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Anna Posfai - One of the best experts on this subject based on the ideXlab platform.

Aihua Xia - One of the best experts on this subject based on the ideXlab platform.

  • On moderate deviations in Poisson Approximation
    Journal of Applied Probability, 2020
    Co-Authors: Qingwei Liu, Aihua Xia
    Abstract:

    In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson Approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson Approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}.

  • On Stein's factors for Poisson Approximation in Wasserstein distance with non-linear transportation costs
    arXiv: Probability, 2020
    Co-Authors: Zhong-wei Liao, Aihua Xia
    Abstract:

    We establish various bounds on the solutions to a Stein equation for Poisson Approximation in Wasserstein distance with non-linear transportation costs. The proofs are a refinement of those in [Barbour and Xia (2006)] using the results in [Liu and Ma (2009)]. As a corollary, we obtain an estimate of Poisson Approximation error measured in L^2-Wasserstein distance.

  • stein s method for conditional compound Poisson Approximation
    Statistics & Probability Letters, 2015
    Co-Authors: H.l. Gan, Aihua Xia
    Abstract:

    Abstract The occurrence of rare events can often be well described by a compound Poisson distribution. However, one can only start modelling the occurrence of rare events after such events have happened, thus a conditional compound Poisson distribution is more appropriate in applications. In this note, we develop Stein’s method for conditional compound Poisson Approximation. Several applications are given to demonstrate the advantage of the direct approach of Stein’s method.

  • Stein’s method for conditional compound Poisson Approximation
    Statistics & Probability Letters, 2015
    Co-Authors: H.l. Gan, Aihua Xia
    Abstract:

    Abstract The occurrence of rare events can often be well described by a compound Poisson distribution. However, one can only start modelling the occurrence of rare events after such events have happened, thus a conditional compound Poisson distribution is more appropriate in applications. In this note, we develop Stein’s method for conditional compound Poisson Approximation. Several applications are given to demonstrate the advantage of the direct approach of Stein’s method.

  • On Stein's factors for Poisson Approximation in Wasserstein distance
    Bernoulli, 2006
    Co-Authors: Andrew Barbour, Aihua Xia
    Abstract:

    We provide a probabilistic proof of various Stein's factors for Poisson Approximation in terms of the Wasserstein distance.

K. Teerapabolarn - One of the best experts on this subject based on the ideXlab platform.

Oliver Johnson - One of the best experts on this subject based on the ideXlab platform.

  • Relaxation of monotone coupling conditions: Poisson Approximation and beyond
    Journal of Applied Probability, 2018
    Co-Authors: Fraser Daly, Oliver Johnson
    Abstract:

    It is well-known that assumptions of monotonicity in size-bias couplings may be used to prove simple, yet powerful, Poisson Approximation results. Here we show how these assumptions may be relaxed, establishing explicit Poisson Approximation bounds (depending on the first two moments only) for random variables which satisfy an approximate version of these monotonicity conditions. These are shown to be effective for models where an underlying random variable of interest is contaminated with noise. We also give explicit Poisson Approximation bounds for sums of associated or negatively associated random variables. Applications are given to epidemic models, extremes, and random sampling. Finally, we also show how similar techniques may be used to relax the assumptions needed in a Poincar\'e inequality and in a normal Approximation result.

  • Compound Poisson Approximation via Information Functionals
    Electronic Journal of Probability, 2010
    Co-Authors: Andrew Barbour, Oliver Johnson, Ioannis Kontoyiannis, Mokshay Madiman
    Abstract:

    An information-theoretic development is given for the problem of compound Poisson Approximation, which parallels earlier treatments for Gaussian and Poisson Approximation. Nonasymptotic bounds are derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance between their sum and the compound Poisson distribution is derived, based on the data-processing property of relative entropy and earlier Poisson Approximation results. When the summands have arbitrary distributions, corresponding bounds are derived in terms of the total variation distance. The main technical ingredient is the introduction of two "information functionals,'' and the analysis of their properties. These information functionals play a role analogous to that of the classical Fisher information in normal Approximation. Detailed comparisons are made between the resulting inequalities and related bounds.

  • fisher information compound Poisson Approximation and the Poisson channel
    International Symposium on Information Theory, 2007
    Co-Authors: Mokshay Madiman, Oliver Johnson, Ioannis Kontoyiannis
    Abstract:

    Fisher information plays a fundamental role in the analysis of Gaussian noise channels and in the study of Gaussian Approximations in probability and statistics. For discrete random variables, the scaled Fisher information plays an analogous role in the context of Poisson Approximation. Our first results show that it also admits a minimum mean squared error characterization with respect to the Poisson channel, and that it satisfies a monotonicity property that parallels the monotonicity recently established for the central limit theorem in terms of Fisher information. We next turn to the more general case of compound Poisson distributions on the nonnegative integers, and we introduce two new "local information quantities" to play the role of Fisher information in this context. We show that they satisfy subadditivity properties similar to those of classical Fisher information, we derive a minimum mean squared error characterization, and we explore their utility for obtaining compound Poisson Approximation bounds.

  • FisherInformation, Compound Poisson Approximation, and thePoissonChannel
    2007
    Co-Authors: Oliver Johnson, Joannis Kontoyiannis
    Abstract:

    Fisher information playsa fundamental rolein theanalysis ofGaussian noisechannels andinthestudyof Gaussian Approximations inprobability andstatistics. Fordis- crete randomvariables, thescaled Fisher information plays an analogous role inthecontext ofPoisson Approximation. Ourfirst results showthatitalsoadmits aminimummeansquared error characterization withrespect tothePoisson channel, andthatit satisfies amonotonicity property thatparallels themonotonicity recently established forthecentral limit theoremintermsof Fisher information. We nextturntothemoregeneral caseof compoundPoisson distributions on thenonnegative integers, andwe introduce twonew"local information quantities" to playtheroleofFisher information inthiscontext. We show thattheysatisfy subadditivity properties similar tothoseof classical Fisher information, wederive aminimummeansquared errorcharacterization, andweexplore their utility forobtaining compound Poisson Approximation bounds.

  • ISIT - Fisher Information, Compound Poisson Approximation, and the Poisson Channel
    2007 IEEE International Symposium on Information Theory, 2007
    Co-Authors: Mokshay Madiman, Oliver Johnson, Ioannis Kontoyiannis
    Abstract:

    Fisher information plays a fundamental role in the analysis of Gaussian noise channels and in the study of Gaussian Approximations in probability and statistics. For discrete random variables, the scaled Fisher information plays an analogous role in the context of Poisson Approximation. Our first results show that it also admits a minimum mean squared error characterization with respect to the Poisson channel, and that it satisfies a monotonicity property that parallels the monotonicity recently established for the central limit theorem in terms of Fisher information. We next turn to the more general case of compound Poisson distributions on the nonnegative integers, and we introduce two new "local information quantities" to play the role of Fisher information in this context. We show that they satisfy subadditivity properties similar to those of classical Fisher information, we derive a minimum mean squared error characterization, and we explore their utility for obtaining compound Poisson Approximation bounds.

Bero Roos - One of the best experts on this subject based on the ideXlab platform.

  • improvements in the Poisson Approximation of mixed Poisson distributions
    Journal of Statistical Planning and Inference, 2003
    Co-Authors: Bero Roos
    Abstract:

    We consider the Approximation of mixed Poisson distributions by Poisson laws and also by related finite signed measures of higher order. Upper bounds and asymptotic relations are given for several distances. Even in the case of the Poisson Approximation with respect to the total variation distance, our bounds have better order than those given in the literature. In particular, our results hold under weaker moment conditions for the mixing random variable. As an example, we consider the Approximation of the negative binomial distribution, which enables us to prove the sharpness of a constant in the upper bound of the total variation distance. The main tool is an integral formula for the difference of the counting densities of a Poisson distribution and a related finite signed measure.

  • Poisson Approximation of multivariate Poisson mixtures
    Journal of Applied Probability, 2003
    Co-Authors: Bero Roos
    Abstract:

    We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson Approximation of generalized multinomial distributions.

  • Metric Multivariate Poisson Approximation of the Generalized Multinomial Distribution
    Theory of Probability & Its Applications, 1999
    Co-Authors: Bero Roos
    Abstract:

    The aim of this paper is to introduce the multivariate Charlier B expansion to the metric multivariate Poisson Approximation of a generalized multinomial distribution considered by Barbour [J. Appl. Probab., 25 (1988), pp. 175--184] and Deheuvels and Pfeifer [J. Multivariate Anal., 25 (1988), pp. 65--89]. Bounds for the total variation and the point metric are given.