Notes for Intel® oneAPI Math Kernel Library Vector Statistics

ID 772987
Date 12/04/2020
Document Table of Contents

Distribution Moments Test

Test Purpose

The test verifies that sample moments of a given distribution agree with theoretical moments. Sample mean (first order moment) and sample variance (central moment of the second order) are considered as stable responses.

First Level Test

The generated random number sequence is used to compute the sample mean M and the sample variance D that are of an asymptotically normal distribution. Proceeding from this asymptotic, p-values pM and pD are found using the values of M and D.

Second Level Test

The first level test is run ten times, each run producing a pair of p-values pM and pD, j = 1, 2, ... , 10. The Kolmogorov-Smirnov goodness-of-fit test with Anderson-Darling’s statistics is applied to the obtained p-values pjM, j = 1, 2, ... , 10. If the resulting p-value pM < 0.05 or pM > 0.95, the test is considered failed for the sample mean. The same procedure is performed for p-values pjD, j = 1, 2, ... , 10, and if p-value pD < 0.05 or pD > 0.95, the test is considered failed for the sample variance.

Interpreting Final Results 

Ten runs of the second level test provide the percentage FAILM of failed tests for the sample mean and the percentage FAILD of failed tests for the sample variance. The final result of the test is the percentage FAIL = max(FAILM, FAILD ). The value of FAIL < 50% is considered acceptable.