Compound Poisson Approximation Research Articles

Approximations for a conditional two-dimensional scan statistic

Let X i, j ,1⩽ i⩽ n 1,1⩽ j⩽ n 2, be a sequence of independent and identically distributed nonnegative integer valued random variables. The observation X i, j denotes the number of events that have occurred in the i, jth location in a two dimensional rectangular region R. For 2⩽m i⩽n i−1, i=1,2 , the two dimensional discrete scan statistic is defined as the maximum number of events in any of the m 1 by m 2 consecutive rectangular windows in that region. Conditional on the total number of events that have occurred in R, we refer to this scan statistic as the conditional two-dimensional scan statistic. Two-dimensional scan statistics have been extensively used in many areas of science to analyze the occurrence of observed clusters of events in space. Since this scan statistic is based on highly dependent consecutive subsequences of observed data, accurate approximations for its distributions are of great value. In this article, based on the scanning window representation of the scan statistic, accurate product-type, Poisson and a compound Poisson approximations are investigated. Moreover, accurate approximations for the expected size of the scan statistic are derived. Numerical results are presented to evaluate the performance of the approximations discussed in this article.

Total variation asymptotics for sums of independent integer random variables

Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the $Z_j:$ an appropriate number of moments and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for signed compound Poisson approximation.

Remarks on compound Poisson approximation of Gaussian random sequences

Let {X i, i⩾1} be a sequence of m-dependent stationary standard Gaussian random variables and u n, n∈ N some positive constants. In this note we generalise results of Raab (Extremes 1(3) (1999) 29.), who considered compound Poisson approximation for W n=∑ i=1 n 1 {X i>u n} the number of exceedances above the level u n . More precisely, the main result concerns an upper asymptotic bound for the total variational distance d TV (W n, CP(λ ∗)) where CP(λ ∗) = d N 1+2N 2+⋯+rN r, with 2⩽ r⩽2 m and N i = d Poi(λ i), λ i>0 are independent Poisson random variables.

Compound Poisson approximation for the distribution of extremes

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

Compound Poisson approximation for the distribution of extremes

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

Estimate of the Accuracy of the Compound Poisson Approximation for the Distribution of the Number of Matching Patterns

Let $X_1,\ldots,X_m$ and $Y_1,\ldots,Y_n$ be two sequences of independent identically distributed random variables taking on values $1, 2,\ldots\;$. By means of a particular version of the Stein method we construct an estimate of the accuracy of approximation for the distribution of the number of matching patterns of outcomes $X_i,\ldots,X_{i+s-1}$ of a given length~s in the first sequence with the patterns of outcomes $Y_j,\ldots,Y_{j+s-1}$ in the second sequence. The approximating distribution is the distribution of the sum of Poisson number of independent random variables with geometric distribution.

Compound Poisson Approximation for Multiple Runs in a Markov Chain

We consider a sequence X1, ..., Xn of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r ≥ 1. We study the overlapping appearances of runs of ki consecutive i's, for all i = 1, ..., r, in the sequence X1,..., Xn. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.

Modeling claim exceedances over thresholds

In this paper, we consider a simple risk model and study the occurrences of clusters of threshold exceedances by the individual claims. The statistic used to study the model is the discrete multiple scan statistic. A compound Poisson approximation is established and certain asymptotic results are obtained for both the risk model and a similar in nature financial problem. Finally, we review two typical examples from areas of applied science where the outcomes of this paper may have beneficial impact.

On the number of overflown urns and excess balls in an allocation model with limited urn capacity

In the present paper, we consider an equiprobable allocation urn model with m urns of limited capacity and study the distribution of the number of overflown urns and the number of excess balls (i.e those assigned to already filled up urns) after n balls have been assigned. The exact distribution of the variables of interest are studied by a Markov chain imbedding method; a Poisson and compound Poisson approximation is discussed as well along with estimates of their rates of convergence. Certain limit laws as n, m→∞ are also established under appropriate conditions on n, m and urns’ capacities.

Compound Poisson approximation: a user's guide

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecu- lar sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration.We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods.

Compound Poisson approximation for long increasing sequences

Consider a sequence X 1,…,X n of independent random variables with the same continuous distribution and the event X i-r+1 < ⋯ < X i of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

Compound Poisson approximation for long increasing sequences

Consider a sequence X1,…,Xn of independent random variables with the same continuous distribution and the event Xi-r+1 < ⋯ < Xi of the appearance of an increasing sequence with length r, for i=r,…,n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend.

On the Number of Lost Customers in Stationary Loss Systems in the Light Traffic Case

For the stationary loss systems M/M/m/K and GI/M/m/K, we study two quantities: the number of lost customers during the time interval (0,t] (the first system only), and the number of lost customers among the first n customers to arrive (both systems). We derive explicit bounds for the total variation distances between the distributions of these quantities and compound Poisson–geometric distributions. The bounds are small in the light traffic case, i.e., when the loss of a customer is a “rare” event. To prove our results, we show that the studied quantities can be interpreted as accumulated rewards of stationary renewal reward processes, embedded into the queue length process or the process of queue lengths immediately before arrivals of new customers, and apply general results by Erhardsson on compound Poisson approximation for renewal reward processes.

A bound for the distribution of the sum of discrete associated or negatively associated random variables

Let $X_1, X_2,\dots, X_n$ be a sequence of integer-valued random variables that are either associated or negatively associated.We present a simple upper bound for the distance between the distribution of the sumof $X_i$ and a sum of $n$ independent randomvariables with the same marginals as $X_i$. An upper bound useful for establishing a compound Poisson approximation for $\Sigma_{i=1}^nX_i$ is also provided. The new bounds are of the same order as the much acclaimed Stein–Chen bound.

On Compound Poisson Approximation for Sequence Matching

Consider sequences {Xi}mi=1 and {Yj}nj=1 of independent random variables, taking values in a finite alphabet, and assume that the variables X1, X2, … and Y1, Y2, … follow the distributions μ and v, respectively. Two variables Xi and Yj are said to match if Xi = Yj. Let the number of matching subsequences of length k between the two sequences, when r, 0 [les ] r < k, mismatches are allowed, be denoted by W.In this paper we use Stein's method to bound the total variation distance between the distribution of W and a suitably chosen compound Poisson distribution. To derive rates of convergence, the case where E[W] stays bounded away from infinity, and the case where E[W] → ∞ as m, n → ∞, have to be treated separately. Under the assumption that ln n/ln(mn) → ρ ∈ (0, 1), we give conditions on the rate at which k → ∞, and on the distributions μ and v, for which the variation distance tends to zero.

Estimating Stein's Constants for Compound Poisson Approximation

Stein's method for compound Poisson approximation was introduced by Barbour, Chen and Loh. One difficulty in applying the method is that the bounds on the solutions of the Stein equation are by no means as good as for Poisson approximation. We show that, for the Kolmogorov metric and under a condition on the parameters of the approximating compound Poisson distribution, bounds comparable with those obtained for the Poisson distribution can be recovered.

On stationary renewal reward processes where most rewards are zero

We consider a stationary version of a renewal reward process, i.e., a renewal process where a random variable called a reward is associated with each renewal. The rewards are nonnegative and I.I.D., but each reward may depend on the distance to the next renewal. We give an explicit bound for the total variation distance between the distribution of the accumulated reward over the interval (0,L] and a compound Poisson distribution. The bound depends in its simplest form only on the first two joint moments of T and Y (or I{Y > 0}), where T is the distance between successive renewals and Y is the reward. If T and Y are independent, and LE(Y) (or LP(Y > 0)) is bounded or Y binary valued, then the bound is O(E(Y)) as E(Y) → 0 (or O(P(Y > 0)) as P(Y > 0) → 0). To prove our result we generalize a Poisson approximation theorem for point processes by Barbour and Brown, derived using Stein's method and Palm theory, to the case of compound Poisson approximation, and combine this theorem with suitable couplings.

Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains

Compound Poisson approximation for counts of rare patterns in Markov chains and extreme sojourns in birth-death chains

Compound Poisson approximation and the clustering of random points

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλ i be decreasing.

Compound Poisson approximation and the clustering of random points

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλi be decreasing.