Compound Poisson approximations for sums of discrete nonlattice variables

Čekanavičius, Vydas; Wang, Y. H.

doi:10.1017/s0001867800012167

Cited by 7 publications

(6 citation statements)

References 20 publications

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“…Remarkably it was just one of many auxiliary results used for the proof of the first uniform Kolmogorov theorem. Lemma 7.2 is Lemma 4.3 from [46]. Theorem 7.1 is a special case of Theorem 1 from [55].…”

Section: Bibliographical Notesmentioning

confidence: 98%

Approximation Methods in Probability Theory

Čekanavičius

2016

Universitext

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Section: Bibliographical Notesmentioning

confidence: 98%

Approximation Methods in Probability Theory

Čekanavičius

2016

Universitext

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“…Kolmogorov ([59], formula (30)) has applied (93) without formulating it explicitly. Presman [83] was probably the first to formulate (93) explicitly and present its proof.…”

Section: Accuracy Of Cp Approximationmentioning

confidence: 99%

“…Presman [83] has evaluated d T V (ν n ; π λ ) (and hence d T V (S n ; Y )) using ( 93) and (30). Michel [70] has applied (93) and the Barbour-Eagleson estimate (29).…”

Section: Accuracy Of Cp Approximationmentioning

confidence: 99%

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Poisson approximation

Novak¹

2019

Preprint

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We overview results on the topic of Poisson approximation that are missed in existing surveys. The main attention is paid to the problem of Poisson approximation to the distribution of a sum of Bernoulli and, more generally, non-negative integervalued random variables.We do not restrict ourselves to a particular method, and overview the whole range of issues including the general limit theorem, estimates of the accuracy of approximation, asymptotic expansions, etc. Related results on the accuracy of compound Poisson approximation are presented as well.We indicate a number of open problems and discuss directions of further research.

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“…On the other hand, multinomial sampling can be reduced via poissoniza-tion to the corresponding Poisson sampling scheme which provides a good approximation to the former (in case of sparse asymptotics, see, e.g., Mnatsakanov and Klassen, 2000;van Es et al, 2003). Alternative approximation results are presented in (Čekanavičius, 1999;Čekanavičius and Wang, 2003;Zaitsev, 2005 and references therein). Khmaladze (1988) introduced a general framework called large number of rare events (LNRE).…”

Section: Large Number Of Rare Eventsmentioning

confidence: 99%

Nonparametric criteria for sparse contingency tables

Samusenko¹

2012

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In the dissertation, the problem of nonparametric testing for sparse contingency tables is addressed. Statistical inference problems caused by sparsity of contingency tables are widely discussed in the literature. Traditionally, the expected (under null the hypothesis) frequency is required to exceed 5 in almost all cells of the contingency table. If this condition is violated, the χ 2 approximations of goodnessof-fit statistics may be inaccurate and the table is said to be sparse. Several techniques have been proposed to tackle the problem: exact tests, alternative approximations, parametric and nonparametric bootstrap, Bayes approach and other methods. However they all are not applicable or have some limitations in nonparametric statistical inference of very sparse contingency tables. In the dissertation, it is shown that, for sparse categorical data, the likelihood ratio statistic and Pearson's χ 2 statistic may become noninformative: they do not anymore measure the goodness-of-fit of null hypotheses to data. Thus, they can be inconsistent even in cases where a simple consistent test does exist. An improvement of the classical criteria for sparse contingency tables is proposed. The improvement is achieved by grouping and smoothing of sparse categorical data by making use of a new sparse asymptotics model relying on (extended) empirical Bayes approach. Under general conditions, the consistency of the proposed criteria based on grouping is proved. Finite-sample behavior of the criteria is investigated via Monte Carlo simulations. The dissertation consists of four parts including Introduction, 4 chapters, General conclusions, References and Appendices. The introduction reveals the importance of the scientific problem, describes the purpose and tasks of the thesis, research methodology, scientific novelty, the practical significance of results The introduction ends in presenting the author's publications on the subject of the defended dissertation, offering the material of made presentations in conferences. In Chapter 1, an overview of the problem is presented and basic definitions are introduced. Chapter 2 demonstrates the inconsistency of classical tests in case of (very) sparse categorical data for both multinomial and Poisson sampling scheme. In Chapter 3, extended Bayes model is introduced. It provides a basis for smoothing and grouping of sparse (nominal) data. The consistency of criteria based on grouping is proved. Finite-sample behavior of the classical and proposed criteria is studied in Chapter 4. Details of the computer simulation results are given in Appendix. H 0-the null hypothesis; H 1-the alternative. Abbreviations LN RE-Large Number of Rare Events; M CM C-Markov chain Monte Carlo.

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Compound Poisson approximations for sums of discrete nonlattice variables

Cited by 7 publications

References 20 publications

Approximation Methods in Probability Theory

Approximation Methods in Probability Theory

Poisson approximation

Nonparametric criteria for sparse contingency tables

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