# NAG FL Interfaceg08chf (gofstat_​anddar)

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## 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 <nag.h>
 double g08chf_ (const Integer *n, const logical *issort, double y[], Integer *ifail)
The routine may be called by the names g08chf or nagf_nonpar_gofstat_anddar.

## 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):
 $Y=F(X) ,$
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.

## 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: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}>1$.
2: $\mathbf{issort}$Logical Input
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) array Input/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: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ 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 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

Not applicable.

## 8Parallelism and Performance

g08chf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

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)