The investigation of statistic distributions of the Cochran test for the means shift detection

The statistic distributions of the modified Cochran test for the means shift detection have been investigated by methods of statistical simulation at the different distributions of observations. The considered test is destined for testing the hypothesis H : E[x ] = E[x ] =... = E[x ], i.e., for checking that all observations x , x ,...,x , recorded in order to their appearance, have the same expectations. A competing hypothesis (alternative) is the mean shift change by a jump after the first « observations (« + n = «). Note that a series of tests for the null hypothesis is carried out with the partition of the original sample into two parts of unequal sizes. The Cochran test statistics, obtained for different partitions, have the form n« (xl - XR ) K = - 1 n _ -Г Z (xi - x) n -1 i=1 _ 1 «1 _ 1 « _ 1 « L = Z i , R = Z i , = Z i . « i=1 n i=« 1 « i=1 The distribution of the test statistics is approximated by the chi-square distribution with one degree of freedom when the assumption of the normal distribution of random values x^, x2,.»«, x« is true. The convergence rate of the statistic distribution to the %2(1) distribution has been estimated: the goodness-of-fit of the % (1) distribution with empirical distributions of considered statistic begins from the sample size n = 60. The simulations of Cochran statistic distributions have confirmed their high stability to the deviation of the observed distribution from the normal law. It has been shown, that if the distribution of observed random values is symmetrical, then change of its kurtosis does not lead to significant deviation of the statistic distribution from the % (1) distribution. Exceptions are in the cases of distributions with the too heavy tails. Moreover, bimodal or asymmetrical distributions of observations practically do not influence on the distribution of the Cochran statistic. The investigations have confirmed high stability of the Cochran statistic distributions to the het-erosсedasticity. A comparative analysis of the power of the Cochran test with the power of the Fisher test has been performed. It has been shown, that the Fisher test for homogeneity of means which is less stable to the heterosсedasticity than considered Cochran test, is practically identical with the latter by power. The Abbe test has been compared by power with the Cochran test for the different alternatives and different ways of sample division into two subsamples. It has been shown that the closer the ratio of subsamples sizes to 1:1, the higher power of Cochran's test. If there is a linear trend in data, power of the Cochran test is higher than the power of the Abbe test even when the subsamples sizes are at a ratio of nine to one. In the case of nonlinear sinusoidal trend, the Cochran test begins to concede to the Abbe test in power for the subsamples sizes ratio of four to one. The investigations have shown that abrupt decrease in power of the Cochran test begins from the subsamples sizes ratio of about 7:3. The case of a mixed trend (the sum of sinusoidal and linear trends) has been also considered. Note that the regularities obtained for the case of sinusoidal trend are generally the same for the case of mixed trend. Thus, it is shown that if one wants to identify the trend of arbitrary form, then the division of the sample for the Cochran test should be such that the less size of the subsamples would be at least a third of the size of the entire sample. The following conclusion based on the analysis of the properties of the modified Cochran test has been made: application of this test to practical problems of detecting shift of means, expressed as jump or as a trend, in sequences of observations is expedient.

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