Umerical methods and algorithms of calculation of exact distributions of non-parametrical criteria statistical hypotheses

Аuthors

Agamirov L. V.^1*, Agamirov V. L.^2**, Vestyak V. A.^3***

1. Russian State Technological University named after K.E. Tsiolkovsky, MATI, 3, Orshanskaya str., Moscow, 121552, Russia
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia
3. ,

*e-mail: mmk@mati.ru
**e-mail: avl095@mail.ru
***e-mail: kaf311@mai.ru

Abstract

One of the important tasks of statistical analysis is calculation of percentage points of hypotheses testing criteria. The peculiarity of this task for nonparametric ranking criteria is the need for accurate calculation of percentage points of distribution as a result of low accuracy of crude approximations at a small scope of survey, while nonparametric criteria are particularly effective when working with a small sampling. Inappropriateness of using for this purpose statistically common normal or other approximations [4-6] is accounted for by the fact that well-known ranking criteria (sign test, Wilcoxon sign rank test, Wilcoxon two-sample test, Kruskal-Wallis test, etc.) refer to the area of nonparametric statistics, while for whatever reasons no assumptions are made about the kind of hypothetical function of distribution of the random variable under study. With technical problems, especially when analyzing the results of mechanical tests, such a situation occurs due to a small amount of tests, considerable scattering of properties due to structural heterogeneity of constructional materials and a large variability of external factors during the tests. Thus distribution approximations of nonparametric criteria statistics, while for the reasons given above these criteria are particularly effective with a small sampling.
In many cases it is impossible to increase sampling population when testing expensive materials, structural components and full-scale objects. In addition, if it is achieved, it is most appropriate to use the traditional parametric statistical analysis which is always more effective as opposed to nonparametric methods.
That is why accurate calculation of distribution of nonparametric criteria statistics, which this paper deals with, is a topical task.
Within the framework if this paper there have been developed algorithms of accurate calculation of percentage points of sign test [2], Wilcoxon sign rank test, Wilcoxon two-sample test, Kruskal-Wallis test.
For example, exact critical values of Wilcoxon two- sample test statistic are calculated with the use of frequency generating function which under null hypothesis looks as follows [3]:

Another way of accurate calculation of Wilcoxon distribution statistic is the use of the following recurrent formula [2]:

with the following initial conditions:

In this paper we suggest several algorithms of calculation of exact distribution of ranking criteria statistics.
The first algorithm is based on formula (1) and assumes development of an algorithm of automated multiplication and division of polynoms which is realized in a standard scientific software package SSPLIB coded in FORTRAN [7, 8]. This article contains the authors programs of calculating exact distributions of ranking criteria. The second algorithm is based on models (2), (3) and is realized in Mathcad environment.

Conclusions

1. As applied to technical problems that arise when analyzing the results of mechanical tests which are characterized by a small scope of survey and a large data scattering, it is appropriate to use exact distributions of nonparametric ranking criteria in order to increase validity of conclusions about the independence of different sampled populations.
   
2. There have been considered mathematical models based on frequency generating functions and recurrent equations with constraints for Wilcoxon sign rank test and Wilcoxon two-sample test; there have been developed algorithms and presented programs in software environments Mathcad and Basic with an open code. 
3. For k-random criteria there has been suggested a modification of Wilcoxon two-sample test made by pair comparison of each sampling with each other; there has also been developed an algorithm and a calculation program. For k-random Kruskal-Wallis test there has been developed an algorithm of direct enumeration of rank permutation; there is also given an open program code applicable for subsequent recursive optimization of cyclic procedures

Keywords:

ranking criteria, nonparametric criteria, statistical analysis, mechanical tests, material, material characteristics

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