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goodness.tex
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\documentclass[draft]{article}
\usepackage{makeidx}
%\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}}}
\makeindex
\addtolength{\oddsidemargin}{-.55in}
\addtolength{\evensidemargin}{-.55in}
\addtolength{\textwidth}{.1in}
\addtolength{\marginparwidth}{.45in}
\addtolength{\topmargin}{-.25in}
\addtolength{\textheight}{.5in}
\def\libname{{\tt cdhc}}
\def\returns#1{\sffamily\slshape Returns \(\mathsf{#1}\).}
\def\function#1#2{\centerline{%
\protect\index{#1}
\framebox[.9\marginparwidth][l]{\vbox{\noindent\textsf{#2}}}}
\vspace{.5\baselineskip}}
%\def\function#1#2{\marginpar{%
% \protect\index{#1}
% \framebox[.9\marginparwidth][l]{\vbox{\textsf{#2}}}\hfill}}
\newenvironment{example}{%
\vspace{\baselineskip}
\par\noindent\hrulefill\par
\noindent{\em Example:}}{%
\par\noindent\hrulefill\par
\vspace{\baselineskip}}
\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}