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On the asymptotic behavior of the deficiency of some statistical estimators based on samples with random sizes

Bening VE

Due to the stochastic character of the intensities of information flows in high performance information systems, the size of data available for the statistical analysis can be often regarded as random. The purpose of this paper is to present some means for the comparison of the quality of estimators constructed from samples with random sizes with that of estimators constructed from samples with non-random sizes. As this means it is proposed to use the deficiency. It can be an illustrative characteristic of a possible loss of the accuracy of statistical inference if a random-size-sample is erroneously regarded as a sample with non-random size. It is heuristically shown that if the asymptotic distribution of the sample size normalized by its expectation is not degenerate, then the deficiency of a statistic constructed from a sample with random size whose expectation equals n with respect to the same statistic constructed as if the sample size was non-random and equal to n, grows almost linearly as n grows. A non-trivial behavior of the deficiency is possible only if the random sample size is asymptotically degenerate. This is the case considered in the paper where the deficiencies of statistics constructed from samples whose sizes have the Poisson, binomial and special three-point distributions, respectively, are considered. Some basic results dealing with some properties of estimators based on the samples with random sizes are also presented.