NAG C Library Function Document

1Purpose

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

2Specification

 #include #include
 double nag_anderson_darling_stat (Integer n, Nag_Boolean issort, double y[], NagError *fail)

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:    $\mathbf{issort}$Nag_BooleanInput
On entry: set ${\mathbf{issort}}=\mathrm{Nag_TRUE}$ if the observations are sorted in ascending order; otherwise the function will sort the observations.
3:    $\mathbf{y}\left[{\mathbf{n}}\right]$doubleInput/Output
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$, the $n$ observations.
On exit: if ${\mathbf{issort}}=\mathrm{Nag_FALSE}$, the data sorted in ascending order; otherwise the array is unchanged.
Constraint: if ${\mathbf{issort}}=\mathrm{Nag_TRUE}$, the values must be sorted in ascending order. Each ${y}_{i}$ must lie in the interval $\left(0,1\right)$.
4:    $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_BOUND
The data in y must lie in the interval $\left(0,1\right)$.
NE_INT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}>1$.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NOT_INCREASING
${\mathbf{issort}}=\mathrm{Nag_TRUE}$ and the data in y is not sorted in ascending order.

Not applicable.

8Parallelism and Performance

nag_anderson_darling_stat (g08chc) 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 function. 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 (g08chce.c)

10.2Program Data

Program Data (g08chce.d)

10.3Program Results

Program Results (g08chce.r)