Compound Poisson Research Articles

Nonlinear Functional Analysis of Boundary Value Problems 2013

and Applied Analysis 3 solve a class of boundary value problems for second-order singular differential equations. (iii) In the paper titled “Existence and uniqueness of solution to nonlinear boundary value problems with sign-changing Green’s function,” by using the cone theory and the Banach contraction mapping principle, some existence and uniqueness results are established for a nonlinear higher-order differential equation boundary value problem with signchanging Green’s function. (iv) In the paper titled “Onboundedness and attractiveness of nonlinear switched delay systems,” the authors study the boundedness and attractiveness problems for a class of nonlinear switched delay systems. Some sufficient conditions are established to guarantee the system’s boundedness. The authors also work out the region where the solution will remain and the relationship between the initial function and the bounded region. (v) In the paper titled “Unbounded solutions of asymmetric oscillator,” the authors establish sufficient conditions for the existence of unbounded solutions of a nonlinear differential equation with a continuous, bounded, and periodic source function. (6) Stochastic Differential Equation Models (i) In the paper titled “A stochastic string with a compound poisson process,” the authors introduce a compound Poisson process with a constant jump intensity and random jump-size to capture information burst and the resulting discontinuous path and derive the no-arbitrage condition on the drift of instantaneous forward rates in the compound model. With the proposedmodel, the impact of random jump on the price of the zero coupon bond is shown in the paper. (ii) In the paper titled “Equilibrium asset and option pricing under jump diffusion model with stochastic volatility,” the authors study the equity premium and option pricing under a jump-diffusion model with stochastic volatility. The pricing kernel is established and the exact expression for the option price is derived by using the Fourier transformation method. (7) Fixed-Point Theory (i) In the paper titled “Generalizations of fixed-point theorems of Altman and Rothe types,” the authors present some extensions of the Altman and Rothe type fixed-point theorems. Some new fixed-point theorems for continuous operators are established by use of the theory of topological degree. (ii) In the paper titled “Commonfixed-points for weak contractive mappings in ordered metric space with applications,” the authors establish some new common fixed-point theorems satisfying a weak contractive condition in the framework of partially ordered metric spaces. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

Normal and compound poisson approximations for pattern occurrences in NGS reads.

Next generation sequencing (NGS) technologies are now widely used in many biological studies. In NGS, sequence reads are randomly sampled from the genome sequence of interest. Most computational approaches for NGS data first map the reads to the genome and then analyze the data based on the mapped reads. Since many organisms have unknown genome sequences and many reads cannot be uniquely mapped to the genomes even if the genome sequences are known, alternative analytical methods are needed for the study of NGS data. Here we suggest using word patterns to analyze NGS data. Word pattern counting (the study of the probabilistic distribution of the number of occurrences of word patterns in one or multiple long sequences) has played an important role in molecular sequence analysis. However, no studies are available on the distribution of the number of occurrences of word patterns in NGS reads. In this article, we build probabilistic models for the background sequence and the sampling process of the sequence reads from the genome. Based on the models, we provide normal and compound Poisson approximations for the number of occurrences of word patterns from the sequence reads, with bounds on the approximation error. The main challenge is to consider the randomness in generating the long background sequence, as well as in the sampling of the reads using NGS. We show the accuracy of these approximations under a variety of conditions for different patterns with various characteristics. Under realistic assumptions, the compound Poisson approximation seems to outperform the normal approximation in most situations. These approximate distributions can be used to evaluate the statistical significance of the occurrence of patterns from NGS data. The theory and the computational algorithm for calculating the approximate distributions are then used to analyze ChIP-Seq data using transcription factor GABP. Software is available online (www-rcf.usc.edu/∼fsun/Programs/NGS_motif_power/NGS_motif_power.html). In addition, Supplementary Material can be found online (www.liebertonline.com/cmb).

Asymptotic optimization for stochastic models based on a compound poisson process

This paper considers the possibility of using a multidimensional compound Poisson process controlled by a continuous-time Markov chain in mathematical decision-making models. A definition of this process is presented and examples are given that illustrate its use in formalizing the concepts of "uncertainty" and "risk" and constructing risk functions and objective functions for the corresponding optimization problems. Some approaches are proposed to solve these problems, in particular, a direct analytic approach consisting of finding explicit formulas for a risk function and a method of approximate solution based on limit theorems of stochastic process theory.

Note on nonuniform estimate for compound poisson approximations to 2-runs

We obtain nonuniform estimates depending on the interval length for Poisson and compound Poisson approximations to 2-runs. We calculate exact constants for the estimates in the Wasserstein metric.

Random partitioning over a sparse contingency table

The present article investigates a class of random partitioning distributions of a positive integer. This class is called the limiting conditional compound poisson (LCCP) distribution and characterized by the law of small numbers. Accordingly the LCCP distribution explains the limiting behavior of counts on a sparse contingency table by the frequencies of frequencies. The LCCP distribution is constructed via some combinations of conditioning and limiting, and this view reveals that the LCCP distribution is a subclass of several known classes that depend on a Bell polynomial. It follows that the limiting behavior of a Bell polynomial provides new asymptotics for a sparse contingency table. Also the Neyman Type A distribution and the Thomas distribution are revisited as the basis of the sparsity.

Compound poisson approximations for sums of 1-dependent random variables. II

Compound poisson approximations for sums of 1-dependent random variables. II

Minimizing Soft Errors in TCAM Devices: A Probabilistic Approach to Determining Scrubbing Intervals

Ternary content addressable memory (TCAM) is more susceptible to soft errors than static random access memory (SRAM). The large di/dt issue during comparison operation reduces operating voltage ranges, which in turn reduces soft error immunity. The tight structural coupling of TCAM comparison circuits and memory cells does not allow for an interleaving design scheme in mitigating soft errors. Regular scrubbing of stored content can greatly mitigate the reliability issue caused by soft errors. However, frequent scrubbing can also affect device performance. The scrubbing interval should be determined to facilitate both reliability and performance. This paper proposes a novel, model-based approach that includes both single-bit upsets (SBUs) and multi-cell upsets (MCUs) to determine the scrubbing interval by predictive and probabilistic failure rate analysis. This model uses the compound Poisson (CP) process to count clustered random events, which are common phenomena of soft errors in technologies that use chips under 90 nm. The 20 M TCAM with 90-nm CMOS technology was tested with 180-MeV neutron strikes. The scrubbing interval determined based on the proposed model is applied to the TCAM test results. The failure probabilities based on the CP model showed 31% overestimation on average compared to the same from the test data. Such overestimation is mainly due to the independent upset assumption in the proposed model and can enable use of the model as worst case analysis. The worst case comparison with the test data showed 1.7% overestimation, which can tell the proposed model is effective in predicting upper-bound soft error reliability.

Modeling heterogeneity for bivariate survival data by the compound poisson distribution

We propose bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by compound Poisson distribution. We assume that the bivariate survival data follow bivariate Weibull of Hanagal (9). There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behavior (which is unknown and random) of patients follows a known frailty distribution. These are the situations which motivate to study this particular model. We propose two stage maximum likelihood estimation procedure for the parameters in the proposed model and develop large sample tests for testing no frailty and significance of regression parameters.

A note on the perturbed compound poisson risk model with a threshold dividend strategy

In this paper, we consider the Perturbed Compound Poisson Risk Model with a threshold dividend strategy (PCT). Integro-differential equations (IDE) for its Gerber-Shiu functions and dividend payments function are stated. We maily focus on deriving the boundary conditions to solve these equations.

Regression models for infant mortality data in Norwegian siblings, using a compound Poisson frailty distribution with random scale

The power variance function distributions, which include the gamma and compound Poisson (CP) distributions among others, are commonly used in frailty models for family data. In a previous paper, we presented a frailty model constructed by randomizing the scale parameter in a CP distribution. When combined with a parametric baseline hazard, this yields a model with heterogeneity on both the individual and the family level and a subgroup with zero frailty, corresponding to people not experiencing the event. In this paper, we discuss covariates in the model. Depending on where the covariates are inserted in the model, one may have proportional hazards at the individual level, the family level, and a larger group level (for covariates shared by many families, e.g. ethnic groups) or get accelerated failure times. Each of these alternatives gives a specific interpretation of the covariate effects. An application to data infant mortality in siblings from the Medical Birth Registry of Norway is included. We compare the results for some of the different covariate modeling options.

Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X +(u) be the risk reserve just before ruin, and Y +(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function m δ(u) =E[e−δ τ(u) w(X +(u), Y +(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that m δ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for m δ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired m δ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

Approximations for the Gerber-Shiu expected discounted penalty function in the compound poisson risk model

In the classical risk model with initial capital u, let τ(u) be the time of ruin, X+(u) be the risk reserve just before ruin, and Y+(u) be the deficit at ruin. Gerber and Shiu (1998) defined the function mδ(u) =E[e−δ τ(u)w(X+(u), Y+(u)) 1 (τ(u) < ∞)], where δ ≥ 0 can be interpreted as a force of interest and w(r,s) as a penalty function, meaning that mδ(u) is the expected discounted penalty payable at ruin. This function is known to satisfy a defective renewal equation, but easy explicit formulae for mδ(u) are only available for certain special cases for the claim size distribution. Approximations thus arise by approximating the desired mδ(u) by that associated with one of these special cases. In this paper a functional approach is taken, giving rise to first-order correction terms for the above approximations.

A NOTE VIA DIAGONALITY OF THE 2 ◊ 2 BHATTACHARYYA MATRICES

In this paper, we consider characterizations based on the Bhattacharyya matrices. We characterize, under certain constraint, dis- tributions such as normal, compound poisson and gamma via the diago- nality of the 2 ◊ 2 Bhattacharyya matrix.

The distribution of the dividend payments in the compound poisson risk model perturbed by diffusion

We consider a diffusion perturbed classical compound Poisson risk model in the presence of a constant dividend barrier. An integro-differential equation with certain boundary conditions for the n-th moment of the discounted dividend payments prior to ruin is derived and solved. Its solution can be expressed in terms of the expected discounted penalty (Gerber-Shiu) functions due to oscillation in the corresponding perturbed risk model without a barrier. When the discount factor δ is zero, we show that all the results can be expressed in terms of the non-ruin probability in the perturbed risk model without a barrier.

Decompounding poisson random sums: Recursively truncated estimates in the discrete case

Given a sample from a discrete compound Poisson distribution, we consider variants of plug-in and likelihood estimators for the corresponding base distribution. These proceed recursively with an intermediate truncation step. We discuss the asymptotic behaviour of the estimators and give some numerical examples. Both procedures compare favourably with the straightforward and the naively projected plug-in estimator that we introduced in Buchmann and Grubel (2003,The Annals of Statistics,31, 1054–1074).

First-exit times for compound poisson processes for some types of positive and negative jumps

We consider the one-sided and the two-sided first-exit problem for a compound Poisson process with linear deterministic decrease between positive and negative jumps. This process (X(t)) t≥0 occurs as the workload process of a single-server queueing system with random workload removal, which we denote by M/G u /G d /1, where G u (G d ) stands for the distribution of the upward (downward) jumps; other applications are to cash management, dams, and several related fields. Under various conditions on G u and G d (assuming e.g. that one of them is hyperexponential, Erlang or Coxian), we derive the joint distribution of τ y =inf{t≥0|X(t)∉(0,y)}, y>0, and X(τ y ) as well as that of T=inf{t≥0|X(t)≤0} and X(T). We also determine the distribution of sup{X(t)|0≤t≤T}.

SIMULATING LEVEL CROSSING PROBABILITIES BY IMPORTANCE SAMPLING FOR NON-DECREASING COMPOUND POISSON PROCESSES WITH BOUNDED JUMPS AND A NEGATIVE DRIFT

Article SIMULATING LEVEL CROSSING PROBABILITIES BY IMPORTANCE SAMPLING FOR NON-DECREASING COMPOUND POISSON PROCESSES WITH BOUNDED JUMPS AND A NEGATIVE DRIFT was published on February 1, 2001 in the journal Statistics & Risk Modeling (volume 19, issue 2).

A compound poisson process for relaxing the molecular clock.

The molecular clock hypothesis remains an important conceptual and analytical tool in evolutionary biology despite the repeated observation that the clock hypothesis does not perfectly explain observed DNA sequence variation. We introduce a parametric model that relaxes the molecular clock by allowing rates to vary across lineages according to a compound Poisson process. Events of substitution rate change are placed onto a phylogenetic tree according to a Poisson process. When an event of substitution rate change occurs, the current rate of substitution is modified by a gamma-distributed random variable. Parameters of the model can be estimated using Bayesian inference. We use Markov chain Monte Carlo integration to evaluate the posterior probability distribution because the posterior probability involves high dimensional integrals and summations. Specifically, we use the Metropolis-Hastings-Green algorithm with 11 different move types to evaluate the posterior distribution. We demonstrate the method by analyzing a complete mtDNA sequence data set from 23 mammals. The model presented here has several potential advantages over other models that have been proposed to relax the clock because it is parametric and does not assume that rates change only at speciation events. This model should prove useful for estimating divergence times when substitution rates vary across lineages.

A random review replacement model for a system subject to compound poisson shocks

We present a transform–free analysis of the following model. The state of the system is initially 0 and thereafter increases jumpwise due to compound Poisson shocks. Each shock increases the state by a random amount. The system is inspected at random points in time. If the state is above a threshold at an inspection, the system is replaced, otherwise no action is taken. Each replacement instantaneously brings the state back to 0. (Existing models assume either exponential interinspection times or discrete shock magnitudes.) This model can be applied to reliability, inventory, and queueing problems.Interpretations are given throughout to make the results easier to understand and to apply

The infinite server queue with arrivals generated by a non-homogeneous compound poisson process and heterogeneous customers

We consider the infinite server queue where customers arrive in groups of various sizes according to a non-homogeneous Poisson process. The customers in each group may be of different types and also may have dependent service times. Assuming that the system is initially empty, we obtain the probability generating function of the transient distribution of the numbers of customers of various types present in the system, thus extending recent relevant results. At the same time the probability generating function of the corresponding departure counting process is derived as well.