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Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_stat_normal_scores_exact (g01da)

Purpose

nag_stat_normal_scores_exact (g01da) computes a set of Normal scores, i.e., the expected values of an ordered set of independent observations from a Normal distribution with mean 0.0$0.0$ and standard deviation 1.0$1.0$.

Syntax

[pp, errest, ifail] = g01da(n, etol)
[pp, errest, ifail] = nag_stat_normal_scores_exact(n, etol)

Description

If a sample of n$n$ observations from any distribution (which may be denoted by x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$), is sorted into ascending order, the r$r$th smallest value in the sample is often referred to as the r$r$th ‘order statistic’, sometimes denoted by x(r)${x}_{\left(r\right)}$ (see Kendall and Stuart (1969)).
The order statistics therefore have the property
 x(1) ≤ x(2) ≤ … ≤ x(n). $x(1)≤x(2)≤…≤x(n).$
(If n = 2r + 1$n=2r+1$, xr + 1${x}_{r+1}$ is the sample median.)
For samples originating from a known distribution, the distribution of each order statistic in a sample of given size may be determined. In particular, the expected values of the order statistics may be found by integration. If the sample arises from a Normal distribution, the expected values of the order statistics are referred to as the ‘Normal scores’. The Normal scores provide a set of reference values against which the order statistics of an actual data sample of the same size may be compared, to provide an indication of Normality for the sample (see nag_stat_plot_scatter_normal (g01ah)). Normal scores have other applications; for instance, they are sometimes used as alternatives to ranks in nonparametric testing procedures.
nag_stat_normal_scores_exact (g01da) computes the r$r$th Normal score for a given sample size n$n$ as
 ∞ E(x(r)) = ∫ xrdGr, − ∞
$E(x(r))=∫-∞∞xrdGr,$
where
 xr dGr = (Arr − 1 (1 − Ar)n − r d Ar)/(β (r,n − r + 1)),  Ar = 1/(sqrt(2π )) ∫ e − t2 / 2dt,  r = 1,2, … ,n, − ∞
$dGr=Arr- 1 (1-Ar)n-r d Ar β (r,n-r+ 1) , Ar=12π ∫-∞xre-t2/2 dt, r= 1,2,…,n,$
and β$\beta$ denotes the complete beta function.
The function attempts to evaluate the scores so that the estimated error in each score is less than the value etol specified by you. All integrations are performed in parallel and arranged so as to give good speed and reasonable accuracy.

References

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
n$n$, the size of the set.
Constraint: n > 0${\mathbf{n}}>0$.
2:     etol – double scalar
The maximum value for the estimated absolute error in the computed scores.
Constraint: etol > 0.0${\mathbf{etol}}>0.0$.

None.

work iw

Output Parameters

1:     pp(n) – double array
The Normal scores. pp(i)${\mathbf{pp}}\left(\mathit{i}\right)$ contains the value E(x(i))$E\left({x}_{\left(\mathit{i}\right)}\right)$, for i = 1,2,,n$\mathit{i}=1,2,\dots ,n$.
2:     errest – double scalar
A computed estimate of the maximum error in the computed scores (see Section [Accuracy]).
3:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, etol ≤ 0.0${\mathbf{etol}}\le 0.0$.
W ifail = 3${\mathbf{ifail}}=3$
The function was unable to estimate the scores with estimated error less than etol. The best result obtained is returned together with the associated value of errest.
ifail = 4${\mathbf{ifail}}=4$
 On entry, if n is even, iw < 3 × n / 2$\mathit{iw}<3×{\mathbf{n}}/2$; or if n is odd, iw < 3 × (n − 1) / 2$\mathit{iw}<3×\left({\mathbf{n}}-1\right)/2$.

Accuracy

Errors are introduced by evaluation of the functions dGr$d{G}_{r}$ and errors in the numerical integration process. Errors are also introduced by the approximation of the true infinite range of integration by a finite range [a,b]$\left[a,b\right]$ but a$a$ and b$b$ are chosen so that this effect is of lower order than that of the other two factors. In order to estimate the maximum error the functions dGr$d{G}_{r}$ are also integrated over the range [a,b]$\left[a,b\right]$. nag_stat_normal_scores_exact (g01da) returns the estimated maximum error as
errest = max [max (|a|,|b|) ×
 ( b ) ∫ dGr − 1.0 a
]
.
r
$errest=maxr [ max(|a|,|b|)× | ∫ab dGr-1.0 | ] .$

The time taken by nag_stat_normal_scores_exact (g01da) depends on etol and n. For a given value of etol the timing varies approximately linearly with n.

Example

function nag_stat_normal_scores_exact_example
n = int64(5);
etol = 0.001;
[pp, errest, ifail] = nag_stat_normal_scores_exact(n, etol)

pp =

-1.1630
-0.4950
0
0.4950
1.1630

errest =

9.0800e-09

ifail =

0

function g01da_example
n = int64(5);
etol = 0.001;
[pp, errest, ifail] = g01da(n, etol)

pp =

-1.1630
-0.4950
0
0.4950
1.1630

errest =

9.0800e-09

ifail =

0