The waiting time and related statistics in the multinomial scheme: a survey

Abstract

The sequential allocation of particles in the multinomial scheme is considered. The multinomial trials are conducted until the frequences of k, 1 < k < N, outcomes become for the first time no less thanthe corresponding levels. Decomposable statistics (DS) are studied, where N is the number of outcomes, QJ, 1 < j < N t are functions of an integer-valued argument and fy-is the frequency of the jth outcome at the stopping time. The exact and asymptotic results related to L^k and their various specializations are presented and some applications to the statistical inference for the multinomial scheme are given.

Decomposable statistics in inverse urn problems

“…'s, and this property is used for obtaining the general representation of the ch.f. of DS (3). This fact has first been noted in [3], where the equiprobable polynomial scheme has been studied.…”

“…of DS (3). This fact has first been noted in [3], where the equiprobable polynomial scheme has been studied. The frequencies (the numbers of occurrences) of the corresponding outcomes ηι,.,.,ηΝ at the stopping time (after z/ m (7V, k) trials) are proved to be conditionally independent, truncated at the point m Poisson r.v.…”

“…The following problem gives a typical and most common example of this kind in the literature: what is the distribution of the number of particles v m (N, k) required to be thrown into Ν cells in order to get for the first time k cells containing no less than m particles each? A great amount of publications deals with the stopping time (or waiting time) v m (N,k) and related problems (see [2,6]). …”

On waiting time in the Markov-Pólya scheme

“…The complete realization of this program for the scheme of sampling with replacement {i.e. : for c = 0) was carried out in [7] (see also the survey [8]); for the scheme of sampling without replacement (e = -l). this was done in [10]; the case of the degenerated level (P{u = m} = t for some m > 1) was investigated thoroughly in [11].…”

“…We note that in the general case there are at least k zero levels with probability [8]): it can be said that the investigations of this scheme from the viewpoint of inverse problems are in a most complete state. Similar results on tile waiting time for tile scheme of sampling without replacement (the case where c = -l) are obtail~ed in the authors" paper [10].…”