Generalized poisson distributions

Poisson distributions evidently can be treated in a unified manner by dealing with the G.P. which contains a characteristic parameterq. The properties of the PEBL, long recognized as significant in a variety of statistical applications like telephone trunking, queueing, etc., may be deduced by passage to the limitq = 1.     

Approximation of aggregate claims distributions by compound poisson distributions

New error bounds are derived for the approximation of aggregate claims distributions by compound Poisson distributions. These approximations can be recommended in most cases in which the normal approximation fails.     

Exponential bounds for generalized poisson distributions

Exponential estimates are constructed for the probabilities of large deviations of Poisson random sums. Supported by the Russian Foundation for Fundamental Research (grant No. 97-01-00271) and the Russian Humanitarian Science Foundation (grant No. 97-02-02235). Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow, Russia, 1996, Part II.     

Improved recursions for some compound poisson distributions

Calculations by means of the well-known recursion for the compound Poisson distribution are in general time-consuming since each probability depends on all the preceding ones. It is shown that for some claim size distributions a more efficient recursion can be derived.     

Further modified forms of binomial and poisson distributions

Summary Some new type of modifications of binomial and Poisson distributions, are discussed. First, we consider Bernoulli trials of lengthn with success ratep up to time whenm times of successes occur, and then, changing the success rate to γp, we continue the remaining trial. The distribution of number of successes is called the modified binomial distribution. The Poisson limit (n tends to infinity andp tends to 0, keepingnp=λ) of the modified binomial is called the modified Poisson distribution. The probability functions of modified binomial and Poisson distributions are given (Section 1). A new concept of (m, γ)-modification is introduced and fundamental theorem which gives the relations between the factorial moments of any probability function and the factorial moments of its (m, γ)-modification, is presented. Then some lower order moments of the modified binomial and Poisson distributions are given explicitly (Section 2). The modified Poisson ofm=2 is fitted to the distribution of number of children for Japanese women in some age group. The fitting procedure is also presented (Section 3). Some historical sketch concerning the modification and generalization of binomial and Poisson distributions is given in Appendix. The Institute of Statistical Mathematics     

ESTIMATING OF ORDER-RESTRICTED LOCATION PARAMETERS OF TWO-EXPONENTIAL DISTRIBUTIONS UNDER MULTIPLE TYPE- Ⅱ CENSORING

In this article, Bayes estimation of location parameters under restriction is broughtforth. Since Bayes estimator is closely connected with the first value of order statistics that canbe observed, it is possible to consider “complete data” method, through which the pseudo-value of first order statistics and pseudo-right censored samples can he obtained. Thus the results under Type- Ⅱ right censoring can be used directly to get more accurate estimators by Bayes method.     

Estimating parameters in one-way analysis of covariance model with short-tailed symmetric error distributions

We consider one-way analysis of covariance (ANCOVA) model with a single covariate when the distribution of error terms are short-tailed symmetric. The maximum likelihood (ML) estimators of the parameters are intractable. We, therefore, employ a simple method known as modified maximum likelihood (MML) to derive the estimators of the model parameters. The method is based on linearization of the intractable terms in likelihood equations. Incorporating these linearizations in the maximum likelihood, we get the modified likelihood equations. Then the MML estimators which are the solutions of these modified equations are obtained. Computer simulations were performed to investigate the efficiencies of the proposed estimators. The simulation results show that the proposed estimators are remarkably efficient compared with the conventional least squares (LS) estimators.     

Bounds for the difference between median and mean of gamma and poisson distributions

We use elementary methods and the Poisson-Gamma relationship to obtain bounds for the difference between median and mean of Gamma and Poisson distributions.     

Some properties of a two-parameter family of poisson distributions of orderk

We study the problem of simulating the process of detecting a weighted sum of independent Poisson random variables. We investigate the properties of the resulting compound Poisson distribution: the analytic form of the probability function and recursion formulas for computing it, moments and semi-invariants, asymptotics of the distribution, and recursion relations for the derivatives with respect to the parameters. We give the results of model computations showing the set structure of the distribution. One figure. Bibliography: 8 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 46–54.     

Estimating a bounded parameter for symmetric distributions

For the problem of estimating under squared error loss the parameter of a symmetric distribution which is subject to an interval constraint, we develop general theory which provides improvements on various types of inadmissible procedures, such as maximum likelihood procedures. The applications and further developments given include: (i) symmetric location families such as the exponential power family including double-exponential and normal, Student and Cauchy, a Logistic type family, and scale mixture of normals in cases where the variance is lower bounded; (ii) symmetric exponential families such as those related to a Binomial(n,p) model with bounded |p−1/2| and to a Beta(α + θ, α −θ) model; and (iii) symmetric location distributions truncated to an interval (−c,c). Finally, several of the dominance results are studied with respect to model departures yielding robustness results, and specific findings are given for scale mixture of normals and truncated distributions. Research supported by NSERC of Canada.     

MULTIVARIATE SURVIVAL DISTRIBUTIONS OF AGE AND RESIDUAL LIFETIME PROCESSES IN NONHOMOGENEOUS POISSON PROCESS

Let{N(t),t≥0} be the nonhomogeneous Poisson process with cumulative intensity parameter Λ(t),{δt,t≥0} the age process,and {yi,t≥0) the residual lifetime process.In the present paper the expressions of n-dimensional survival distribution functions of the processes{δi} and {yi}, and their Lebesgue decompositions are derived.     

Estimating cumulative incidence functions when the life distributions are constrained

In competing risks studies, the Kaplan-Meier estimators of the distribution functions (DFs) of lifetimes and the corresponding estimators of cumulative incidence functions (CIFs) are used widely when no prior information is available for these distributions. In some cases better estimators of the DFs of lifetimes are available when they obey some inequality constraints, e.g., if two lifetimes are stochastically or uniformly stochastically ordered, or some functional of a DF obeys an inequality in an empirical likelihood estimation procedure. If the restricted estimator of a lifetime differs from the unrestricted one, then the usual estimators of the CIFs will not add up to the lifetime estimator. In this paper we show how to estimate the CIFs in this case. These estimators are shown to be strongly uniformly consistent. In all cases we consider, when the inequality constraints are strict the asymptotic properties of the restricted and the unrestricted estimators are the same, thus providing the asymptotic properties of the restricted estimators essentially “free of charge”. We give an example to illustrate our procedure.     

Estimating the demand distributions of single-period items having frequent stockouts

Very often, the service level of a single-period newsboy-type product is set at such a low level that: (i) stockouts occur in the majority of the periods, and (ii) a large right-hand side of the empirical demand distribution is never observable. This paper reports a practical approach for estimating the periodic-demand distribution of such a product. The approach has three components: (i) using the non-parametric ‘product limit’ method to estimate the fractiles of the observable left-hand side of the empirical distribution; (ii) using a subjective approach and an ‘extrapolation of hourly sales’ approach to ‘fill in’ the missing right-hand side of the empirical distribution; (iii) fitting the estimates obtained in the preceding two components to a Tocher curve — which can handle the diversity of shapes of a realistic demand distribution and is also computationally very convenient for subsequent calculations for production/inventory decisions. The entire approach is shown to be simpler but more powerful than existing alternatives for the problem.     

Transform-free equations for the stationary waiting time distributions in the queue with poisson arrivals and bulk services

We study a queue with Poisson arrivals and a bulk service rule with two thresholdsN andm on the group sizes. No group of fewer thanN or more thanm customers is served. When the number of available customers is betweenN andm, all customers are served together. The principal results deal with the joint stationary distribution of the waiting time of an arriving customer and of the size of the group in which he is eventually served. After prior computation of the stationary queue length density, the evaluation of the waiting time distribution is reduced to the solution of a system of linear differential equations and a single integral equation. The process describing the waiting time is in general non-Markovian.University of Delaware, Applied Mathematics Institute, Technical Report No. 95 B, May 1984.This research was supported by Grant No. ECS-8205404 of the National Science Foundation and by a Senior U.S. Scientist Award of the Alexander von Humboldt Foundation.     

Estimating the parameters of a truncated normal distribution

Let ø(x) be a truncated normal pdf over the interval [a,b], that is, assume ø(x)=exp[-(x–μ)²/2σ²]/∝^b_aexp[-(x–μ)²/2σ²]dx for - ∞<a?x?b?< + ∞ and zero elsewhere. Suppose that X₁,X₂,…,X_n is a random sample of size n from this truncated distribution. Using known properties of exponential families of distributions and the system of Legendre polynomials over the interval [-1,1], we examine the maximum likelihood estimation of the parameters μ and σ².