Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.     

Contre-exemple au théorème de P.C. Consul sur la factorisation des distributions de Poisson généralisées

This note exhibits two independent random variables on integers, X₁ and X₂, such that neither X₁ nor X₂ has a generalized Poisson distribution, but X₁ + X₂ has. This contradicts statements made by Professor Consul in his recent book.     

Modified slashed generalized exponential distribution

Abstract

In this article, we introduce a new distribution for modeling positive data sets with high kurtosis, the modified slashed generalized exponential distribution. The new model can be seen as a modified version of the slashed generalized exponential distribution. It arises as a quotient of two independent random variables, one being a generalized exponential distribution in the numerator and a power of the exponential distribution in the denominator. We studied various structural properties (such as the stochastic representation, density function, hazard rate function and moments) and discuss moment and maximum likelihood estimating approaches. Two real data sets are considered in which the utility of the new model in the analysis with high kurtosis is illustrated.     

Property of bivariate poisson distribution and its application to stochastic processes

Assuming that the random vectors X ₁ and X ₂ have independent bivariate Poisson distributions, the conditional distribution of X ₁ given X ₁?+?X ₂?=?n is obtained. The conditional distribution turns out to be a finite mixture of distributions involving univariate binomial distributions and the mixing proportions are based on a bivariate Poisson (BVP) distribution. The result is used to establish two properties of a bivariate Poisson stochastic process which are the bivariate extensions of the properties for a Poisson process given by Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, Academic Press, New York.     

The distribution of the product of two independent generalized trapezoidal random variables

ABSTRACT

In this article, we derive the probability density function (pdf) of the product of two independent generalized trapezoidal random variables having different supports, in closed form, by considering all possible cases. We also show that the results for the product of two triangular and uniform random variables follow as special cases of our main result. As an illustration, we obtain pdf of product for a suitably constrained set of parameters and plot some graphs using MATLAB, which express variation in pdf with change in different parameters of the generalized trapezoidal distribution.     

Complete convergence for arrays of row-wise ND random variables under sub-linear expectations

In this paper, complete convergence for arrays of row-wise ND random variables under sub-linear expectations is studied. As applications, the complete convergence theorems of weighted sums for negatively dependent random variables have been generalized to the sub-linear expectation space context. We extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Ko.     

Analysis of survival data by an exponential-generalized Poisson distribution

In this paper, we consider the distribution of life length of a series system with random number of components, say Z. Considering the distribution of Z as generalized Poisson, an exponential-generalized Poisson (EGP) distribution is developed. The generalized Poisson distribution is a generalization of the Poisson distribution having one extra parameter. The structural properties of the resulting distribution are presented and the maximum likelihood estimation of the parameters is investigated. Extensive simulation studies are carried out to study the performance of the estimates. The score test is developed to test the importance of the extra parameter. For illustration, two real data sets are examined and it is shown that the EGP model, presented here, fits better than the exponential–Poisson distribution.     

Order statistics from non-identical power function random variables

In this paper, we derive some recurrence relations satisfied by the single and the product moments of order statistics arising from n independent and non-identically distributed power function random variables. These recurrence relations will enable one to compute all the single and the product moments of all order statistics in a simple recursive manner. The results for the multiple-outlier model are deduced as special cases. The results are further generalized to the case of truncated power function random variables.     

Comparison of Records and Maxima

We establish best upper bounds on the expected differences of records and sample maxima, and kth records and kth maxima based on sequences of independent random variables with identical continuous distribution and finite variance. The bounds are expressed in terms of the standard deviation units of the parent distribution. We also provide conditions for attaining the bounds.     

Estimation of the Reliability of a Stress-Strength System from Power Lindley Distributions

In this paper, we are interested in the estimation of the reliability parameter R = P(X > Y) where X, a component strength, and Y, a component stress, are independent power Lindley random variables. The point and interval estimation of R, based on maximum likelihood, nonparametric and parametric bootstrap methods, are developed. The performance of the point estimate and confidence interval of R under the considered estimation methods is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the proposed procedure.     

A note on infinite-armed Bernoulli bandit problems with generalized beta prior distributions

A bandit problem with infinitely many Bernoulli arms is considered. The parameters of Bernoulli arms are independent and identically distributed random variables from a generalized beta distributionG3B(a, b, λ) witha, b>0 and 0<λ<2. Under the generalized beta prior distributions, we first derive the asymptotic expected failure rates ofk-failure strategies, and then obtain a lower bound for the expected failure rate over all strategies investigated in Berry et al. (1997). The asymptotic expected failure rates for the other three strategies studied in Berry et al. (1997) are also included. Numerical estimations for a variety of generalized beta prior distributions are presented to illustrate the performances of these strategies.     

A multivariate extension of poisson's theorem

For n ≥ 1, let X_nl,…, X_nn be independent integer-valued random variables, and define S_n = X_nl+···+X_nn. In a recent paper, we obtained a simple proof for the convergence of the distribution of S_n to a Poisson distribution under very general conditions. In this paper, we extend that result to the multidimensional case.     

Sum,product and ratio of Pareto and gamma variables

The exact distributions of X+Y, X Y and X/(X+Y) are studied when X and Y are independent Pareto and gamma random variables. Applications are discussed, to real problems in clinical trials, computer networks and economics.     

On some moving-average processes with exponentially distributed innovations

Summary In this paper we discusse the stationary sequence of random variables which are formed from an independent identically distributed sequence, according to the moving-average model of ordern. Some properties of the process are considered. The joint bivariate exponential distribution is given, as well as the distribution of the sum.     

Estimation of P(Y < X) for the Three-Parameter Generalized Exponential Distribution

This article considers the estimation of R = P(Y < X) when X and Y are distributed as two independent three-parameter generalized exponential (GE) random variables with different shape parameters but having the same location and scale parameters. A modified maximum likelihood method and a Bayesian technique are used to estimate R on the basis of independent complete samples. The Bayes estimator cannot be obtained in explicit form, and therefore it has been determined using an importance sampling procedure. An analysis of a real life data set is presented for illustrative purposes.     

The type I distribution of the ratio of independent “Weibullized” generalized beta-prime variables

In this paper we introduce the distribution of , with c >  0, where X _i , i =  1, 2, are independent generalized beta-prime-distributed random variables, and establish a closed form expression of its density. This distribution has as its limiting case the generalized beta type I distribution recently introduced by Nadarajah and Kotz (2004). Due to the presence of several parameters the density can take a wide variety of shapes.      

Classes of power semicircle laws that are randomly weighted average distributions

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n independent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of size n?1, for each n. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.     

Integer-valued autoregressive models for counts showing underdispersion

The Poisson distribution is a simple and popular model for count-data random variables, but it suffers from the equidispersion requirement, which is often not met in practice. While models for overdispersed counts have been discussed intensively in the literature, the opposite phenomenon, underdispersion, has received only little attention, especially in a time series context. We start with a detailed survey of distribution models allowing for underdispersion, discuss their properties and highlight possible disadvantages. After having identified two model families with attractive properties as well as only two model parameters, we combine these models with the INAR(1) model (integer-valued autoregressive), which is particularly well suited to obtain auotocorrelated counts with underdispersion. Properties of the resulting stationary INAR(1) models and approaches for parameter estimation are considered, as well as possible extensions to higher order autoregressions. Three real-data examples illustrate the application of the models in practice.     

Note on sums of a random number of random variables with exponential family distributions

With the help of the result that exponential-type families are determined by their mean value functions it is shown that stochastic independence of the random variables S_N and N-S_N characterizes the Poisson and Bernoulli distributions simultaneously.     

Estimation of the Jump Size Density in a Mixed Compound Poisson Process

In this paper, we consider a mixed compound Poisson process, that is, a random sum of independent and identically distributed (i.i.d.) random variables where the number of terms is a Poisson process with random intensity. We study nonparametric estimators of the jump density by specific deconvolution methods. Firstly, assuming that the random intensity has exponential distribution with unknown expectation, we propose two types of estimators based on the observation of an i.i.d. sample. Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of the random intensity, we propose two non‐parametric estimators of the jump density based on the joint observation of the number of jumps and the random sum of jumps. Risks bounds are provided, leading to unusual rates for one of the two estimators. The methods are implemented and compared via simulations.