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}.     

Explosion time for some Laplace transforms of the Wishart process

In this article, we focus upon a family of matrix valued stochastic processes and study the problem of determining the smallest time such that their Laplace transforms become infinite. In particular, we concentrate upon the class of Wishart processes, which have proved to be very useful in different applications by their ability in describing non-trivial dependence. Thanks to this remarkable property we are able to explain the behavior of the explosion times for the Laplace transforms of the Wishart process and its time integral in terms of the relative importance of the involved factors and their correlations.     

BAYESIAN ANALYSIS FOR THE SUPERPOSITION OF TWO DEPENDENT NONHOMOGENEOUS POISSON PROCESSES

ABSTRACT

A Bayesian analysis for the superposition of two dependent nonhomogenous Poisson processes is studied by means of a bivariate Poisson distribution. This particular distribution presents a new likelihood function which takes into account the correlation between the two nonhomogenous Poisson processes. A numerical example using Markov Chain Monte Carlo method with data augmentation is considered.     

ASYMPTOTIC NORMALITY OF GOODNESS-OF-FIT STATISTICS FOR SPARSE POISSON DATA

Goodness-of-fit tests for discrete data and models with parameters to be estimated are usually based on Pearson's χ2 or the Likelihood Ratio Statistic. Both are included in the family of Power-Divergence Statistics SDλ which are asymptotically χ2 distributed for the usual sampling schemes. We derive a limiting standard normal distribution for a standardization Tλ of SDλ under Poisson sampling by considering an approach with an increasing number of cells. In contrast to the χ2 asymptotics we do not require an increase of all expected values and thus meet the situation when data are sparse. Our limit result is useful even if a bootstrap test is used, because it implies that the statistic Tλ should be bootstrapped and not the sum SDλ. The peculiarity of our approach is that the models under test only specify associations. Hence we have to deal with an infinite number of nuisance parameters. We illustrate our approach with an application.     

A method for computing the autocovariance of renewal processes

In this paper we derive formulae for the autocovariance functions of renewal and renewal reward processes. The derivation is based on a Poissonization technique of a renewal process. The formulae are expressed in the form of Laplace transforms. In some cases we may invert the Laplace transforms analytically, but in general we have to invert them numerically.     

POISSON REGRESSION WITH A PERIODIC FUNCTION

ABSTRACT

Let {y_t } be a Poisson-like process with the mean μ _t which is a periodic function of time t. We discuss how to fit this type of data set using quasi-likelihood method. Our method provides a new avenue to fit a time series data when the usual assumption of stationarity and homogeneous residual variances are invalid. We show that the estimators obtained are strongly consistent and also asymptotically normal.     

LOG-LINEAR MODELS FOR MEAN AND DISPERSION IN MIXED POISSON REGRESSION MODELS

This paper is concerned with the analysis of repeated measures count data overdispersed relative to a Poisson distribution, with the overdispersion possibly heterogeneous. To accommodate the overdispersion, the Poisson random variable is compounded with a gamma random variable, and both the mean of the Poisson and the variance of the gamma are modelled using log linear models. Maximum likelihood estimates (MLE) are then obtained. The paper also gives extended quasi-likelihood estimates for a more general class of compounding distributions which are shown to be approximations to the MLEs obtained for the gamma case. The theory is illustrated by modelling the determination of asbestos fibre intensity on membrane filters mounted on microscope slides.     

Numerical method for means of linear Hawkes processes

Abstract

Linear Hawkes processes are widely used in many fields and means are the basic and critical information of them. However, there is little research on linear Hawkes processes’ means. In this paper, we present a numerical method based on the Laplace transform and inverse Laplace transform for means of linear Hawkes processes. The advantage of this method is that whatever the kernel function is, we can always obtain the numerical solutions of means for a linear Hawkes process. In addition, this numerical method provides the basic information of linear Hawkes processes by means. As an application, the numerical method is applied in a WeChat network model.     

A note on non-negative continuous time processes

Summary.  Recently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models.     

A density function connected with a non-negative self-decomposable random variable

The innovation random variable for a non-negative self-decomposable random variable can have a compound Poisson distribution. In this case, we provide the density function for the compounded variable. When it does not have a compound Poisson representation, there is a straightforward and easily available compound Poisson approximation for which the density function of the compounded variable is also available. These results can be used in the simulation of Ornstein–Uhlenbeck type processes with given marginal distributions. Previously, simulation of such processes used the inverse of the corresponding tail Lévy measure. We show this approach corresponds to the use of an inverse cdf method of a certain distribution. With knowledge of this distribution and hence density function, the sampling procedure is open to direct sampling methods.     

SOJOURN TIMES IN NON-HOMOGENEOUS QBD PROCESSES WITH PROCESSOR SHARING

We study sojourn times of customers in a processor sharing queue with a service rate that varies over time, depending on the number of customers and on the state of a random environment. An explicit expression is derived for the Laplace–Stieltjes transform of the sojourn time conditional on the state upon arrival and the amount of work brought into the system. Particular attention is paid to the conditional mean sojourn time of a customer as a function of his required amount of work, and we establish the existence of an asymptote as the amount of work tends to infinity. The method of random time change is then extended to include the possibility of a varying service rate. By means of this method, we explain the well-established proportionality between the conditional mean sojourn time and required amount of work in processor sharing queues without random environment. Based on numerical experiments, we propose an approximation for the conditional mean sojourn time. Although first presented for exponentially distributed service requirements, the analysis is shown to extend to phase-type services. The service discipline of discriminatory processor sharing is also shown to fall within the framework.     

NONPARAMETRIC META‐ANALYSIS FOR MINIMAL‐REPAIR SYSTEMS

We consider the situation that repair times of several identically structured technical systems are observed. As an example of such data we discuss the Boeing air conditioner data, consisting of successive failures of the air conditioning system of each member of a fleet of Boeing jet airplanes. The repairing process is assumed to be performed according to a minimal‐repair strategy. This reflects the idea that only those operations are accomplished that are absolutely necessary to restart the system after a failure. The ‘after‐repair‐state’ of the system is the same as it was shortly before the failure. Clearly, the observed repair times contain valuable information about the repair times of an identically structured system put into operation in the future. Thus, for statistical analysis and prediction, it is certainly favourable to take into account all repair times from each system. The resulting pooled sample is used to construct nonparametric prediction intervals for repair times of a future minimal‐repair system. To illustrate our results we apply them to the above‐mentioned data set. As expected, the maximum coverage probabilities of prediction intervals based on two samples exceed those based on one sample. We show that the relative gain for a two‐sample prediction over a one‐sample prediction can be substantial. One of the advantages of the present approach is that it allows nonparametric prediction intervals to be constructed directly. This provides a beneficial alternative to existing nonparametric methods for minimal‐repair systems that construct prediction intervals via the asymptotic distribution of quantile estimators. Moreover, the prediction intervals presented here are exact regardless of the sample size.     

First hitting time of integral diffusions and applications

We study the first hitting time of integral functionals of time-homogeneous diffusions, and characterize their Laplace transforms through a stochastic time change. We obtain explicit expressions of the Laplace transforms for the geometric Brownian motion (GBM) and the mean-reverting GBM process. We also introduce a novel probability identity based on an independent exponential randomization and obtain explicit Laplace transforms of the price of arithmetic Asian options and other derivative prices that non-linearly depend on the integral diffusions. Numerical examples are given to demonstrate the accuracy and efficiency of the proposed method.     

Bounds of moment generating functions of some life distributions

In this article we show that if a life has new better than used in expectation (NBUE) ageing property and if the mean life is finite then the moment generating function exists and is finite. In fact, the moment generating function is shown to be bounded above by that of the exponential distribution with the same mean. Analogous results are also proven for two much bigger families of life distribution, namely, the new better than renewal used in expectation (NBRUE) and the renewal new is better than used in expectation (RNBUE) and the renewal new better than renewal used in expectation (RNBRUE), provided that the life has finite two moments. Further, stronger results are also obtained for the smaller new better than used version of the above classes.     

The discrete delta and nabla Mittag-Leffler distributions

In this paper, we extend Bernstein theorem by using basic tools of calculus on time scales, and, as a further application of it, the discrete nabla and delta Mittag-Leffler distributions are introduced here with respect to their Laplace transforms on the discrete time scale. For these discrete distributions, infinite divisibility and geometric infinite divisibility are proved along with some statistical properties. The delta and nabla Mittag-Leffler processes are defined.     

Estimating the intensity function of a power law process at the current time: time truncated case

The power law process, a nonhomogeneous Poisson process with intensity function µ(t) = (β/θ)(t/θ) , is frequently used to model the occurence of events in time. Often, an important quantity is the value of the intensity function at the current time, that is, the time when data collection is ceased. In this article, the problem of estimating this quantity is addressed when the data are time truncated, that is, when data collection is stopped at a predetermined time T. The class of multiples of the conditional MLE is suggested, and some members are analyzed. In addition, the class of estimators formed by first performing a preliminary test of significance on the parameter β is analyzed. Expressions for the bias and MSE of these estimators are derived and evaluated for several values of the parameters     

A discrete distribution associated with a pure birth process

A discrete distribution associated with a pure birth process starting with no individuals, with birth rates λ ⁿ =λ forn=0, 2, …,m−1 and λ ⁿ forn≥m is considered in this paper. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions and others. Using this representation, the mean and the k-th factorial moments of the distribution are obtained. Some nice characterizations of this distribution are also given.     

New Explicit Examples of Fixed Points of Poisson Shot Noise Transforms

The main objective of this paper is to establish a close relation between the fixed points of Poisson shot noise transforms and perpetuities of a special type. With this relation it is shown that the gamma distributions, the generalized positive Linnik distributions, and the S2 distributions are fixed points of Poisson shot noise transforms. The paper also proves that log‐convexity of the response functions is not needed for non‐negative Poisson shot noise distributions to be self‐decomposable. Finally, the problems of existence and uniqueness of the above mentioned perpetuities are investigated.     

Integral Representation of a Distribution Associated with a Pure Birth Process

ABSTRACT

This article deals with a distribution associated with a pure birth process starting with no individuals, with birth rates λ _n  = λ for n = 0, 2,…, m ? 1 and λ _n  = μ for n ≥ m. The probability mass function is expressed in terms of an integral that is very convenient for computing probabilities, moments, generating functions, and others. Using this representation, the kth factorial moments of the distribution is obtained. Some other forms of this distribution are also given.