Decomposable statistics in an inverse allocation problem

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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.

“…The waiting time v (N,k) in the scheme with random levels was intensively studied then by Ivanov, Ivchenko and Protasov [4], Ivanov [2] and Ivchenko [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

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

Ivchenko¹

1993

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“…( ;v) of the model were obtained. This approach was also used in study of limit distributions of the general DS's in the equiprobable polynomial scheme with random levels [4]. Similar results for the GUS with the waiting time i/ m (N,k) were derived in [11].…”

Section: Introductionmentioning

confidence: 84%

“…. , fc/v) to the state at the next birth is given by (4). Let ξι(ί) be the number of births in the Aj-process in the interval (0./.1 (£j(Q) = 0) and…”

Section: Embedding the Urn Scheme In The Birth Processmentioning

confidence: 99%

“…Many special results on DS (3) in the inverse problems were recently obtained (see [4,11]). To obtain these results, the method of embedding the urn scheme in the corresponding stochastic birth process was developed.…”

Section: Introductionmentioning

confidence: 99%

“…The sequence of random vectors (77(71) = (ηι(η),...,η Ν (π)), n = 0,1,2,...} forms an TV-dimensional Markov chain with transitions only of the form (&!,...,&.,·,..., &/v) -> (ki,...,kj + l,...,fcyv), J = !,.·.,whose probabilities are given by vector(4). case the trials are not needed at all and z/(TV, k) = 0.…”

mentioning

confidence: 99%

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Decomposable statistics in inverse urn problems

Ivchenko¹,

Ivanov²

1995

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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.

Decomposable statistics and stopping times in sampling without replacement

Ivanov¹,

Ivchenko²

1997

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In the framework of the generalized urn scheme, we consider the sampling without replacement. From an urn containing n balls of each of N colours, balls are drawn consequently with equal probabilities and independently of each other. Each ball drawn is not returned to the urn. The sampling of balls is stopped if the numbers of balls of any k different colours drawn from the urn exceed for the first time the levels chosen before the sampling begins. We investigate the statistics of the form where g is a function of an integer-value argument and η/ is the number of balls of the y'th colour drawn before the sampling is stopped. We obtain a complete description of the class of limit distributions of the statistics L/v* as Ν -» «>, η = n(N) and k = k(N).