# NAG Library Routine Document

## 1Purpose

g08chf calculates the Anderson–Darling goodness-of-fit test statistic.

## 2Specification

Fortran Interface
 Function g08chf ( n, y,
 Real (Kind=nag_wp) :: g08chf Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: y(n) Logical, Intent (In) :: issort
#include <nagmk26.h>
 double g08chf_ (const Integer *n, const logical *issort, double y[], Integer *ifail)

## 3Description

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.

## 4References

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

## 5Arguments

1:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
2:     $\mathbf{issort}$ – LogicalInput
On entry: set ${\mathbf{issort}}=\mathrm{.TRUE.}$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
3:     $\mathbf{y}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
On exit: if ${\mathbf{issort}}=\mathrm{.FALSE.}$, the data sorted in ascending order; otherwise the array is unchanged.
Constraint: if ${\mathbf{issort}}=\mathrm{.TRUE.}$, the values must be sorted in ascending order. Each ${y}_{i}$ must lie in the interval $\left(0,1\right)$.
4:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
${\mathbf{ifail}}=3$
${\mathbf{issort}}=\mathrm{.TRUE.}$ and the data in y is not sorted in ascending order.
${\mathbf{ifail}}=9$
The data in y must lie in the interval $\left(0,1\right)$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

g08chf is not threaded in any implementation.

None.

## 10Example

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.

### 10.1Program Text

Program Text (g08chfe.f90)

### 10.2Program Data

Program Data (g08chfe.d)

### 10.3Program Results

Program Results (g08chfe.r)