Randomized decomposable statistics in the scheme of independent allocating particles into boxes

Mirakhmedov, Sherzod M.

doi:10.1515/dma.1992.2.1.91

Cited by 6 publications

(5 citation statements)

References 10 publications

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“…Special formulations of this Formula (2.5) show up in literature; see e.g. Holst (1979), Quine and Robinson (1984), Mirakhmedov (1985Mirakhmedov ( , 1987Mirakhmedov ( , 1992.…”

Section: N Ppmentioning

confidence: 99%

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On Edgeworth Expansions in Generalized Urn Models

2012

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Abstract. The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a "sum of functions of such frequencies", which are also called "decomposable statistics." Applying these results to urn models such as with-and withoutreplacement sampling schemes as well as the multicolor Pólya-Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without replacement scheme, and for the so-called Dixon statistic that is useful in comparing 2 samples.

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“…Special formulations of this Formula (2.5) show up in literature; see e.g. Holst (1979), Quine and Robinson (1984), Mirakhmedov (1985Mirakhmedov ( , 1987Mirakhmedov ( , 1992.…”

Section: N Ppmentioning

confidence: 99%

“…It should be mentioned that we are dealing with triangular arrays where all the parameters of a Under some mild conditions, one can show that as n and () N N n (2.3) Quine and Robinson (1984), Mirakhmedov (1985Mirakhmedov ( , 1987Mirakhmedov ( , 1992.…”

Section: Introductionmentioning

confidence: 99%

On Edgeworth Expansions in Generalized Urn Models

2012

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“…If the jumps of random walk had the same law as | log W | and were independent of the sequence (23) would follow from Corollary 2.3 [21]. In the present situation where the aforementioned independence is absent the proof given by Mikosch and Resnick still applies except that the relation…”

Section: Proof Of Theorem 12mentioning

confidence: 82%

“…Recall that some infinite random allocation schemes in nonrandom environment were also investigated in [20,22,23]. It should be emphasized that infinite allocation schemes radically differ from the classical allocation scheme with finitely many positive frequencies (see monograph [19] for more detail).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Weak Convergence of Finite-Dimensional Distributions of the Number of Empty Boxes in the Bernoulli Sieve

Iksanov¹,

Marynych²,

Vatutin³

2015

Theory Probab. Appl.

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The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0, 1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0, 1). Assuming that the number of points is equal to n we investigate the weak convergence, as n → ∞, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random walk. d → m(f ), as t → ∞. Consequently, the families X t t≥0 and m t (f ) t≥0 are tight on D and [0, ∞), respectively. Now Prohorov's theorem ensures that these are relatively compact. By Lemma 3.20 [24], the family m t is tight (hence relatively compact) on M p ([0, ∞) × (0, ∞]). Then the Cartesian product X t , m s t,s≥0 is relatively compact on D × M p ([0, ∞) × (0, ∞]) which implies that the family X t , m t t>0 is tight on D × M p ([0, ∞) × (0, ∞]). It remains to note that all subsequential limits of the collection X t , m t t>0 are equal in distribution, for (59) holds for any function f ∈ C K ([0, ∞) × (0, ∞]).

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“….4)The asymptotical normality result (2.2) follows from Theorem 1 ofMirakhmedov (1992). Proof of (2.3) is given by Mirakhmedov (2022).…”

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confidence: 87%

On the asymptotic properties of a certain class of goodness-of-fit tests associated with multinomial distributions

Mirakhmedov

2021

Communications in Statistics - Theory and Methods

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The object of study is the problem of testing for uniformity of the multinomial distribution. We consider tests based on symmetric statistics, defined as the sum of some function of cell-frequencies. Mainly, attention is focused on the class of power divergence statistics, in particular, on the chi-square and log-likelihood ratio statistics. The main issue of the article is to study the asymptotic properties of tests at the concept of an intermediate setting in terms of so called  -intermediate asymptotic efficiency due to Ivchenko and Mirakhmedov (1995), when the asymptotic power of tests are bounded away from zero and one, while sequences of alternatives converge to the hypothesis, but not too fast.

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Randomized decomposable statistics in the scheme of independent allocating particles into boxes

Cited by 6 publications

References 10 publications

On Edgeworth Expansions in Generalized Urn Models

On Edgeworth Expansions in Generalized Urn Models

Weak Convergence of Finite-Dimensional Distributions of the Number of Empty Boxes in the Bernoulli Sieve

On the asymptotic properties of a certain class of goodness-of-fit tests associated with multinomial distributions

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