g08ed:: Nonparametric Statistics (NAG Toolbox)

nag_nonpar_randtest_gaps (g08ed) may be used in two different modes:

(i)	a single call to nag_nonpar_randtest_gaps (g08ed) which computes all test statistics after counting the gaps;
(ii)	multiple calls to nag_nonpar_randtest_gaps (g08ed) with the final test statistics only being computed in the last call.

The term gap is used to describe the distance between two numbers in the sequence that lie in the interval

(r_{l}, r_{u})

. That is, a gap ends at

x_{j}

r_{l} \leq x_{j} \leq r_{u}

. The next gap then begins at

x_{j + 1}

. The interval

(r_{l}, r_{u})

should lie within the region of all possible numbers. For example if the test is carried out on a sequence of

(0, 1)

random numbers then the interval

(r_{l}, r_{u})

must be contained in the whole interval

(0, 1)

. Let

t_{len}

be the length of the interval which specifies all possible numbers.

nag_nonpar_randtest_gaps (g08ed) counts the number of gaps of different lengths. Let

c_{i}

denote the number of gaps of length

i

, for

i = 1, 2, \dots, k - 1

. The number of gaps of length

k

or greater is then denoted by

c_{k}

. An unfinished gap at the end of a sequence is not counted unless the sequence is part of an initial or intermediate call to nag_nonpar_randtest_gaps (g08ed) (i.e., unless there is another call to nag_nonpar_randtest_gaps (g08ed) to follow) in which case the unfinished gap is used. The following is a trivial example.

Suppose we called nag_nonpar_randtest_gaps (g08ed) twice (i.e., with

cl ='F'

and then with

cl ='L'

) with the following two sequences and with

rlo = 0.30

and

rup = 0.60

( $0.20$ $0.40$ $0.45$ $0.40$ $0.15$ $0.75$ $0.95$ $0.23$ ) and
( $0.27$ $0.40$ $0.25$ $0.10$ $0.34$ $0.39$ $0.61$ $0.12$ ).

When the counting of gaps is complete nag_nonpar_randtest_gaps (g08ed) computes the expected values of the counts. An approximate

χ^{2}

statistic with

k

degrees of freedom is computed where

X^{2} = \frac{\sum_{i = 1}^{k} {(c_{i} - e_{i})}^{2}}{e_{i}},

where

$e_{i} = ngaps \times p \times {(1 - p)}^{i - 1}$ , if $i < k$ ;
$e_{i} = ngaps \times {(1 - p)}^{i - 1}$ , if $i = k$ ;
$ngaps =$ the number of gaps found and
$p = (r_{u} - r_{l}) / t_{len}$ .

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

Further Comments

Example

function g08ed_example


fprintf('g08ed example results\n\n');

% Initialize the base generator to a repeatable sequence
seed  = [int64(324213)];
genid = int64(1);
subid = int64(1);
[state, ifail] = g05kf( ...
                        genid, subid, seed);

m      = int64(0);
rlo    = 0.4;
rup    = 0.6;
totlen = 1;
ncount = zeros(10, 1, 'int64');
ngaps  = int64(0);
nsampl = 5;
n      = int64(100);
cl    = 'F';

for i=1:nsampl
  % Generate a sample from U(0,1)
  [state, x, ifail] = g05sq( ...
                             n, 0, 1, state);
  % Process the sample
  [ngaps, ncount, ex, chi, df, prob, ifail] = ...
  g08ed( ...
         cl, x, m, rlo, rup, totlen, ngaps, ncount);
  % Adjust CL
  cl = 'I';
  if i==nsampl-1
     cl = 'L';
  end
end

fprintf('Total number of gaps found = %d\n', ngaps);
fprintf('\n%33s\n', 'Count');
head = sprintf('%7d', [0:8]);
head = sprintf('%s     >8', head);
fprintf('%s\n', head);
fprintf('%7d', ncount);

fprintf('\n\n%34s\n', 'Expect');
fprintf('%s\n', head);
fprintf('%7.1f', ex);
fprintf('\n\n');

fprintf('Chisq = %10.4f\n', chi);
fprintf('DF    = %7.1f\n', df);
fprintf('Prob  = %10.4f\n', prob);

g08ed example results

Total number of gaps found = 99

                            Count
      0      1      2      3      4      5      6      7      8     >8
     22     11     10     13      6     12      4      6      2     13

                            Expect
      0      1      2      3      4      5      6      7      8     >8
   19.8   15.8   12.7   10.1    8.1    6.5    5.2    4.2    3.3   13.3

Chisq =     9.9540
DF    =     9.0
Prob  =     0.3542

NAG Toolbox: nag_nonpar_randtest_gaps (g08ed)

▸▿ Contents

Purpose

Syntax

Description