On the asymptotic distribution of themaximumorder statistic in a sample with randomsize

V. I. Pagurova

M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Abstract: The asymptotic distribution of the normalized maximum is investigated under the assumption that the random sample size is representable as a sum of $n$ independent identically distributed random variables. This paper generalizes the results of Pagurova V. // Statistical methods of estimating and testing of hypoteses. – Perm, 2005. P. 104–113 where the sample size was Poisson-distributed with a parameter $n$. For a one-parameter family of distributions depending on an unknown location parameter, the rate of convergence of the distribution of the normalized maximum to the limit law is investigated. The classes of distributions with exponential and power-type tails are considered.

Keywords: randomly indexed maximum; one-parameter family of distributions; convergence rate.