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

NAG Library Function Document

nag_anderson_darling_stat (g08chc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

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

2  Specification

#include <nag.h>
#include <nagg08.h>
double  nag_anderson_darling_stat (Integer n, Nag_Boolean issort, double y[], NagError *fail)

3  Description

Denote by A2 the Anderson–Darling test statistic for n observations y1,y2,,yn of a variable Y assumed to be standard uniform and sorted in ascending order, then:
A2 = -n-S ;
S = i=1 n 2i-1 n lnyi + 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: n>1.
2:     issortNag_BooleanInput
On entry: set issort=Nag_TRUE if the observations are sorted in ascending order; otherwise the function will sort the observations.
3:     y[n]doubleInput/Output
On entry: yi, for i=1,2,,n, the n observations.
On exit: if issort=Nag_FALSE, the data sorted in ascending order; otherwise the array is unchanged.
Constraint: if issort=Nag_TRUE, the values must be sorted in ascending order. Each yi must lie in the interval 0,1.
4:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument value had an illegal value.
The data in y must lie in the interval 0,1.
On entry, n=value.
Constraint: n>1.
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.
issort=Nag_TRUE and the data in y is not sorted in ascending order.

7  Accuracy

Not applicable.

8  Parallelism and Performance

Not applicable.

9  Further Comments


10  Example

This example calculates the A2 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.

10.1  Program Text

Program Text (g08chce.c)

10.2  Program Data

Program Data (g08chce.d)

10.3  Program Results

Program Results (g08chce.r)

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

© The Numerical Algorithms Group Ltd, Oxford, UK. 2014