TRANSFER THEOREM CONCERNING ASYMPTOTIC EXPANSIONS FOR THE DISTRIBUTION FUNCTIONS OF STATISTICS BASED ON SAMPLES WITH RANDOM SIZES

摘要

In the paper, we discuss the transformation of the asymptotic expansion for the distribution of a statistic admitting Edgeworth expansion if the sample size is replaced by a random variable.We demonstrate that all those statistics that are regarded as asymptotically normal in the classical sense, become asymptotically Laplace or Student if the sample size is random. Thus, the Laplace and Student distributions may be used as an asymptotic approximation in descriptive statistics being a convenient heavy-tailed alternative to stable laws.