This class provides methods to compute several types of EDF goodness-of-fit test statistics and to apply certain transformations to a set of observations. More...

Classes
class	OutcomeCategoriesChi2
	This class helps managing the partitions of possible outcomes into categories for applying chi-square tests. More...

Transforming the observations
static DoubleArrayList	unifTransform (DoubleArrayList data, ContinuousDistribution dist)
	Applies the probability integral transformation \(U_i = F (V_i)\) for \(i = 0, 1, …, n-1\), where \(F\) is a continuous distribution function, and returns the result as an array of length \(n\). More...

static DoubleArrayList	unifTransform (DoubleArrayList data, DiscreteDistribution dist)
	Applies the transformation \(U_i = F (V_i)\) for \(i = 0, 1, …, n-1\), where \(F\) is a discrete distribution function, and returns the result as an array of length \(n\). More...

static void	diff (IntArrayList sortedData, IntArrayList spacings, int n1, int n2, int a, int b)
	Assumes that the real-valued observations \(U_0,…,U_{n-1}\) contained in `sortedData` are already sorted in increasing order and computes the differences between the successive observations. More...

static void	diff (DoubleArrayList sortedData, DoubleArrayList spacings, int n1, int n2, double a, double b)
	Same as method diff(IntArrayList,IntArrayList,int,int,int,int), but for the continuous case. More...

static void	iterateSpacings (DoubleArrayList data, DoubleArrayList spacings)
	Applies one iteration of the iterated spacings transformation [112], [226] . More...

static void	powerRatios (DoubleArrayList sortedData)
	Applies the power ratios transformation \(W\) described in section 8.4 of Stephens [226] . More...

Computing EDF test statistics
static double	EPSILONAD = Num.DBL_EPSILON/2
	Used by andersonDarling(DoubleArrayList). More...

static double	chi2 (double[] nbExp, int[] count, int smin, int smax)
	Computes and returns the chi-square statistic for the observations \(o_i\) in `count[smin...smax]`, for which the corresponding expected values \(e_i\) are in `nbExp[smin...smax]`. More...

static double	chi2 (OutcomeCategoriesChi2 cat, int[] count)
	Computes and returns the chi-square statistic for the observations \(o_i\) in `count`, for which the corresponding expected values \(e_i\) are in `cat`. More...

static double	chi2 (IntArrayList data, DiscreteDistributionInt dist, int smin, int smax, double minExp, int[] numCat)
	Computes and returns the chi-square statistic for the observations stored in `data`, assuming that these observations follow the discrete distribution `dist`. More...

static double	chi2Equal (double nbExp, int[] count, int smin, int smax)
	Similar to #chi2(double[],int[],int,int), except that the expected number of observations per category is assumed to be the same for all categories, and equal to `nbExp`. More...

static double	chi2Equal (DoubleArrayList data, double minExp)
	Computes the chi-square statistic for a continuous distribution. More...

static double	chi2Equal (DoubleArrayList data)
	Equivalent to `chi2Equal (data, 10)`. More...

static int	scan (DoubleArrayList sortedData, double d)
	Computes and returns the scan statistic \(S_n (d)\), defined in ( scan ). More...

static double	cramerVonMises (DoubleArrayList sortedData)
	Computes and returns the Cramér-von Mises statistic \(W_n^2\) (see [55], [224], [225] ), defined by. More...

static double	watsonG (DoubleArrayList sortedData)
	Computes and returns the Watson statistic \(G_n\) (see [238], [41] ), defined by. More...

static double	watsonU (DoubleArrayList sortedData)
	Computes and returns the Watson statistic \(U_n^2\) (see [55], [224], [225] ), defined by. More...

static double	andersonDarling (DoubleArrayList sortedData)
	Computes and returns the Anderson-Darling statistic \(A_n^2\) (see method #andersonDarling(double[]) ). More...

static double	andersonDarling (double[] sortedData)
	Computes and returns the Anderson-Darling statistic \(A_n^2\) (see [165], [225], [6] ), defined by. More...

static double []	andersonDarling (double[] data, ContinuousDistribution dist)
	Computes the Anderson-Darling statistic \(A_n^2\) and the corresponding \(p\)-value \(p\). More...

static double []	kolmogorovSmirnov (double[] sortedData)
	Computes the Kolmogorov-Smirnov (KS) test statistics \(D_n^+\), \(D_n^-\), and \(D_n\) (see method kolmogorovSmirnov(DoubleArrayList) ). More...

static double []	kolmogorovSmirnov (DoubleArrayList sortedData)
	Computes the Kolmogorov-Smirnov (KS) test statistics \(D_n^+\), \(D_n^-\), and \(D_n\) defined by. More...

static void	kolmogorovSmirnov (double[] data, ContinuousDistribution dist, double[] sval, double[] pval)
	Computes the KolmogorovSmirnov (KS) test statistics and their \(p\)-values. More...

static double []	kolmogorovSmirnovJumpOne (DoubleArrayList sortedData, double a)
	Compute the KS statistics \(D_n^+(a)\) and \(D_n^-(a)\) defined in the description of the method FDist.kolmogorovSmirnovPlusJumpOne, assuming that \(F\) is the uniform distribution over \([0,1]\) and that \(U_{(1)},…,U_{(n)}\) are in `sortedData`. More...

static double	pDisc (double pL, double pR)
	Computes a variant of the \(p\)-value \(p\) whenever a test statistic has a discrete probability distribution. More...

Detailed Description

This class provides methods to compute several types of EDF goodness-of-fit test statistics and to apply certain transformations to a set of observations.

This includes the probability integral transformation \(U_i = F(X_i)\), as well as the power ratio and iterated spacings transformations [226] . Here, \(U_{(0)}, …, U_{(n-1)}\) stand for \(n\) observations \(U_0,…,U_{n-1}\) sorted by increasing order, where \(0\le U_i\le1\).

Note: This class uses the Colt library.

Member Function Documentation

◆ andersonDarling() [1/3]

static double andersonDarling ( DoubleArrayList sortedData )

static

Computes and returns the Anderson-Darling statistic \(A_n^2\) (see method #andersonDarling(double[]) ).

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the Anderson-Darling statistic

◆ andersonDarling() [2/3]

static double andersonDarling ( double [] sortedData )

static

Computes and returns the Anderson-Darling statistic \(A_n^2\) (see [165], [225], [6] ), defined by.

\begin{align*} A_n^2 & = -n -\frac{1}{ n} \sum_{j=0}^{n-1} \left\{ (2j+1)\ln(U_{(j)}) + (2n-1-2j) \ln(1-U_{(j)}) \right\}, \tag{Andar} \end{align*}

assuming that sortedData contains \(U_{(0)},…,U_{(n-1)}\) sorted in increasing order. When computing \(A_n^2\), all observations \(U_i\) are projected on the interval \([\epsilon, 1-\epsilon]\) for some \(\epsilon> 0\), in order to avoid numerical overflow when taking the logarithm of \(U_i\) or \(1-U_i\). The variable EPSILONAD gives the value of \(\epsilon\).

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the Anderson-Darling statistic

◆ andersonDarling() [3/3]

static double [] andersonDarling	(	double []	data,
		ContinuousDistribution	dist
	)

static

Computes the Anderson-Darling statistic \(A_n^2\) and the corresponding \(p\)-value \(p\).

The \(n\) (unsorted) observations in data are assumed to be independent and to come from the continuous distribution dist. Returns the 2-elements array [ \(A_n^2\), \(p\)].

Parameters

data	array of observations
dist	assumed distribution of the observations

Returns: the array \([A_n^2\), \(p]\).

◆ chi2() [1/3]

static double chi2	(	double []	nbExp,
		int []	count,
		int	smin,
		int	smax
	)

static

Computes and returns the chi-square statistic for the observations \(o_i\) in count[smin...smax], for which the corresponding expected values \(e_i\) are in nbExp[smin...smax].

Assuming that \(i\) goes from 1 to \(k\), where \(k =\) smax-smin+1 is the number of categories, the chi-square statistic is defined as

\[ X^2 = \sum_{i=1}^k \frac{(o_i - e_i)^2}{e_i}. \tag{chi-square} \]

Under the hypothesis that the \(e_i\) are the correct expectations and if these \(e_i\) are large enough, \(X^2\) follows approximately the chi-square distribution with \(k-1\) degrees of freedom. If some of the \(e_i\) are too small, one can use OutcomeCategoriesChi2 to regroup categories.

Parameters

nbExp	numbers expected in each category
count	numbers observed in each category
smin	index of the first valid data in `count` and `nbExp`
smax	index of the last valid data in `count` and `nbExp`

Returns: the \(X^2\) statistic

◆ chi2() [2/3]

static double chi2	(	OutcomeCategoriesChi2	cat,
		int []	count
	)

static

Computes and returns the chi-square statistic for the observations \(o_i\) in count, for which the corresponding expected values \(e_i\) are in cat.

This assumes that cat.regroupCategories has been called before to regroup categories in order to make sure that the expected numbers in each category are large enough for the chi-square test.

Parameters

cat	numbers expected in each category
count	numbers observed in each category

Returns: the \(X^2\) statistic

◆ chi2() [3/3]

static double chi2	(	IntArrayList	data,
		DiscreteDistributionInt	dist,
		int	smin,
		int	smax,
		double	minExp,
		int []	numCat
	)

static

Computes and returns the chi-square statistic for the observations stored in data, assuming that these observations follow the discrete distribution dist.

For dist, we assume that there is one set \(S=\{a, a+1,…, b-1, b\}\), where \(a<b\) and \(a\ge0\), for which \(p(s)>0\) if \(s\in S\) and \(p(s)=0\) otherwise.

Generally, it is not possible to divide the integers in intervals satisfying \(nP(a_0\le s< a_1)=nP(a_1\le s< a_2)=\cdots=nP(a_{j-1}\le s< a_j)\) for a discrete distribution, where \(n\) is the sample size, i.e., the number of observations stored into data. To perform a general chi-square test, the method starts from smin and finds the first non-negligible probability \(p(s)\ge\epsilon\), where \(\epsilon=\) DiscreteDistributionInt.EPSILON. It uses smax to allocate an array storing the number of expected observations ( \(np(s)\)) for each \(s\ge\) smin. Starting from \(s=\) smin, the \(np(s)\) terms are computed and the allocated array grows if required until a negligible probability term is found. This gives the number of expected elements for each category, where an outcome category corresponds here to an interval in which sample observations could lie. The categories are regrouped to have at least minExp observations per category. The method then counts the number of samples in each categories and calls #chi2(double[],int[],int,int) to get the chi-square test statistic. If numCat is not null, the number of categories after regrouping is returned in numCat[0]. The number of degrees of freedom is equal to numCat[0]-1. We usually choose minExp = 10.

Parameters

data	observations, not necessarily sorted
dist	assumed probability distribution
smin	estimated minimum value of \(s\) for which \(p(s)>0\)
smax	estimated maximum value of \(s\) for which \(p(s)>0\)
minExp	minimum number of expected observations in each interval
numCat	one-element array that will be filled with the number of categories after regrouping

Returns: the chi-square statistic for a discrete distribution

◆ chi2Equal() [1/3]

static double chi2Equal	(	double	nbExp,
		int []	count,
		int	smin,
		int	smax
	)

static

Similar to #chi2(double[],int[],int,int), except that the expected number of observations per category is assumed to be the same for all categories, and equal to nbExp.

Parameters

nbExp	number of expected observations in each category (or interval)
count	number of counted observations in each category
smin	index of the first valid data in `count` and `nbExp`
smax	index of the last valid data in `count` and `nbExp`

Returns: the \(X^2\) statistic

◆ chi2Equal() [2/3]

static double chi2Equal	(	DoubleArrayList	data,
		double	minExp
	)

static

Computes the chi-square statistic for a continuous distribution.

Here, the equiprobable case can be used. Assuming that data contains observations coming from the uniform distribution, the \([0,1]\) interval is divided into \(1/p\) subintervals, where \(p=\) minExp \(/n\), \(n\) being the sample size, i.e., the number of observations stored in data. For each subinterval, the method counts the number of contained observations and the chi-square statistic is computed using #chi2Equal(double,int[],int,int). We usually choose minExp = 10.

Parameters

data	array of observations in \([0,1)\)
minExp	minimum number of expected observations in each subintervals

Returns: the chi-square statistic for a continuous distribution

◆ chi2Equal() [3/3]

static double chi2Equal ( DoubleArrayList data )

static

Equivalent to chi2Equal (data, 10).

Parameters

data	array of observations in \([0,1)\)

Returns: the chi-square statistic for a continuous distribution

◆ cramerVonMises()

static double cramerVonMises ( DoubleArrayList sortedData )

static

Computes and returns the Cramér-von Mises statistic \(W_n^2\) (see [55], [224], [225] ), defined by.

\[ W_n^2 = \frac{1}{ 12n} + \sum_{j=0}^{n-1} \left(U_{(j)} - \frac{(j+0.5) }{ n}\right)^2, \tag{CraMis} \]

assuming that sortedData contains \(U_{(0)},…,U_{(n-1)}\) sorted in increasing order.

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the Cramér-von Mises statistic

◆ diff() [1/2]

static void diff	(	IntArrayList	sortedData,
		IntArrayList	spacings,
		int	n1,
		int	n2,
		int	a,
		int	b
	)

static

Assumes that the real-valued observations \(U_0,…,U_{n-1}\) contained in sortedData are already sorted in increasing order and computes the differences between the successive observations.

Let \(D\) be the differences returned in spacings. The difference \(U_i - U_{i-1}\) is put in \(D_i\) for n1 < i <= n2, whereas \(U_{n1} - a\) is put into \(D_{n1}\) and \(b - U_{n2}\) is put into \(D_{n2+1}\). The number of observations must be greater or equal than n2, we must have n1 < n2, and n1 and n2 are greater than 0. The size of spacings will be at least \(n+1\) after the call returns.

Parameters

sortedData	array of sorted observations
spacings	pointer to an array object that will be filled with spacings
n1	starting index, in `sortedData`, of the processed observations
n2	ending index, in `sortedData` of the processed observations
a	minimum value of the observations
b	maximum value of the observations

◆ diff() [2/2]

static void diff	(	DoubleArrayList	sortedData,
		DoubleArrayList	spacings,
		int	n1,
		int	n2,
		double	a,
		double	b
	)

static

Same as method diff(IntArrayList,IntArrayList,int,int,int,int), but for the continuous case.

Parameters

sortedData	array of sorted observations
spacings	pointer to an array object that will be filled with spacings
n1	starting index of the processed observations in `sortedData`
n2	ending index, in `sortedData` of the processed observations
a	minimum value of the observations
b	maximum value of the observations

◆ iterateSpacings()

static void iterateSpacings	(	DoubleArrayList	data,
		DoubleArrayList	spacings
	)

static

Applies one iteration of the iterated spacings transformation [112], [226] .

Let \(U\) be the \(n\) observations contained into data, and let \(S\) be the spacings contained into spacings, Assumes that \(S[0..n]\) contains the spacings between \(n\) real numbers \(U_0,…,U_{n-1}\) in the interval \([0,1]\). These spacings are defined by

\[ S_i = U_{(i)} - U_{(i-1)}, \qquad1\le i < n, \]

where \(U_{(0)}=0\), \(U_{(n-1)}=1\), and \(U_{(0)},…,U_{(n-1)}\), are the \(U_i\) sorted in increasing order. These spacings may have been obtained by calling diff(DoubleArrayList,DoubleArrayList,int,int,double,double). This method transforms the spacings into new spacings, by a variant of the method described in section 11 of [177] and also by Stephens [226] : it sorts \(S_0,…,S_n\) to obtain \(S_{(0)} \le S_{(1)} \le S_{(2)} \le\cdots\le S_{(n)}\), computes the weighted differences

\begin{align*} S_0 & = (n+1) S_{(0)}, \\ S_1 & = n (S_{(1)}-S_{(0)}), \\ S_2 & = (n-1) (S_{(2)}-S_{(1)}), \\ & \vdots \\ S_n & = S_{(n)}-S_{(n-1)}, \end{align*}

and computes \(V_i = S_0 + S_1 + \cdots+ S_i\) for \(0\le i < n\). It then returns \(S_0,…,S_n\) in S[0..n] and \(V_1,…,V_n\) in V[1..n].

Under the assumption that the \(U_i\) are i.i.d. \(U (0,1)\), the new \(S_i\) can be considered as a new set of spacings having the same distribution as the original spacings, and the \(V_i\) are a new sample of i.i.d. \(U (0,1)\) random variables, sorted by increasing order.

This transformation is useful to detect clustering in a data set: A pair of observations that are close to each other is transformed into an observation close to zero. A data set with unusually clustered observations is thus transformed to a data set with an accumulation of observations near zero, which is easily detected by the Anderson-Darling GOF test.

Parameters

data	array of observations
spacings	spacings between the observations, will be filled with the new spacings

◆ kolmogorovSmirnov() [1/3]

static double [] kolmogorovSmirnov ( double [] sortedData )

static

Computes the Kolmogorov-Smirnov (KS) test statistics \(D_n^+\), \(D_n^-\), and \(D_n\) (see method kolmogorovSmirnov(DoubleArrayList) ).

Returns the array [ \(D_n^+\), \(D_n^-\), \(D_n\)].

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the array [ \(D_n^+\), \(D_n^-\), \(D_n\)]

◆ kolmogorovSmirnov() [2/3]

static double [] kolmogorovSmirnov ( DoubleArrayList sortedData )

static

Computes the Kolmogorov-Smirnov (KS) test statistics \(D_n^+\), \(D_n^-\), and \(D_n\) defined by.

\begin{align} D_n^+ & = \max_{0\le j\le n-1} \left((j+1)/n - U_{(j)}\right), \tag{DNp} \\ D_n^- & = \max_{0\le j\le n-1} \left(U_{(j)} - j/n\right), \tag{DNm} \\ D_n & = \max (D_n^+, D_n^-). \tag{DN} \end{align}

and returns an array of length 3 that contains [ \(D_n^+\), \(D_n^-\), \(D_n\)]. These statistics compare the empirical distribution of \(U_{(1)},…,U_{(n)}\), which are assumed to be in sortedData, with the uniform distribution over \([0,1]\).

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the array [ \(D_n^+\), \(D_n^-\), \(D_n\)]

◆ kolmogorovSmirnov() [3/3]

static void kolmogorovSmirnov	(	double []	data,
		ContinuousDistribution	dist,
		double []	sval,
		double []	pval
	)

static

Computes the KolmogorovSmirnov (KS) test statistics and their \(p\)-values.

This is to compare the empirical distribution of the (unsorted) observations in data with the theoretical distribution dist. The KS statistics \(D_n^+\), \(D_n^-\) and \(D_n\) are returned in sval[0], sval[1], and sval[2] respectively, and their corresponding \(p\)-values are returned in pval[0], pval[1], and pval[2].

Parameters

data	array of observations to be tested
dist	assumed distribution of the observations
sval	values of the 3 KS statistics
pval	\(p\)-values for the 3 KS statistics

◆ kolmogorovSmirnovJumpOne()

static double [] kolmogorovSmirnovJumpOne	(	DoubleArrayList	sortedData,
		double	a
	)

static

Compute the KS statistics \(D_n^+(a)\) and \(D_n^-(a)\) defined in the description of the method FDist.kolmogorovSmirnovPlusJumpOne, assuming that \(F\) is the uniform distribution over \([0,1]\) and that \(U_{(1)},…,U_{(n)}\) are in sortedData.

Returns the array [ \(D_n^+\), \(D_n^-\)].

Parameters

sortedData	array of sorted real-valued observations in the interval \([0,1]\)
a	size of the jump

Returns: the array [ \(D_n^+\), \(D_n^-\)]

◆ pDisc()

static double pDisc	(	double	pL,
		double	pR
	)

static

Computes a variant of the \(p\)-value \(p\) whenever a test statistic has a discrete probability distribution.

This \(p\)-value is defined as follows:

\begin{align*} p_L & = P[Y \le y] \\ p_R & = P[Y \ge y] \\ p & = \left\{ \begin{array}{l@{qquad}l} p_R, & \mbox{if } p_R < p_L \\ 1 - p_L, \mbox{if } p_R \ge p_L \mbox{ and } p_L < 0.5 \\ 0.5 & \mbox{otherwise.} \end{array} \right. \end{align*}

\[ \begin{array}{rll} p = & p_R, & \qquad\mbox{if } p_R < p_L, \\ p = & 1 - p_L, & \qquad\mbox{if } p_R \ge p_L \mbox{ and } p_L < 0.5, \\ p = & 0.5 & \qquad\mbox{otherwise.} \end{array} \]

The function takes \(p_L\) and \(p_R\) as input and returns \(p\).

Parameters

pL	left \(p\)-value
pR	right \(p\)-value

Returns: the \(p\)-value for a test on a discrete distribution

◆ powerRatios()

static void powerRatios ( DoubleArrayList sortedData )

static

Applies the power ratios transformation \(W\) described in section 8.4 of Stephens [226] .

Let \(U\) be the \(n\) observations contained into sortedData. Assumes that \(U\) contains \(n\) real numbers \(U_{(0)},…,U_{(n-1)}\) from the interval \([0,1]\), already sorted in increasing order, and computes the transformations:

\[ U’_i = (U_{(i)} / U_{(i+1)})^{i+1}, \qquad i=0,…,n-1, \]

with \(U_{(n)} = 1\). These \(U’_i\) are sorted in increasing order and put back in U[1...n]. If the \(U_{(i)}\) are i.i.d. \(U (0,1)\) sorted by increasing order, then the \(U’_i\) are also i.i.d. \(U (0,1)\).

This transformation is useful to detect clustering, as explained in iterateSpacings(DoubleArrayList,DoubleArrayList), except that here a pair of observations close to each other is transformed into an observation close to 1. An accumulation of observations near 1 is also easily detected by the Anderson-Darling GOF test.

Parameters

sortedData sorted array of real-valued observations in the interval \([0,1]\) that will be overwritten with the transformed observations

◆ scan()

static int scan	(	DoubleArrayList	sortedData,
		double	d
	)

static

Computes and returns the scan statistic \(S_n (d)\), defined in ( scan ).

Let \(U\) be the \(n\) observations contained into sortedData. The \(n\) observations in \(U[0..n-1]\) must be real numbers in the interval \([0,1]\), sorted in increasing order. (See FBar.scan for the distribution function of \(S_n (d)\)).

Parameters

sortedData	sorted array of real-valued observations in the interval \([0,1]\)
d	length of the test interval ( \(\in(0,1)\))

Returns: the scan statistic

◆ unifTransform() [1/2]

static DoubleArrayList unifTransform	(	DoubleArrayList	data,
		ContinuousDistribution	dist
	)

static

Applies the probability integral transformation \(U_i = F (V_i)\) for \(i = 0, 1, …, n-1\), where \(F\) is a continuous distribution function, and returns the result as an array of length \(n\).

\(V\) represents the \(n\) observations contained in data, and \(U\), the returned transformed observations. If data contains random variables from the distribution function dist, then the result will contain uniform random variables over \([0,1]\).

Parameters

data	array of observations to be transformed
dist	assumed distribution of the observations

Returns: the array of transformed observations

◆ unifTransform() [2/2]

static DoubleArrayList unifTransform	(	DoubleArrayList	data,
		DiscreteDistribution	dist
	)

static

Applies the transformation \(U_i = F (V_i)\) for \(i = 0, 1, …, n-1\), where \(F\) is a discrete distribution function, and returns the result as an array of length \(n\).

\(V\) represents the \(n\) observations contained in data, and \(U\), the returned transformed observations.

Note: If \(V\) are the values of random variables with distribution function dist, then the result will contain the values of discrete random variables distributed over the set of values taken by dist, not uniform random variables over \([0,1]\).

Parameters

data	array of observations to be transformed
dist	assumed distribution of the observations

Returns: the array of transformed observations

◆ watsonG()

static double watsonG ( DoubleArrayList sortedData )

static

Computes and returns the Watson statistic \(G_n\) (see [238], [41] ), defined by.

\begin{align} G_n & = \sqrt{n} \max_{\Rule{0.0pt}{7.0pt}{0.0pt} 0\le j \le n-1} \left\{ (j+1)/n - U_{(j)} + \overline{U}_n - 1/2 \right\} \tag{WatsonG} \\ & = \sqrt{n}\left(D_n^+ + \overline{U}_n - 1/2\right), \nonumber \end{align}

where \(\overline{U}_n\) is the average of the observations \(U_{(j)}\), assuming that sortedData contains the sorted \(U_{(0)},…,U_{(n-1)}\).

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the Watson statistic \(G_n\)

◆ watsonU()

static double watsonU ( DoubleArrayList sortedData )

static

Computes and returns the Watson statistic \(U_n^2\) (see [55], [224], [225] ), defined by.

\begin{align} W_n^2 & = \frac{1}{ 12n} + \sum_{j=0}^{n-1} \left\{U_{(j)} - \frac{(j + 0.5)}{ n} \right\}^2, \\ U_n^2 & = W_n^2 - n\left(\overline{U}_n - 1/2\right)^2. \tag{WatsonU} \end{align}

where \(\overline{U}_n\) is the average of the observations \(U_{(j)}\), assuming that sortedData contains the sorted \(U_{(0)},…,U_{(n-1)}\).

Parameters

sortedData array of sorted real-valued observations in the interval \([0,1]\)

Returns: the Watson statistic \(U_n^2\)

Member Data Documentation

◆ EPSILONAD

double EPSILONAD = Num.DBL_EPSILON/2

static

Used by andersonDarling(DoubleArrayList).

Num.DBL_EPSILON is usually \(2^{-52}\).

The documentation for this class was generated from the following file:

GofStat.java

Classes

Transforming the observations

Computing EDF test statistics

Detailed Description

Member Function Documentation

◆ andersonDarling() [1/3]

◆ andersonDarling() [2/3]

◆ andersonDarling() [3/3]

◆ chi2() [1/3]

◆ chi2() [2/3]

◆ chi2() [3/3]

◆ chi2Equal() [1/3]

◆ chi2Equal() [2/3]

◆ chi2Equal() [3/3]

◆ cramerVonMises()

◆ diff() [1/2]

◆ diff() [2/2]

◆ iterateSpacings()

◆ kolmogorovSmirnov() [1/3]

◆ kolmogorovSmirnov() [2/3]

◆ kolmogorovSmirnov() [3/3]

◆ kolmogorovSmirnovJumpOne()

◆ pDisc()

◆ powerRatios()

◆ scan()

◆ unifTransform() [1/2]

◆ unifTransform() [2/2]

◆ watsonG()

◆ watsonU()

Member Data Documentation

◆ EPSILONAD