On the asymptotic expansions for the risk function and deficiencies of some statistical estimators based on the samples with random sizes
Statistical regularities of the information flows in contemporary communication, computational and other information systems are characterized by the presence of the so-called ``heavy tails''. The random character of the intensity of the flow of informative events results in that the available sample size (traditionally this is the number of observations registered within a certain time interval) is random. The randomness of the sample size crucially changes the asymptotic properties of the statistical procedures (e.g., estimators). The present paper consists of a number of applications of the deficiency concept, i.e., the number of additional observations required by the less effective procedure and thereby provides a basis for deciding whether or not the price is too high. The deficiency was introduced and initiated in its study by Hodges and Lehmann in 1970 [1]. In this paper asymptotic deficiencies of statistical estimators based on the samples with random sizes are considered. Asymptotic expansions for the risk function of some estimators based on the samples with random sizes are presented.