naginterfaces.library.stat.normal_​scores_​approx

naginterfaces.library.stat.normal_scores_approx(n)

normal_scores_approx calculates an approximation to the set of Normal Scores, i.e., the expected values of an ordered set of independent observations from a Normal distribution with mean $0.0$ and standard deviation $1.0$.

For full information please refer to the NAG Library document for g01db

https://support.nag.com/numeric/nl/nagdoc_30/flhtml/g01/g01dbf.html

Parameters

nint

$n$, the size of the sample.

Returns

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

The Normal scores. $pp[\text{i}-1]$ contains the value $E\left(x_{\left(\text{i}\right)}\right)$, for $\text{i}=1,2,\dots ,n$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

normal_scores_approx is an adaptation of the Applied Statistics Algorithm AS $177.3$, see Royston (1982). If you are particularly concerned with the accuracy with which normal_scores_approx computes the expected values of the order statistics (see Accuracy), then normal_scores_exact() which is more accurate should be used instead at a cost of increased storage and computing time.

Let $x_{\left(1\right)},x_{\left(2\right)},\dots ,x_{\left(n\right)}$ be the order statistics from a random sample of size $n$ from the standard Normal distribution. Defining

$$P_{r,n}=\Phi (-E\left(x_{\left(r\right)}\right))$$

and

$$Q_{r,n}=\frac{r-ϵ}{n+\gamma }\text{,}r=1,2,\dots ,n\text{,}$$

where $E\left(x_{\left(r\right)}\right)$ is the expected value of $x_{\left(r\right)}$, the current function approximates the Normal upper tail area corresponding to $E\left(x_{\left(r\right)}\right)$ as,

$${\tilde{P}}_{\text{r},n}=Q_{\text{r},n}+\frac{{\delta }_1}{n}{Q}_{\text{r},n}^{\lambda }+\frac{{\delta }_2}{n}{Q}_{\text{r},n}^{2\lambda }-C_{\text{r},n}\text{.}$$

, for $\text{r}=1,2,\dots ,3$, and $r\ge 4$. Estimates of $ϵ$, $\gamma$, ${\delta }_1$, ${\delta }_2$ and $\lambda$ are obtained. A small correction $C_{r,n}$ to ${\tilde{P}}_{r,n}$ is necessary when $r\le 7$ and $n\le 20$.

The approximation to $E\left(X_{\left(r\right)}\right)$ is thus given by

$$E\left(x_{\left(r\right)}\right)=-{\Phi }^{-1}\left({\tilde{P}}_{r,n}\right),r=1,2,\dots ,n\text{.}$$

Values of the inverse Normal probability integral ${\Phi }^{-1}$ are obtained from inv_cdf_normal().

References

Royston, J P, 1982, Algorithm AS 177: expected normal order statistics (exact and approximate), Appl. Statist. (31), 161–165