NAG Library Function Document

nag_anderson_darling_stat (g08chc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

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

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:

A^{2} = - n - S;

where:

S = \sum_{i = 1}^{n} \frac{2 i - 1}{n} [\ln y_{i} + \ln (1 - y_{n - i + 1})] .

When observations of a random variable

X

are non-uniformly distributed, the probability integral transformation (PIT):

Y = F (X),

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: $n$ – IntegerInput: On entry: $n$ , the number of observations.

Constraint: $n > 1$ .
2: $issort$ – Nag_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: $y_{i}$ , for $i = 1, 2, \dots, n$ , 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 $y_{i}$ must lie in the interval $(0, 1)$ .
4: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_BOUND: The data in y must lie in the interval $(0, 1)$ .
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 1$ .
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.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_NOT_INCREASING: $issort = Nag_TRUE$ and the data in y is not sorted in ascending order.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 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.

NAG Library Function Documentnag_anderson_darling_stat (g08chc)