nag_anderson_darling_stat (g08chc) (PDF version)
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NAG C Library Manual

NAG Library Function Documentnag_anderson_darling_stat (g08chc)

1  Purpose

nag_anderson_darling_stat (g08chc) calculates the Anderson–Darling goodness-of-fit test statistic.

2  Specification

 #include #include
 double nag_anderson_darling_stat (Integer n, Nag_Boolean issort, double y[], NagError *fail)

3  Description

Denote by ${A}^{2}$ the Anderson–Darling test statistic for $n$ observations ${y}_{1},{y}_{2},\dots ,{y}_{n}$ of a variable $Y$ assumed to be standard uniform and sorted in ascending order, then:
 $A2 = -n-S ;$
where:
 $S = ∑ i=1 n 2i-1 n ln⁡yi + ln 1- y n-i+1 .$
When observations of a random variable $X$ are non-uniformly distributed, the probability integral transformation (PIT):
 $Y=FX ,$
where $F$ is the cumulative distribution function of the distribution of interest, yields a uniformly distributed random variable $Y$. The PIT is true only if all parameters of a distribution are known as opposed to estimated; otherwise it is an approximation.

4  References

Anderson T W and Darling D A (1952) Asymptotic theory of certain ‘goodness-of-fit’ criteria based on stochastic processes Annals of Mathematical Statistics 23 193–212

5  Arguments

1:     nIntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
2:     issortNag_BooleanInput
On entry: set ${\mathbf{issort}}=\mathrm{Nag_TRUE}$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
3:     y[n]doubleInput/Output
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
On exit: if ${\mathbf{issort}}=\mathrm{Nag_FALSE}$, the data sorted in ascending order; otherwise the array is unchanged.
Constraint: if ${\mathbf{issort}}=\mathrm{Nag_TRUE}$, the values must be sorted in ascending order. Each ${y}_{i}$ must lie in the interval $\left(0,1\right)$.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_BAD_PARAM
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_BOUND
The data in y must lie in the interval $\left(0,1\right)$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
NE_NOT_INCREASING
${\mathbf{issort}}=\mathrm{Nag_TRUE}$ and the data in y is not sorted in ascending order.

Not applicable.

None.

9  Example

This example calculates the ${A}^{2}$ statistic for data assumed to arise from an exponential distribution with a sample parameter estimate and simulates its $p$-value using the NAG basic random number generator.

9.1  Program Text

Program Text (g08chce.c)

9.2  Program Data

Program Data (g08chce.d)

9.3  Program Results

Program Results (g08chce.r)

nag_anderson_darling_stat (g08chc) (PDF version)
g08 Chapter Contents
g08 Chapter Introduction
NAG C Library Manual