Calculation of the deficiency of some statistical estimators constructed from samples with random sizes
The purpose of this paper is to use deficiency to compare estimators constructed from samples with random size with those constructed from samples with non-random size. The deficiency can be a characteristic of a possible loss of 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 of expectation $n$ with respect to the same statistic constructed as if the sample size were non-random and equal to $n$, grows almost linearly as $n$ grows. A non-trivial behavior of the deficiency is only possible if the random sample size is asymptotically degenerate. This is the case considered in this paper, where we study the deficiencies of statistics constructed from samples whose sizes have the Poisson, binomial and special three-point distributions. We also give some basic properties of estimators based on samples with random sizes.