8&126.6&121.1& 16.7&133.4& 77.1& 35.7&129.9& 76.1& 12.1\\
126.5& 77.3& 35.4&129.1& 83.1& 32.7&134.4& 87.0& 30.1&130.7& 90.1& 28.4\\
130.9& 90.7& 16.5&132.0& 94.5& 12.7&136.8& 96.7&  5.5&137.7& 98.0&  2.5\\
157.8& 99.9&  3.0&187.1& 98.1&  3.2&190.6& 92.1&  3.2&185.4& 93.1&  4.0\\
186.6& 92.2&  3.6&185.9& 91.7&  3.8&184.3& 92.1&  4.3&188.2& 91.2&  3.3\\
104.4&145.1&  6.3&104.9&145.0& 18.4&101.5&148.4&  5.4&102.4&148.7&  5.4\\
123.4&128.9& 26.0&123.8&135.1& 22.3&127.0&133.8& 35.2&109.6&145.9& 24.1\\
112.4&145.0&  6.9&133.1&144.8& 61.0&139.4&143.1& 20.6&140.4&143.6&  6.5\\
184.1& 88.2&  2.8&183.5& 88.5&  4.8&183.0& 88.0&  5.4&176.1& 91.0&  4.3\\
175.6& 90.2&  4.0&173.8& 89.9&  3.2&164.9& 93.7&  2.8&163.0& 95.3&  4.9\\
163.2& 94.1&  3.5&162.4& 94.5&  2.9&161.5& 94.9&  2.4&162.2& 94.3&  3.3\\
161.0& 94.7&  2.1&157.7& 95.7&  2.0&154.9& 96.2&  3.9&154.6& 92.7&  5.0\\
152.9& 93.7&  2.3&153.2& 93.2&  2.2&168.2& 73.0& 67.7&151.6& 93.0&  2.9\\
151.4& 93.4&  2.4&157.6& 67.2& 56.3&149.4& 63.0& 39.4&149.4& 64.3& 59.5\\
167.3& 54.6& 42.4&157.4& 51.5& 63.7&181.5& 66.1& 66.6&196.5& 55.2& 69.3\\
189.9& 85.2& 56.9&155.1&149.2& 23.5&154.5&148.4&  9.1&162.9&119.9& 29.9\\
158.4&113.4& 14.9&153.9&108.3& 38.7&156.1&116.0& 31.5&156.5&118.9& 27.8\\
156.8&122.3& 28.5&159.0&126.1& 21.6&161.0&131.9&  2.0&161.3&132.8&  2.6\\
160.6&132.6&  2.3&161.3&134.9&  3.5&159.7&129.8&  3.6&161.7&136.1&  2.6\\
161.1&136.4&  2.0&160.1&133.0&  2.0&159.0&133.6&  2.7&160.0&134.8&  2.6\\
160.2&135.5&  2.2&159.1&136.5&  2.7&154.7&126.8& 30.1&151.9&127.5& 16.6\\
151.3&124.7& 10.4&151.0&127.3& 11.8&150.4&123.0& 32.3&149.6&124.6& 33.5\\
146.2&127.1& 30.5&146.1&127.4& 10.5&144.4&131.8& 13.8&143.3&131.5& 22.8\\
140.6&137.7& 31.7&143.2&125.4& 10.1&127.1&119.9& 14.5&120.7&115.6& 12.0\\
115.3&112.6&  2.2&134.1&105.2&  2.3&134.6&104.1&  3.2&135.6&103.3&  3.0\\
128.9&102.6& 50.6&116.3&106.5&  2.6&104.3&104.0& 50.0&111.5&100.0& 52.2\\
100.5&149.7&  5.2&100.0&145.5&  5.2&100.8&145.0&  6.7&100.9&143.5& 14.0\\
100.3&140.8& 12.7&101.5&120.8& 59.5& 99.3&110.6& 52.0& 99.2&106.0& 45.9\\
102.0&137.1& 18.0&105.4&115.7& 43.5&103.6&134.2&  3.3&103.9&139.4&  4.3\\
102.6&141.6&  7.4&102.0&143.3& 10.1&102.1&144.4& 23.1&103.5&141.3&  8.1\\
102.9&143.8&  5.7&105.7&138.2& 13.3&106.6&135.1& 12.8&108.5&133.2& 11.6\\
105.2&142.3&  6.3&139.7&145.8& 20.0&145.5&148.4&  8.9&146.4&148.4& 27.6\\
105.8&149.8&  4.5& 96.7&149.1&  9.2& 66.5&150.0&  2.3& 55.7&148.5&  5.0\\
 54.7&146.8&  4.0& 57.1&144.0& 21.8& 61.7&145.3& 10.9& 60.1&143.7& 14.9\\
 77.7&144.8& 45.0& 67.2&139.3& 16.4& 80.7&133.2& 43.3& 85.1&133.5& 55.6\\
 94.7&143.7& 10.6& 81.2&125.0& 45.9& 81.9&123.2& 45.2& 83.8&123.1& 35.5\\
 84.8&121.4& 43.6& 82.9&119.2& 44.6& 82.1&116.4& 38.8& 84.3&114.8& 34.9\\
 96.7&142.6& 17.0& 92.0&109.0& 50.4& 96.1&146.6&  2.0& 78.5&102.5& 33.8\\
 78.7&103.0& 51.1& 59.5&107.4& 21.8& 56.5&105.5& 46.5& 64.3&132.1&  5.6\\
152.7&146.7& 19.6&155.8&145.4& 32.3&161.2&138.1&  3.7&161.0&138.1&  2.7\\
162.1&136.9&  2.5&166.2&132.0&  2.5&168.7&133.4&  2.4&169.3&133.7&  7.2\\
 57.9&140.7&  7.0& 57.5&142.3& 11.8& 57.3&141.7&  8.5& 56.0&137.7&  9.5\\
 53.4&139.3&  7.0& 53.1&136.0& 10.5& 54.0&137.7&  6.6& 54.5&136.7&  6.6\\
 53.3&137.8&  8.8& 52.1&139.3& 11.6& 48.0&114.4& 48.2& 44.2&129.6& 36.2\\
 39.4&136.8& 44.9& 42.7&124.0& 43.0& 38.1&134.4& 37.5& 37.1&131.9& 31.5\\
 37.6&125.4& 39.9& 31.2&127.9& 35.5& 40.1&112.2& 51.7& 29.3&118.6& 36.5\\
 23.8&114.5& 40.2&141.0&127.8&  7.8&140.1&127.3& 17.0&140.9&121.4& 36.4\\
135.0&132.3& 19.6&139.3&122.9& 15.0&142.0&117.2& 28.8&140.4&117.2& 20.1\\
\hline
\end{tabular}
\end{center}
\clearpage

\scriptsize
\begin{center}
{\normalsize Table~\thetable (continued).}
\par\vspace{\baselineskip}\par
\begin{tabular}{rrrrrrrrrrrr}
\hline
\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh &\(x\)&\(y\)& dbh \\
\hline
138.5&121.5& 39.3& 28.7&158.8& 37.9& 33.7&162.3& 40.6& 23.1&160.8& 33.0\\
 11.3&158.9& 35.7& 18.2&168.2& 20.6& 21.5&172.3& 22.0& 15.9&168.3& 16.3\\
 15.4&172.8&  5.6& 14.0&174.2&  7.4&  6.8&179.6& 42.3&  6.0&184.1& 43.8\\
  1.6&194.9& 53.0& 43.6&197.3& 48.1& 39.4&195.5& 41.9& 37.1&196.1& 48.0\\
 23.7&193.9& 75.9& 21.5&187.9& 40.4& 27.7&188.7& 40.9& 32.3&178.9& 39.4\\
 32.6&168.6& 40.9& 37.7&176.9& 17.6&107.5&138.5& 17.8&107.9&139.5&  3.7\\
116.5&122.6& 19.0&114.5&127.7& 11.2&115.3&127.4& 27.6&115.3&128.1& 14.5\\
119.0&127.4& 34.4&119.4&127.7& 20.0& 94.7&179.8&  2.9& 89.3&185.0&  7.3\\
 90.8&174.0& 52.7& 95.3&158.4&  8.7& 90.9&162.1&  3.6& 90.2&162.1&  4.6\\
 90.2&161.7& 11.4& 90.6&160.8& 11.0& 93.0&158.0& 18.7& 78.4&172.4&  5.6\\
 76.2&171.4&  2.1& 75.8&171.0&  3.3& 75.7&169.7& 11.5& 82.7&163.5&  2.6\\
 76.7&166.3&  4.4& 74.7&167.1& 18.3&119.4&170.8&  7.5& 74.2&164.3& 17.2\\
 73.9&162.7&  4.6& 81.7&156.7& 32.0& 79.5&156.3& 56.7& 56.8&116.0& 46.0\\
 62.2&137.7&  7.8& 58.2&125.1& 54.9& 54.1&115.5& 45.5& 59.5&138.1&  9.2\\
 58.6&140.3& 13.2& 58.8&141.5& 15.3& 57.9&137.3&  8.5&153.5&159.9&  2.2\\
155.9&183.7& 58.8&160.4&176.6& 47.5&171.3&185.1& 52.2&182.8&187.4& 56.3\\
182.5&196.0& 39.8&176.3&197.7& 38.1&161.9&199.4& 38.9&199.5&179.4&  9.7\\
197.6&176.9&  7.4&196.3&192.4& 22.1&195.7&180.5& 16.9&196.2&177.1&  5.9\\
196.3&176.0& 10.5&193.7&185.8&  9.5&191.7&189.2& 45.9&194.5&173.8& 11.4\\
192.7&177.3&  7.8&188.9&182.1& 14.4&190.1&174.4&  8.3&186.9&179.4& 30.6\\
 26.9&111.3& 44.4& 17.9&111.0& 38.7& 34.4&104.2& 41.5& 31.9&103.2& 34.5\\
 20.6&101.5& 31.8& 14.1&103.1& 39.7&  2.9&122.8& 23.3&  6.4&125.9& 37.7\\
  2.2&142.2& 43.0& 11.7&116.2& 39.2& 14.2&116.5& 40.4& 15.6&118.1& 36.7\\
 13.6&127.4& 48.4& 11.1&134.8& 27.9&  7.2&141.7& 46.4& 12.2&140.1& 38.5\\
 23.0&132.7& 39.4& 30.2&133.9& 50.0& 27.7&136.5& 51.6&  3.4&148.8& 38.7\\
 15.4&145.6& 39.6& 16.7&146.4& 29.1& 24.3&145.7& 44.0&  0.4&175.2& 50.9\\
  0.0&177.5& 50.8&  7.9&151.0& 43.0& 33.2&151.2& 44.5& 36.6&150.6& 29.8\\
 42.2&153.7& 44.3& 24.5&153.4& 51.2& 40.4&179.3& 37.7& 41.0&176.6& 36.8\\
 43.9&182.2& 33.6& 44.7&184.6& 47.9& 45.6&175.2& 32.0& 47.5&175.9& 40.3\\
 51.2&177.9& 42.5& 55.0&159.3& 59.7& 58.0&180.3& 44.2& 54.6&188.7& 30.9\\
 58.9&180.0& 39.5& 63.9&178.6& 48.7& 64.3&178.9& 32.8& 65.6&179.3& 47.2\\
 61.0&184.9& 42.1& 63.1&183.3& 43.8& 86.1&186.9& 30.5& 65.8&194.9& 28.3\\
 90.0&195.1& 10.4& 94.3&196.1& 15.0& 91.9&197.1&  7.4& 86.5&197.4& 15.3\\
 87.5&199.3& 17.5& 93.9&199.2&  5.0& 92.4&199.3& 12.2& 81.8&198.9&  9.0\\
 99.0&158.1&  2.4& 94.1&187.2& 13.7& 95.4&182.9& 13.1& 97.1&168.4& 12.8\\
 79.2&155.6& 27.0& 61.6&158.2&  2.6& 70.3&153.1&  4.9& 79.8&151.8& 35.0\\
110.1&150.4& 23.7&116.1&156.8& 42.9&114.0&165.1& 14.2&103.2&154.4&  3.3\\
112.3&167.0& 28.4&110.4&167.3& 10.0&110.6&166.4&  6.4&107.0&165.0& 22.0\\
105.6&160.6&  4.3&104.0&162.4& 10.0&104.0&166.1&  9.2&103.7&167.2&  3.7\\
108.6&182.1& 66.7&105.7&182.6& 68.0&102.8&169.7& 23.1&101.5&171.8&  5.7\\
100.4&170.5& 11.7&144.1&199.0& 40.4&138.3&197.9& 43.3&142.7&197.2& 60.2\\
118.8&188.0& 55.5&142.3&173.3& 54.1&143.8&156.0& 22.3&145.3&155.6& 21.4\\
151.2&192.2& 55.7&153.7&176.5& 51.4&186.9&174.7& 23.9&181.2&176.9&  5.2\\
181.1&176.1&  7.6&177.2&174.5& 27.8&182.8&162.9& 49.6&180.0&160.2& 51.0\\
189.1&156.3& 50.7&196.9&151.4& 43.4&171.4&161.6& 55.6&169.1&160.0&  4.3\\
162.5&157.3&  2.5&156.7&155.3& 23.5&154.1&150.8&  8.0& 87.7&200.0& 11.7\\
\hline
\end{tabular}
\end{center}
\normalsize
\clearpage

\begin{table}
\caption{
Coefficients for transforming \(\protect\sqrt{b_1}\) to a standard normal
using a Johnson \(S_U\) approximation.}
\centerline{Reproduced from Table~4 of D'Agostino and 
Pearson~\protect\cite{dagostino73}.}
\label{tbl:johnson}
\scriptsize
\begin{center}
\begin{tabular}{rllrllrll}\hline
\multicolumn{1}{c}{\(n\)} & \multicolumn{1}{c}{\(\delta\)} 
& \multicolumn{1}{c}{\(1/\lambda\)} &
\multicolumn{1}{c}{\(n\)} & \multicolumn{1}{c}{\(\delta\)} 
& \multicolumn{1}{c}{\(1/\lambda\)} &
\multicolumn{1}{c}{\(n\)} & \multicolumn{1}{c}{\(\delta\)} 
& \multicolumn{1}{c}{\(1/\lambda\)} \\ \hline
 8 & 5.563 & 0.3030 & 62 & 3.389 & 1.0400 & 260 & 5.757 & 1.1744 \\
 9 & 4.260 & 0.4080 & 64 & 3.420 & 1.0449 & 270 & 5.835 & 1.1761 \\
10 & 3.734 & 0.4794 & 66 & 3.450 & 1.0495 & 280 & 5.946 & 1.1779 \\
   &       &        & 68 & 3.480 & 1.0540 & 290 & 6.039 & 1.1793 \\
11 & 3.447 & 0.5339 & 70 & 3.510 & 1.0581 & 300 & 6.130 & 1.1808 \\
12 & 3.270 & 0.5781 \\
13 & 3.151 & 0.6153 & 72 & 3.540 & 1.0621 & 310 & 6.220 & 1.1821 \\
14 & 3.069 & 0.6473 & 74 & 3.569 & 1.0659 & 320 & 6.308 & 1.1834 \\
15 & 3.010 & 0.6753 & 76 & 3.599 & 1.0695 & 330 & 6.396 & 1.1846 \\
   &       &        & 78 & 3.628 & 1.0730 & 340 & 6.482 & 1.1858 \\
16 & 2.968 & 0.7001 & 80 & 3.657 & 1.0763 & 350 & 6.567 & 1.1868 \\
17 & 2.937 & 0.7224 \\
18 & 2.915 & 0.7426 & 82 & 3.686 & 1.0795 & 360 & 6.651 & 1.1879 \\
19 & 2.900 & 0.7610 & 84 & 3.715 & 1.0825 & 370 & 6.733 & 1.1888 \\
20 & 2.890 & 0.7779 & 86 & 3.744 & 1.0854 & 380 & 6.815 & 1.1897 \\
   &       &        & 88 & 3.772 & 1.0882 & 390 & 6.896 & 1.1906 \\
21 & 2.884 & 0.7934 & 90 & 3.801 & 1.0909 & 400 & 6.976 & 1.1914 \\
22 & 2.882 & 0.8078 \\
23 & 2.882 & 0.8211 & 92 & 3.829 & 1.0934 & 410 & 7.056 & 1.1922 \\
24 & 2.884 & 0.8336 & 94 & 3.857 & 1.0959 & 420 & 7.134 & 1.1929 \\
25 & 2.889 & 0.8452 & 96 & 3.885 & 1.0983 & 430 & 7.211 & 1.1937 \\
   &       &        & 98 & 3.913 & 1.1006 & 440 & 7.288 & 1.1943 \\
26 & 2.895 & 0.8561 & 100 & 3.940 & 1.1028 & 450 & 7.363 & 1.1950 \\
27 & 2.902 & 0.8664 \\
28 & 2.910 & 0.8760 & 105 & 4.009 & 1.1080 & 460 & 7.438 & 1.1956 \\
29 & 2.920 & 0.8851 & 110 & 4.076 & 1.1128 & 470 & 7.513 & 1.1962 \\
30 & 2.930 & 0.8938 & 115 & 4.142 & 1.1172 & 480 & 7.586 & 1.1968 \\
   &       &        & 120 & 4.207 & 1.1212 & 490 & 7.659 & 1.1974 \\
31 & 2.941 & 0.9020 & 125 & 4.272 & 1.1250 & 500 & 7.731 & 1.1959 \\
32 & 2.952 & 0.9097 \\
33 & 2.964 & 0.9171 & 130 & 4.336 & 1.1285 & 520 & 7.873 & 1.1989 \\
34 & 2.977 & 0.9241 & 135 & 4.398 & 1.1318 & 540 & 8.013 & 1.1998 \\
35 & 2.990 & 0.9308 & 140 & 4.460 & 1.1348 & 560 & 8.151 & 1.2007 \\
   &       &        & 145 & 4.521 & 1.1377 & 580 & 8.286 & 1.2015 \\
36 & 3.003 & 0.9372 & 150 & 4.582 & 1.1403 & 600 & 8.419 & 1.2023 \\
37 & 3.016 & 0.9433 \\
38 & 3.030 & 0.9492 & 155 & 4.641 & 1.1428 & 620 & 8.550 & 1.2030 \\
39 & 3.044 & 0.9548 & 160 & 4.700 & 1.1452 & 640 & 8.679 & 1.2036 \\
40 & 3.058 & 0.9601 & 165 & 4.758 & 1.1474 & 660 & 8.806 & 1.2043 \\
   &       &        & 170 & 4.816 & 1.1496 & 680 & 8.931 & 1.2049 \\
41 & 3.073 & 0.9653 & 175 & 4.873 & 1.1516 & 700 & 9.054 & 1.2054 \\
42 & 3.087 & 0.9702 \\
43 & 3.102 & 0.9750 & 180 & 4.929 & 1.1535 & 720 & 9.176 & 1.2060 \\
44 & 3.117 & 0.9795 & 185 & 4.985 & 1.1553 & 740 & 9.297 & 1.2065 \\
45 & 3.131 & 0.9840 & 190 & 5.040 & 1.1570 & 760 & 9.415 & 1.2069 \\
   &       &        & 195 & 5.094 & 1.1586 & 780 & 9.533 & 1.2073 \\
46 & 3.146 & 0.9882 & 200 & 5.148 & 1.1602 & 800 & 9.649 & 1.2078 \\
47 & 3.161 & 0.9923 \\
48 & 3.176 & 0.9963 & 205 & 5.202 & 1.1616 & 820 & 9.763 & 1.2082 \\
49 & 3.192 & 1.0001 & 210 & 5.255 & 1.1631 & 840 & 9.876 & 1.2086 \\
50 & 3.207 & 1.0038 & 215 & 5.307 & 1.1644 & 860 & 9.988 & 1.2089 \\
   &       &        & 220 & 5.359 & 1.1657 & 880 & 10.098 & 1.2093 \\
52 & 3.237 & 1.0108 & 225 & 5.410 & 1.1669 & 900 & 10.208 & 1.2096 \\
54 & 3.268 & 1.0174 \\
56 & 3.298 & 1.0235 & 230 & 5.461 & 1.1681 & 920 & 10.316 & 1.2100 \\
58 & 3.329 & 1.0293 & 235 & 5.511 & 1.1693 & 940 & 10.423 & 1.2103 \\
60 & 3.359 & 1.0348 & 240 & 5.561 & 1.1704 & 960 & 10.529 & 1.2106 \\
   &       &        & 245 & 5.611 & 1.1714 & 980 & 10.634 & 1.2109 \\
   &       &        & 250 & 5.660 & 1.1724 & 1000 & 10.738 & 1.2111 \\
\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Coefficients \(\{a_{n-i+1}\}\) for the Shapiro-Wilk
\(W\) Test for Normality.}
\centerline{Reproduced from Table~5 of Shapiro and Wilk~\cite{shapiro65}.}
\label{tbl:shapiro-wilk-a}
\tiny
\begin{center}
\begin{tabular}{rcccccccccc}\hline
\multicolumn{1}{c}{\(i\)}  & \multicolumn{10}{c}{\(n\)} \\ \hline
  & \multicolumn{1}{c}{2} 
  & \multicolumn{1}{c}{3} 
  & \multicolumn{1}{c}{4} 
  & \multicolumn{1}{c}{5} 
  & \multicolumn{1}{c}{6} 
  & \multicolumn{1}{c}{7} 
  & \multicolumn{1}{c}{8} 
  & \multicolumn{1}{c}{9} 
  & \multicolumn{1}{c}{10} \\ \cline{2-10}
 1&0.7071&0.7071&0.6872&0.6646&0.6431&0.6233&0.6052&0.5888&0.5739\\
 2&    --&0.0000&0.1677&0.2413&0.2806&0.3031&0.3164&0.3244&0.3291\\
 3&    --&    --&   -- &0.0000&0.0875&0.1401&0.1743&0.1976&0.2141\\
 4&    --&    --&   -- &   -- &   -- &0.0000&0.0561&0.0947&0.1224\\
 5&    --&    --&   -- &   -- &   -- &   -- &   -- &0.0000&0.0399\\
\\
  & \multicolumn{1}{c}{11} 
  & \multicolumn{1}{c}{12} 
  & \multicolumn{1}{c}{13} 
  & \multicolumn{1}{c}{14} 
  & \multicolumn{1}{c}{15} 
  & \multicolumn{1}{c}{16} 
  & \multicolumn{1}{c}{17} 
  & \multicolumn{1}{c}{18} 
  & \multicolumn{1}{c}{19} 
  & \multicolumn{1}{c}{20} \\ \cline{2-11}
 1&0.5601&0.5475&0.5359&0.5251&0.5150&0.5056&0.4968&0.4886&0.4808&0.4734\\
 2&0.3315&0.3325&0.3325&0.3318&0.3306&0.3290&0.3273&0.3253&0.3232&0.3211\\
 3&0.2260&0.2347&0.2412&0.2460&0.2495&0.2521&0.2540&0.2553&0.2561&0.2565\\
 4&0.1429&0.1586&0.1707&0.1802&0.1878&0.1939&0.1988&0.2027&0.2059&0.2085\\
 5&0.0695&0.0922&0.1099&0.1240&0.1353&0.1447&0.1524&0.1587&0.1641&0.1686\\
 6&0.0000&0.0303&0.0539&0.0727&0.0880&0.1005&0.1109&0.1197&0.1271&0.1334\\
 7&   -- &   -- &0.0000&0.0240&0.0433&0.0593&0.0725&0.0837&0.0932&0.1013\\
 8&   -- &   -- &   -- &   -- &0.0000&0.0196&0.0359&0.0496&0.0612&0.0711\\
 9&   -- &   -- &   -- &   -- &   -- &   -- &0.0000&0.0163&0.0303&0.0422\\
10&   -- &   -- &   -- &   -- &   -- &   -- &   -- &   -- &0.0000&0.0140\\
\\
  & \multicolumn{1}{c}{21} 
  & \multicolumn{1}{c}{22} 
  & \multicolumn{1}{c}{23} 
  & \multicolumn{1}{c}{24} 
  & \multicolumn{1}{c}{25} 
  & \multicolumn{1}{c}{26} 
  & \multicolumn{1}{c}{27} 
  & \multicolumn{1}{c}{28} 
  & \multicolumn{1}{c}{29} 
  & \multicolumn{1}{c}{30} \\ \cline{2-11}
 1&0.4643&0.4590&0.4542&0.4493&0.4450&0.4407&0.4366&0.4328&0.4291&0.4254\\
 2&0.3185&0.3156&0.3126&0.3098&0.3069&0.3043&0.3018&0.2992&0.2968&0.2944\\
 3&0.2578&0.2571&0.2563&0.2554&0.2543&0.2533&0.2522&0.2510&0.2499&0.2487\\
 4&0.2119&0.2131&0.2139&0.2145&0.2148&0.2151&0.2152&0.2151&0.2150&0.2148\\
 5&0.1736&0.1764&0.1787&0.1807&0.1822&0.1836&0.1848&0.1857&0.1864&0.1870\\
 6&0.1399&0.1443&0.1480&0.1512&0.1539&0.1563&0.1584&0.1601&0.1616&0.1630\\
 7&0.1092&0.1150&0.1201&0.1245&0.1283&0.1316&0.1346&0.1372&0.1395&0.1415\\
 8&0.0804&0.0878&0.0941&0.0997&0.1046&0.1089&0.1128&0.1162&0.1192&0.1219\\
 9&0.0530&0.0618&0.0696&0.0764&0.0823&0.0876&0.0923&0.0965&0.1002&0.1036\\
10&0.0263&0.0368&0.0459&0.0539&0.0610&0.0672&0.0728&0.0778&0.0822&0.0862\\
11&0.0000&0.0122&0.0228&0.0321&0.0403&0.0476&0.0540&0.0598&0.0650&0.0697\\
12&  --  &  --  &0.0000&0.0107&0.0200&0.0284&0.0358&0.0424&0.0483&0.0537\\
13&  --  &  --  &  --  &  --  &0.0000&0.0094&0.0178&0.0253&0.0320&0.0381\\
14&  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0084&0.0159&0.0227\\
15&  --  &  --  &  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0076\\
\\
  & \multicolumn{1}{c}{31} 
  & \multicolumn{1}{c}{32} 
  & \multicolumn{1}{c}{33} 
  & \multicolumn{1}{c}{34} 
  & \multicolumn{1}{c}{35} 
  & \multicolumn{1}{c}{36} 
  & \multicolumn{1}{c}{37} 
  & \multicolumn{1}{c}{38} 
  & \multicolumn{1}{c}{39} 
  & \multicolumn{1}{c}{40} \\ \cline{2-11}
 1&0.4220&0.4188&0.4156&0.4127&0.4096&0.4068&0.4040&0.4015&0.3989&0.3964\\
 2&0.2921&0.2898&0.2876&0.2854&0.2834&0.2813&0.2794&0.2774&0.2755&0.2737\\
 3&0.2475&0.2463&0.2451&0.2439&0.2427&0.2415&0.2403&0.2391&0.2380&0.2368\\
 4&0.2145&0.2141&0.2137&0.2132&0.2127&0.2121&0.2116&0.2110&0.2104&0.2098\\
 5&0.1874&0.1878&0.1880&0.1882&0.1883&0.1883&0.1883&0.1881&0.1880&0.1878\\
 6&0.1641&0.1651&0.1660&0.1667&0.1673&0.1678&0.1683&0.1686&0.1689&0.1691\\
 7&0.1433&0.1449&0.1463&0.1475&0.1487&0.1496&0.1505&0.1513&0.1520&0.1526\\
 8&0.1243&0.1265&0.1284&0.1301&0.1317&0.1331&0.1344&0.1356&0.1366&0.1376\\
 9&0.1066&0.1093&0.1118&0.1140&0.1160&0.1179&0.1196&0.1211&0.1225&0.1237\\
10&0.0899&0.0931&0.0961&0.0988&0.1013&0.1036&0.1056&0.1075&0.1092&0.1108\\
11&0.0739&0.0777&0.0812&0.0844&0.0873&0.0900&0.0924&0.0947&0.0967&0.0986\\
12&0.0585&0.0629&0.0669&0.0706&0.0739&0.0770&0.0798&0.0824&0.0848&0.0870\\
13&0.0435&0.0485&0.0530&0.0572&0.0610&0.0645&0.0677&0.0706&0.0733&0.0759\\
14&0.0289&0.0344&0.0395&0.0441&0.0484&0.0523&0.0559&0.0592&0.0622&0.0651\\
15&0.0144&0.0206&0.0262&0.0314&0.0361&0.0404&0.0444&0.0481&0.0515&0.0546\\
16&0.0000&0.0068&0.0131&0.0187&0.0239&0.0287&0.0331&0.0372&0.0409&0.0444\\
17&  --  &  --  &0.0000&0.0062&0.0119&0.0172&0.0220&0.0264&0.0305&0.0343\\
18&  --  &  --  &  --  &  --  &0.0000&0.0057&0.0110&0.0158&0.0203&0.0244\\
19&  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0053&0.0101&0.0146\\
20&  --  &  --  &  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0049\\
\\
  & \multicolumn{1}{c}{41} 
  & \multicolumn{1}{c}{42} 
  & \multicolumn{1}{c}{43} 
  & \multicolumn{1}{c}{44} 
  & \multicolumn{1}{c}{45} 
  & \multicolumn{1}{c}{46} 
  & \multicolumn{1}{c}{47} 
  & \multicolumn{1}{c}{48} 
  & \multicolumn{1}{c}{49} 
  & \multicolumn{1}{c}{50} \\ \cline{2-11}
 1&0.3940&0.3917&0.3894&0.3872&0.3850&0.3830&0.3808&0.3789&0.3770&0.3964\\
 2&0.2719&0.2701&0.2684&0.2667&0.2651&0.2635&0.2620&0.2604&0.2589&0.2737\\
 3&0.2357&0.2345&0.2334&0.2323&0.2313&0.2302&0.2291&0.2281&0.2271&0.2368\\
 4&0.2091&0.2085&0.2078&0.2072&0.2065&0.2058&0.2052&0.2045&0.2038&0.2098\\
 5&0.1876&0.1874&0.1871&0.1868&0.1865&0.1862&0.1859&0.1855&0.1851&0.1878\\
 6&0.1693&0.1694&0.1695&0.1695&0.1695&0.1695&0.1695&0.1693&0.1692&0.1691\\
 7&0.1531&0.1535&0.1539&0.1542&0.1545&0.1548&0.1550&0.1551&0.1553&0.1554\\%1526\\
 8&0.1384&0.1392&0.1398&0.1405&0.1410&0.1415&0.1420&0.1423&0.1427&0.1430\\%1376\\
 9&0.1249&0.1259&0.1269&0.1278&0.1286&0.1293&0.1300&0.1306&0.1312&0.1317\\%1237\\
10&0.1123&0.1136&0.1149&0.1160&0.1170&0.1180&0.1189&0.1197&0.1205&0.1212\\%1108\\
11&0.1004&0.1020&0.1035&0.1049&0.1062&0.1073&0.1085&0.1095&0.1105&0.1113\\
12&0.0891&0.0909&0.0927&0.0943&0.0959&0.0972&0.0986&0.0998&0.1010&0.1020\\
13&0.0782&0.0804&0.0824&0.0842&0.0860&0.0876&0.0892&0.0906&0.0919&0.0932\\
14&0.0677&0.0701&0.0724&0.0745&0.0765&0.0783&0.0801&0.0817&0.0832&0.0846\\
15&0.0575&0.0602&0.0628&0.0651&0.0673&0.0694&0.0713&0.0731&0.0748&0.0764\\
16&0.0476&0.0506&0.0534&0.0560&0.0584&0.0607&0.0628&0.0648&0.0667&0.0685\\
17&0.0379&0.0411&0.0442&0.0471&0.0497&0.0522&0.0546&0.0568&0.0588&0.0608\\
18&0.0283&0.0318&0.0352&0.0383&0.0412&0.0439&0.0465&0.0489&0.0511&0.0532\\
19&0.0188&0.0227&0.0263&0.0296&0.0328&0.0357&0.0385&0.0411&0.0436&0.0459\\
20&0.0094&0.0136&0.0175&0.0211&0.0245&0.0277&0.0307&0.0335&0.0361&0.0386\\
21&  --  &0.0045&0.0087&0.0126&0.0163&0.0197&0.0229&0.0259&0.0288&0.0314\\
22&  --  &  --  &0.0000&0.0042&0.0081&0.0118&0.0153&0.0185&0.0215&0.0244\\
23&  --  &  --  &  --  &  --  &0.0000&0.0039&0.0076&0.0111&0.0143&0.0174\\
24&  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0037&0.0071&0.0104\\
25&  --  &  --  &  --  &  --  &  --  &  --  &  --  &  --  &0.0000&0.0035\\
\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Critical Values of the Shapiro-Wilk \(W\) for Testing Normality.}
\centerline{Reproduced from Table~6 of Shapiro and Wilk~\cite{shapiro65}.}
\label{tbl:w-test}
\begin{center}
\begin{tabular}{rlllll}\hline
\(n\) & \multicolumn{5}{c}{\(\alpha\)} \\ \cline{2-6}
& 0.01 & 0.02 & 0.05 & 0.10 & 0.50 \\ \hline
3 & 0.753 & 0.756 & 0.767 & 0.789 & 0.959 \\ \hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Critcal Values of the Shapiro-Wilk \(W\) for Testing Exponentiality.}
\centerline{Reproduced from Table~1 of Shapiro and Wilk~\cite{shapiro72}.}
\label{tbl:w-test-e}
\scriptsize
\begin{center}
\begin{tabular}{r%
                 @{\extracolsep{3pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
                 @{\extracolsep{2pt}}l%
}\hline
\(n\) & \multicolumn{11}{c}{\(\alpha\)} \\ \cline{2-12}
  &\multicolumn{1}{c}{0.005}
  &\multicolumn{1}{c}{0.01}
  &\multicolumn{1}{c}{0.025}
  &\multicolumn{1}{c}{0.05 }
  &\multicolumn{1}{c}{0.10 }
  &\multicolumn{1}{c}{0.50 }
  &\multicolumn{1}{c}{0.90 }
  &\multicolumn{1}{c}{0.95 }
  &\multicolumn{1}{c}{0.975}
  &\multicolumn{1}{c}{0.99 }
  &\multicolumn{1}{c}{0.995}\\ \hline
 3&.2519&.2538&.2596&.2697&.2915&.5714&.9709&.9926&.9981&.9997&.99993\\
 4&.1241&.1302&.1434&.1604&.1891&.3768&.7514&.8581&.9236&.9680&.9837\\
\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Coefficients \(\{b_{n-i+1}\}\) for the Shapiro-Francia
\(W'\) Test for Normality.}
\centerline{Reproduced from Table~1 of Shapiro and Wilk~\cite{shapiro72b}.}
\label{tbl:shapiro-francia-b}
%\begin{center}
%\begin{tabular}
%\hline
%\end{tabular}
%\end{center}
\normalsize
\end{table}
\clearpage

\begin{table}
\caption{Percentage Points for \(W'\) Test Statistic}
\centerline{Reproduced from Table~1 of Shapirio and Francia~\cite{shapiro72b}.}
\label{tbl:w-prime-test}
\scriptsize
\begin{center}
\begin{tabular}{l@{\extracolsep{1pt}}r%
                 @{\extracolsep{1pt}}r%
                 @{\extracolsep{1pt}}r%
                 @{\extracolsep{1pt}}rrrrrrrr}\hline
\(n\) & \multicolumn{11}{c}{\(P\)} \\ \cline{2-12}
&
\multicolumn{1}{l}{0.01} & 
\multicolumn{1}{l}{0.05} & 
\multicolumn{1}{l}{0.10} & 
\multicolumn{1}{l}{0.15} & 
\multicolumn{1}{l}{0.20} & 
\multicolumn{1}{l}{0.50} & 
\multicolumn{1}{l}{0.80} & 
\multicolumn{1}{l}{0.85} & 
\multicolumn{1}{l}{0.90} & 
\multicolumn{1}{l}{0.95} & 
\multicolumn{1}{l}{0.99}\\
\hline
35&0.919&0.943&0.952&0.956&0.964&0.976&0.982&0.985&0.987&0.989&0.992\\
50& .935& .953& .963& .968& .971& .981& .987& .988& .990& .991& .994\\
\\
51&0.935&0.954&0.964&0.968&0.971&0.981&0.988&0.989&0.990&0.992&0.994\\
53& .938& .957& .964& .969& .972& .982& .988& .989& .990& .992& .994\\
55& .940& .958& .965& .971& .973& .983& .988& .990& .991& .992& .994\\
57& .944& .961& .966& .971& .974& .983& .989& .990& .991& .992& .994\\
59& .945& .962& .967& .972& .975& .983& .989& .990& .991& .992& .994\\
\\
61&0.947&0.963&0.968&0.973&0.975&0.984&0.990&0.990&0.991&0.992&0.994\\
63& .947& .964& .970& .973& .976& .984& .990& .991& .992& .993& .994\\
65& .948& .965& .971& .974& .976& .985& .990& .991& .992& .993& .995\\
67& .950& .966& .971& .974& .977& .985& .990& .991& .992& .993& .995\\
69& .951& .966& .972& .976& .978& .986& .990& .991& .992& .993& .995\\
\\
71&0.953&0.967&0.972&0.976&0.978&0.986&0.990&0.991&0.992&0.994&0.995\\
73& .956& .968& .973& .976& .979& .986& .991& .992& .993& .994& .995\\
75& .956& .969& .973& .976& .979& .986& .991& .992& .993& .994& .995\\
77& .957& .969& .974& .977& .980& .987& .991& .992& .993& .994& .996\\
79& .957& .970& .975& .978& .980& .987& .991& .992& .993& .994& .996\\
\\
81&0.958&0.970&0.975&0.979&0.981&0.987&0.992&0.992&0.993&0.994&0.996\\
83& .960& .971& .976& .979& .981& .988& .992& .992& .993& .994& .996\\
85& .961& .972& .977& .980& .981& .988& .992& .992& .993& .994& .996\\
87& .961& .972& .977& .980& .982& .988& .992& .993& .994& .994& .996\\
89& .961& .972& .977& .981& .982& .988& .992& .993& .994& .995& .996\\
\\
91&0.962&0.973&0.978&0.981&0.983&0.989&0.992&0.993&0.994&0.995&0.996\\
93& .963& .973& .979& .981& .983& .989& .992& .993& .994& .995& .996\\
95& .965& .974& .979& .981& .983& .989& .993& .993& .994& .995& .996\\
97& .965& .975& .979& .982& .984& .989& .993& .993& .994& .995& .996\\
99& .967& .976& .980& .982& .984& .989& .993& .994& .994& .995& .996\\
\hline
\end{tabular}
\end{center}
\normalsize
\end{table}
\clearpage

\end{document}