Poisson Approximations for Sum of Bernoulli Random Variables and its Application to Ewens Sampling Formula

Abstract This paper is concerned with developing low variance simulation estimators of probabilities related to the sum of Bernoulli random variables. It shows how to utilize an identity used in the Chen-Stein approach to bounding Poisson approximations to obtain low variance estimators. Applications and numerical examples in such areas as pattern occurrences, generalized coupon collecting, system reliability, and multivariate normals are presented. We also consider the problem of estimating the probability that a positive linear combination of Bernoulli random variables is greater than some specified value, and present a simulation estimator that is always less than the Markov inequality bound on that probability.

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On a variance related to the Ewens sampling formula

Nonlinear Analysis Modelling and Control ◽

10.15388/na.16.4.14088 ◽

2011 ◽

Vol 16 (4) ◽

pp. 453-466 ◽

Cited By ~ 2

Author(s):

Eugenijus Manstavičius ◽

Žydrūnas Žilinskas

Keyword(s):

Upper Bound ◽

Random Variables ◽

Integral Operators ◽

Search Tree ◽

Binary Search ◽

Binary Search Tree ◽

Dependent Random Variables ◽

Best Constant ◽

Ewens Sampling Formula ◽

Sampling Formula

A one-parameter multivariate distribution, called the Ewens sampling formula, was introduced in 1972 to model the mutation phenomenon in genetics. The case discussed in this note goes back to Lynch’s theorem in the random binary search tree theory. We examine an additive statistics, being a sum of dependent random variables, and find an upper bound of its variance in terms of the sum of variances of summands. The asymptotically best constant in this estimate is established as the dimension increases. The approach is based on approximation of the extremal eigenvalues of appropriate integral operators and matrices.

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Central limit theorem for the prefix exchange distance under Ewens sampling formula

Discrete Mathematics ◽

10.1016/j.disc.2020.112206 ◽

2021 ◽

Vol 344 (2) ◽

pp. 112206

Author(s):

Simona Grusea ◽

Anthony Labarre

Keyword(s):

Central Limit Theorem ◽

Limit Theorem ◽

Central Limit ◽

Ewens Sampling Formula ◽

Sampling Formula

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On improvements of the order of approximation in the Poisson limit theorem

Journal of Applied Probability ◽

10.1017/s0021900200103808 ◽

1996 ◽

Vol 33 (01) ◽

pp. 146-155 ◽

Cited By ~ 1

Author(s):

K. Borovkov ◽

D. Pfeifer

Keyword(s):

Limit Theorem ◽

Random Variables ◽

Poisson Approximation ◽

Order Of Approximation ◽

Operator Semigroups ◽

Bernoulli Random Variables ◽

Usual Rate ◽

Poisson Limit ◽

Poisson Measures ◽

Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

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