Decomposable statistics in inverse urn problems

Abstract

We consider the following urn model. At the beginning of trials the urn contains a given number of balls of N colours. In each trial, a ball is randomly drawn from the urn, independently of the results of the other trials. If we draw a ball of some colour, then before the next trial the number of balls of this colour in the urn is changed according to a certain rule. Before the trials, a level Vj is determined for each colour j, j = 1, . . . , N\ we assume that v\ , . . . , z/yy are independent integer-valued random variables. The drawings stop when the numbers of balls of arbitrary k colours attain or exceed the corresponding levels for the first time. Decomposable statistics (DS) are studied, where g\ , . . . , g N are functions of the integer argument and η\ , . . . , η Ν are the numbers of balls of corresponding colours at the stopping time. The characteristic functions of DS Z/w* are obtained by embedding the urn scheme in a suitable Markov pure birth process. Some special cases including sampling with and without replacement and Polya sampling are considered.

On waiting time in the Markov-Pólya scheme

“…where a scheme with degenerated levels is considered: for a scheme with random levels, this approach yields a general integral representation of the characteristic function (ch.f.) of the DS (4) [9], which allows us to analyze their limit distributions as N ~ c~.…”

“…With the help of representation (6), in [9] the following formula for the characteristic function of an arhitrary DS (4) is given:…”

Decomposable statistics and stopping times in sampling without replacement

“…As pointed out in [5], representations (4)-(5) have the following probabilistic interpretation. Consider the set , 7 = , -.,-, ίθ, 7 = l, ...,Vof independent for each value of Θ e (/?,!]…”

“…To find the limit distributions of the decomposable statistic L Nk , we will use the following integral representation of its characteristic function (see [5], Theorem 2) r l…”

“…In its turn, the scheme of sampling without replacement described above can be considered as the so-called generalized urn model studied in [5,11]. In its turn, the scheme of sampling without replacement described above can be considered as the so-called generalized urn model studied in [5,11].…”