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

The waiting time and related statistics in the multinomial scheme: a survey -- 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 than the corresponding levels. Decomposable statistics (DS) are studied, where N is the number of outcomes, QJ, 1 < j < Nt 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. 1. INTRODUCTION Consider an infinite sequence-of the multinomial trials with TV outcomes whose probabilities are pi, . . . ,, Pi + . . · + PN = 1. It is assumed that before the beginning of the trials for each outcome ; a certain level z/, is established, so that V\,...,VN are independent non-negative integer-valued random variables (r.v.'s). The trials are conducted until the frequencies of k outcomes (the numbers of occurrences of the outcomes) for the first time become no less than the corresponding preassigned levels. We denote this stopping time by i/(JV, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Discrete Mathematics and Applications de Gruyter

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

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Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0924-9265
eISSN
1569-3929
DOI
10.1515/dma.1993.3.5.451
Publisher site
See Article on Publisher Site

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 than the corresponding levels. Decomposable statistics (DS) are studied, where N is the number of outcomes, QJ, 1 < j < Nt 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. 1. INTRODUCTION Consider an infinite sequence-of the multinomial trials with TV outcomes whose probabilities are pi, . . . ,, Pi + . . · + PN = 1. It is assumed that before the beginning of the trials for each outcome ; a certain level z/, is established, so that V\,...,VN are independent non-negative integer-valued random variables (r.v.'s). The trials are conducted until the frequencies of k outcomes (the numbers of occurrences of the outcomes) for the first time become no less than the corresponding preassigned levels. We denote this stopping time by i/(JV,

Journal

Discrete Mathematics and Applications – de Gruyter

Published: Jan 1, 1993

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