naginterfaces.library.nonpar.gofstat_​anddar

naginterfaces.library.nonpar.gofstat_anddar(issort, y)

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

For full information please refer to the NAG Library document for g08ch

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g08/g08chf.html

Parameters

issortbool

Set $issort=True$ if the observations are sorted in ascending order; otherwise the function will sort the observations.

yfloat, array-like, shape $\left(n\right)$

$y_{\text{i}}$, for $\text{i}=1,2,\dots ,n$, the $n$ observations.

Returns

gofsfloat

The Anderson–Darling goodness-of-fit test statistic.

yfloat, ndarray, shape $\left(n\right)$

If $issort=False$, the data sorted in ascending order; otherwise the array is unchanged.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n>1$.

(errno $3$)

$issort=True$ and the data in $y$ is not sorted in ascending order.

(errno $9$)

The data in $y$ must lie in the interval $(0,1)$.

Notes

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:

$$A^2=-n-S\text{;}$$

where:

$$S=\sum _1^n\frac{2i-1}{n}[ln\left(y_i\right)+ln(1-y_{n-i+1})]\text{.}$$

When observations of a random variable $X$ are non-uniformly distributed, the probability integral transformation (PIT):

$$Y=F\left(X\right)\text{,}$$

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.

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