g01da:: Simple Calculations on Statistical Data (NAG Toolbox)

If a sample of

n

observations from any distribution (which may be denoted by

x_{1}, x_{2}, \dots, x_{n}

), is sorted into ascending order, the

r

th smallest value in the sample is often referred to as the

r

th ‘order statistic’, sometimes denoted by

x_{(r)}

(see Kendall and Stuart (1969)).

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

th Normal score for a given sample size

n

E (x_{(r)}) = \int_{- \infty}^{\infty} x_{r} d G_{r},

where

d G_{r} = \frac{A_{r}^{r - 1} {(1 - A_{r})}^{n - r} d A_{r}}{β (r, n - r + 1)}, A_{r} = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{x_{r}} e^{- t^{2} / 2} d t, r = 1, 2, \dots, n,

and

β

denotes the complete beta function.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

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

Accuracy

Errors are introduced by evaluation of the functions

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]

but

a

and

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

d G_{r}

are also integrated over the range

[a, b]

. nag_stat_normal_scores_exact (g01da) returns the estimated maximum error as

errest = \max_{r} [\max (|a|, |b|) \times |\int_{a}^{b} d G_{r} - 1.0|] .

Further Comments

Example

function g01da_example


fprintf('g01da example results\n\n');

n = int64(15);
etol = 0.001;

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

fprintf('Set size                = %5d\n', n);
fprintf('Error tolerance (input) = %13.3e\n', etol);
fprintf('Error estimate (output) = %13.3e\n', errest);
fprintf('Normal scores\n');
fprintf('   %10.3f%10.3f%10.3f%10.3f%10.3f\n',pp(1:n));

g01da_plot;



function g01da_plot
  % This produces a Q-Q plot for a randomly generated set of data.
  % The normal scores have been calculated using g01da and the sample
  % quantiles obtained by sorting the observed data using m01ca.

  % Initialize the base generator
  seed = [int64(2324213)];
  genid = int64(1);
  subid = int64(1);

  [state, ifail] = g05kf( ...
                          genid, subid, seed);

  % Generate 50 variates from a Normal distribution
  n = int64(50);
  [state, x, ifail] = g05sk( ...
                             n, 0, 1, state);
  % Sort x
  m1 = int64(1);
  [x, ifail] = m01ca(x, m1, 'Ascending');

  % Calculate normal scores
  etol = 0.001;
  [pp, errest, ifail] = g01da(n, etol);

  fig1 = figure;
  hold on
  plot(pp,x,'+r',[-2.25 2.25],[-2.25 2.25],'black');
  axis([-2.5 2.5 -3 2.5]);
  xlabel('Normal scores');
  ylabel('sample Quantiles');
  title({'Q-Q plot for a random set of data','using exact normal scores'});
  hold off;

g01da example results

Set size                =    15
Error tolerance (input) =     1.000e-03
Error estimate (output) =     2.218e-08
Normal scores
       -1.736    -1.248    -0.948    -0.715    -0.516
       -0.335    -0.165     0.000     0.165     0.335
        0.516     0.715     0.948     1.248     1.736

This shows a Q-Q plot for a randomly generated set of data. The normal scores have been calculated using nag_stat_normal_scores_exact (g01da) and the sample quantiles obtained by sorting the observed data using nag_sort_realvec_sort (m01ca). A reference line at

y = x

is also shown.

On entry,	if n is even, $iw < 3 \times n / 2$ ;
or	if n is odd, $iw < 3 \times (n - 1) / 2$ .

NAG Toolbox: nag_stat_normal_scores_exact (g01da)

▸▿ Contents

Purpose

Syntax

Description