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_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.
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)