Compound Poisson Approximation Research Articles

Total variation distance and compound poisson approximations for random sums

We derive new upper bounds for the total variation distance between compound Poisson distributions as well as between a random sum and a compound Poisson distribution, and as a result we also obtain upper bounds for the total variation distance between compound Poisson distributions and a sum of independent random variables. These bounds are generalizations and refinements of some well-known bounds in the literature. We also derive upper bounds for the total variation distance between negative binomial distributions of order k and between the negative binomial distributions of order k and compound Poisson distributions. Upper bounds for the total variation distances between the number of success runs of length k in binary Markovian trials and its limiting distributions for several enumeration schemes, are also given.

On Poisson Approximation

The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. Among authors who contributed to the topic are Prokhorov, Kolmogorov, LeCam, Shorgin, Barbour, Hall, Deheuvels, Pfeifer, Roos, and many others. From a practical point of view, the problem has important applications in insurance, reliability theory, extreme value theory, etc. From a theoretical point of view, the topic provides insights into Kolmogorov’s problem concerning the accuracy of approximation of the distribution of a sum of independent random variables by infinitely divisible laws. The task of establishing an estimate of the accuracy of Poisson approximation with a correct (the best possible) constant at the leading term remained open for decades. We present a solution to that problem in the case where the accuracy of approximation is evaluated in terms of the point metric. We generalise and sharpen the corresponding inequalities established by preceding authors. A new result is established for the intensively studied topic of compound Poisson approximation to the distribution of a sum of integer-valued r.v.s.

Compound Poisson Approximations to Sums of Extrema of Bernoulli Variables

Compound Poisson Approximations to Sums of Extrema of Bernoulli Variables

Compound Poisson approximation

We overview the results on the topic of compound Poisson approximation to the distribution of a sum Sn=X1+⋯+Xn of (possibly dependent) random variables. We indicate a number of open problems and discuss directions of further research.

Malliavin calculus for marked binomial processes and applications

We develop stochastic analysis tools for marked binomial processes (MBP) that are the discrete analogues of the marked Poisson processes. They include in particular: (i) the statement of a chaos decomposition for square-integrable functionals of MBP, (ii) the design of a tailor-made Malliavin calculus of variations, (iii) the statement of the analogues of Stroock’s, Clark’s and Mehler’s formulas. We provide our formalism with two applications: (App1) studying the (compound) Poisson approximation of MBP functional by combining it with the Chen-Stein method and (App2) solving an optimal hedging problem in the trinomial model.

Spectral-free estimation of Lévy densities in high-frequency regime

We construct an estimator of the Lévy density of a pure jump Lévy process, possibly of infinite variation, from the discrete observation of one trajectory at high frequency. The novelty of our procedure is that we directly estimate the Lévy density relying on a pathwise strategy, whereas existing procedures rely on spectral techniques. By taking advantage of a compound Poisson approximation, we circumvent the use of spectral techniques and in particular of the Lévy–Khintchine formula. A linear wavelet estimator is built and its performance is studied in terms of Lp loss functions, p≥1, over Besov balls. We recover classical nonparametric rates for finite variation Lévy processes and for a nonparametric class of symmetric infinite variation Lévy processes. We show that the procedure is robust when the estimation set gets close to the critical value 0 and also discuss its robustness to the presence of a Brownian part.

Measuring dependence in the Wasserstein distance for Bayesian nonparametric models

The proposal and study of dependent Bayesian nonparametric models has been one of the most active research lines in the last two decades, with random vectors of measures representing a natural and popular tool to define them. Nonetheless, a principled approach to understand and quantify the associated dependence structure is still missing. We devise a general, and not model-specific, framework to achieve this task for random measure based models, which consists in: (a) quantify dependence of a random vector of probabilities in terms of closeness to exchangeability, which corresponds to the maximally dependent coupling with the same marginal distributions, that is, the comonotonic vector; (b) recast the problem in terms of the underlying random measures (in the same Fréchet class) and quantify the closeness to comonotonicity; (c) define a distance based on the Wasserstein metric, which is ideally suited for spaces of measures, to measure the dependence in a principled way. Several results, which represent the very first in the area, are obtained. In particular, useful bounds in terms of the underlying Lévy intensities are derived relying on compound Poisson approximations. These are then specialized to popular models in the Bayesian literature leading to interesting insights.

Fast and exact quantification of motif occurrences in biological sequences

BackgroundIdentification of motifs and quantification of their occurrences are important for the study of genetic diseases, gene evolution, transcription sites, and other biological mechanisms. Exact formulae for estimating count distributions of motifs under Markovian assumptions have high computational complexity and are impractical to be used on large motif sets. Approximated formulae, e.g. based on compound Poisson, are faster, but reliable p value calculation remains challenging. Here, we introduce ‘motif_prob’, a fast implementation of an exact formula for motif count distribution through progressive approximation with arbitrary precision. Our implementation speeds up the exact calculation, usually impractical, making it feasible and posit to substitute currently employed heuristics.ResultsWe implement motif_prob in both Perl and C+ + languages, using an efficient error-bound iterative process for the exact formula, providing comparison with state-of-the-art tools (e.g. MoSDi) in terms of precision, run time benchmarks, along with a real-world use case on bacterial motif characterization. Our software is able to process a million of motifs (13–31 bases) over genome lengths of 5 million bases within the minute on a regular laptop, and the run times for both the Perl and C+ + code are several orders of magnitude smaller (50–1000× faster) than MoSDi, even when using their fast compound Poisson approximation (60–120× faster). In the real-world use cases, we first show the consistency of motif_prob with MoSDi, and then how the p-value quantification is crucial for enrichment quantification when bacteria have different GC content, using motifs found in antimicrobial resistance genes. The software and the code sources are available under the MIT license at https://github.com/DataIntellSystLab/motif_prob.ConclusionsThe motif_prob software is a multi-platform and efficient open source solution for calculating exact frequency distributions of motifs. It can be integrated with motif discovery/characterization tools for quantifying enrichment and deviation from expected frequency ranges with exact p values, without loss in data processing efficiency.

Average‐tempered stable subordinators with applications

AbstractIn this article, the running average of a subordinator with a tempered stable distribution is considered. We investigate a family of previously unexplored infinite‐activity subordinators induced by the probability distribution of the running average process and determine their jump intensity measures. Special cases including gamma processes and inverse Gaussian processes are discussed. Then we derive easily implementable formulas for the distribution functions, cumulants, and moments, as well as provide explicit estimates for their asymptotic behaviors. Numerical experiments are conducted for illustrating the applicability and efficiency of the proposed formulas. Two important extensions of the running average process and its equi‐distributed subordinator are subsequently examined, with concrete applications to structural degradation modeling with memory and financial derivatives pricing in the presence of enhanced asymmetric leptokurtic feature, where their advantages relative to various existing models are highlighted together with the mention of Euler discretization and compound Poisson approximation techniques.

Compound Poisson approximation for regularly varying fields with application to sequence alignment

The article determines the asymptotic shape of the extremal clusters in stationary regularly varying random fields. To deduce this result, we present a general framework for the Poisson approximation of point processes on Polish spaces which appears to be of independent interest. We further introduce a novel and convenient concept of anchoring of the extremal clusters for regularly varying sequences and fields. Together with the Poissonian approximation theory, this allows for a concise description of the limiting behavior of random fields in this setting. We apply this theory to shed entirely new light on the classical problem of evaluating local alignments of biological sequences.

Compound Poisson Approximations in $$\ell _p$$-norm for Sums of Weakly Dependent Vectors

The distribution of the sum of 1-dependent lattice vectors with supports on coordinate axes is approximated by a multivariate compound Poisson distribution and by signed compound Poisson measure. The local and $$\ell _\alpha $$ -norms are used to obtain the error bounds. The Heinrich method is used for the proofs.

DNA Motif Match Statistics Without Poisson Approximation.

Transcription factors (TFs) play a crucial role in gene regulation by binding to specific regulatory sequences. The sequence motifs recognized by a TF can be described in terms of position frequency matrices. Searching for motif matches with a given position frequency matrix is achieved by employing a predefined score cutoff and subsequently counting the number of matches above this cutoff. In this article, we approximate the distribution of the number of motif matches based on a novel dynamic programming approach, which accounts for higher order sequence background (e.g., as is characteristic for CpG islands) and overlapping motif matches on both DNA strands. A comparison with our previously published compound Poisson approximation and a binomial approximation demonstrates that in particular for relaxed score thresholds, the dynamic programming approach yields more accurate results.

Asymptotics for the sum of three state Markov dependent random variables

The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $n\in \mathbb {N}$ periods. The accuracy of order $O(n^{-1})$ and $O(n^{-1/2})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O(n^{-1})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium.

Compound Poisson approximation of subgraph counts in stochastic block models with multiple edges

Abstract We use the Stein‒Chen method to obtain compound Poisson approximations for the distribution of the number of subgraphs in a generalised stochastic block model which are isomorphic to some fixed graph. This model generalises the classical stochastic block model to allow for the possibility of multiple edges between vertices. We treat the case that the fixed graph is a simple graph and that it has multiple edges. The former results apply when the fixed graph is a member of the class of strictly balanced graphs and the latter results apply to a suitable generalisation of this class to graphs with multiple edges. We also consider a further generalisation of the model to pseudo-graphs, which may include self-loops as well as multiple edges, and establish a parameter regime in the multiple edge stochastic block model in which Poisson approximations are valid. The results are applied to obtain Poisson and compound Poisson approximations (in different regimes) for subgraph counts in the Poisson stochastic block model and degree corrected stochastic block model of Karrer and Newman (2011).

COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

In this paper, we focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type $(a,b,0)$ class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein's identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.

Asymptotic results for the multiple scan statistic

AbstractThe contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

On magic factors in Stein’s method for compound Poisson approximation

One major obstacle in applications of Stein’s method for compound Poisson approximation is the availability of so-called magic factors (bounds on the solution of the Stein equation) with favourable dependence on the parameters of the approximating compound Poisson random variable. In general, the best such bounds have an exponential dependence on these parameters, though in certain situations better bounds are available. In this paper, we extend the region for which well-behaved magic factors are available for compound Poisson approximation in the Kolmogorov metric, allowing useful compound Poisson approximation theorems to be established in some regimes where they were previously unavailable. To illustrate the advantages offered by these new bounds, we consider applications to runs, reliability systems, Poisson mixtures and sums of independent random variables.

Refined total variation bounds in the multivariate and compound Poisson approximation

We consider the approximation of a convolution of possibly different probability measures by (compound) Poisson distributions and also by related signed measures of higher order. We present new total variation bounds having a better structure than those from the literature. A numerical example illustrates the usefulness of the bounds, and an application in the Poisson process approximation is given. The proofs use arguments from Kerstan (Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964) 173-179) and Roos (J. Multivariate Anal. 69 (1999) 120-134) in combination with new smoothness inequalities, which could be of independent interest.

Phương pháp hàm đặc trưng cho một số định lí giới hạn trong xác suất

The main purpose of this article is to use Characteristic functions method to solve some approximation problems in probability such as Compound Poisson approximation, Gamma approximation, and Laplace approximation. The received results are extensions and generalizations of some known results.

Approximation of Symmetric Three-State Markov Chain by Compound Poisson Law

The sum of symmetric three-point Markov dependent random variables is approximated by compound Poisson distribution. It is proved that the accuracy of approximation is the same as in compound Poisson approximation to the sum of independent symmetrized Bernoulli variables. Second-order approximations are constructed. The upper-bound and lower-bound estimates are obtained for the total variation, Wasserstein, and local norms.