Family Of Discrete Distributions Research Articles

A Family of Discrete Distributions with Infinite Moments

A Family of Discrete Distributions with Infinite Moments

Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability

In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions.

On a stochastic order induced by an extension of Panjer\u2019s family of discrete distributions

We factorize probability mass functions of discrete distributions belonging to Panjer’s family and to its certain extensions to define a stochastic order on the space of distributions supported on {mathbb {N}}_0. Main properties of this order are presented. Comparison of some well-known distributions with respect to this order allows to generate new families of distributions that satisfy various recurrrence relations. The recursion formula for the probabilities of corresponding compound distributions for one such family is derived. Applications to various domains of reliability theory are provided.

A New Family of Discrete Distributions with Mathematical Properties, Characterizations, Bayesian and Non-Bayesian Estimation Methods

In this work, we propose and study a new family of discrete distributions. Many useful mathematical properties, such as ordinary moments, moment generating function, cumulant generating function, probability generating function, central moment, and dispersion index are derived. Some special discrete versions are presented. A certain special case is discussed graphically and numerically. The hazard rate function of the new class can be “decreasing”, “upside down”, “increasing”, and “decreasing-constant-increasing (U-shape)”. Some useful characterization results based on the conditional expectation of certain function of the random variable and in terms of the hazard function are derived and presented. Bayesian and non-Bayesian methods of estimation are considered. The Bayesian estimation procedure under the squared error loss function is discussed. Markov chain Monte Carlo simulation studies for comparing non-Bayesian and Bayesian estimations are performed using the Gibbs sampler and Metropolis–Hastings algorithm. Four applications to real data sets are employed for comparing the Bayesian and non-Bayesian methods. The importance and flexibility of the new discrete class is illustrated by means of four real data applications.

The T\u2013R {Y} power series family of probability distributions

A new family of univariate probability distributions called the T − R {Y} power series family of probability distributions is introduced in this paper by compounding the T − R {Y} family of distributions and the power series family of discrete distributions. A treatment of the general mathematical properties of the new family is carried out and some sub-families of the new family are specified to depict the broadness of the new family. The maximum likelihood method of parameter estimation is suggested for the estimation of the parameters of the new family of distributions. A special member of the new family called the Gumbel–Weibull–{logistic}–Poisson (GUWELOP) distribution is defined and found to exhibit both unimodal and bimodal shapes. The GUWELOG distribution is further applied to a real multi-modal data set to buttress its applicability.

A bivariate discrete inverse resilience family of distributions with resilience marginals

ABSTRACT In this paper, a new bivariate discrete generalized exponential distribution, whose marginals are discrete generalized exponential distributions, is studied. It is observed that the proposed bivariate distribution is a flexible distribution whose cumulative distribution function has an analytical structure. In addition, a new bivariate geometric distribution can be obtained as a special case. We study different properties of this distribution and propose estimation of its parameters. We will see that the maximum of the variables involved in the proposed bivariate distribution defines some new classes of univariate discrete distributions, which are interesting in their own sake, and can be used to analyze some Reliability systems whose components are positive dependent. Some important futures of this new univariate family of discrete distributions are also studied in details. In addition, a general class of bivariate discrete distributions, whose marginals are exponentiated discrete distributions, is introduced. Moreover, the analysis of two real bivariate data sets is performed to indicate the effectiveness of the proposed models. Finally, we conclude the paper.

A generalization of discrete Weibull distribution

ABSTRACTIn this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.

Generation of New Families of Discrete Distributions

New families of discrete distributions are obtained by inverting non-negative integer-valued discrete distributions. Two methods, namely P-inverting and Q-inverting are proposed. The “ P-inverse” is generated from a non-increasing probability mass function (PMF), while “ Q-inverse” results in a non-increasing PMF. Some relations between the original and the generated distributions are established. Models are shown to give good fits for some well-known datasets.

Sharp bounds on the expectations of L-statistics expressed in the Gini mean difference units

ABSTRACTWe describe a method of calculating sharp lower and upper bounds on the expectations of arbitrary, properly centered L-statistics expressed in the Gini mean difference units of the original i.i.d. observations. Precise values of bounds are derived for the single-order statistics, their differences, and some examples of L-estimators. We also present the families of discrete distributions which attain the bounds, possibly in the limit.

On independent events in families of discrete distributions

Discrete spaces of elementary events with families of probabilities depending on a parameter ë are considered. We introduce the notion of events independent relative to a family of distributions and the notion of families of distributions without independent events. A simple necessary condition of the existence of events independent relative to the family of power series distributions is given. It permits to investigate a question on the existence of events independent relative to these families of distributions. Examples of families of distributions having independent events, families without independent events, and also the families not belonging to both classes are given.

Solving the Multidimensional Assignment Problem by a Cross-Entropy method

The Multidimensional Assignment Problem (MAP) is a higher dimensional version of the linear assignment problem, where we find tuples of elements from given sets, such that the total cost of the tuples is minimal. The MAP has many recognized applications such as data association, target tracking, and resource planning. While the linear assignment problem is solvable in polynomial time, the MAP is NP-hard. In this work, we develop a new approach based on the Cross-Entropy (CE) methods for solving the MAP. Exploiting the special structure of the MAP, we propose an appropriate family of discrete distributions on the feasible set of the MAP that allow us to design an efficient and scalable CE algorithm. The efficiency and scalability of our method are proved via several tests on large-scale problems with up to 5 dimensions and 20 elements in each dimension, which is equivalent to a 0---1 linear program with 3.2 millions binary variables and 100 constraints.

The Generalized Power Series Distributions And Their Application

Necessity of the use of more rich family discrete distributions is practically observable. In this paper, we study about new family of discrete distributions, as Inflated Parameter distributions. Then , we use the application of these distributions for data modeling of Insurance claims and indicate that these distributions have more appropriate approximation then corresponding distributions.

A note on incomplete and boundedly complete families of discrete distributions

Abstract We present a general result giving us families of incomplete and boundedly complete families of discrete distributions. For such families, the classes of unbiased estimators of zero with finite variance and of parametric functions which will have uniformly minimum variance unbiased estimators with finite variance are explicitly characterized. The general result allows us to construct a large number of families of incomplete and boundedly complete families of discrete distributions. Several new examples of such families are described.

Robust Inference Against Imperfect Ranking in Ranked Set Sampling

This article develops robust statistical inference against imperfect ranking in a ranked set sample data obtained from a family of discrete distributions. We provide a class of disparity estimators that minimizes the disparities between the empirical and model distributions. It is shown that the Hellinger disparity estimator in this class is the most stable estimator in the neighborhood of an epsilon-contaminated judgment ranking model. When the true model is equal to assumed model and there is no ranking error, all disparity estimators are asymptotically efficient. A simulation study is provided to discuss the finite sample properties of the estimator. The proposed procedure is applied to a ranked set sample data in a water quality study.

Sums of exchangeable Bernoulli random variables for family and litter frequency data

We describe new families of discrete distributions that are used to model sums of exchangeable Bernoulli random variables. These discrete distributions can be parameterized in terms of their range, mean, variance, and shape parameters. These models are fitted to an example involving mortality rates for children in a survey of families in Brazil. The methods illustrate that mortality rates in this survey increase with family size and that the correlation of within-family mortality status also depends on the family size. These methods are also applied to a laboratory study of birth defects in mice.

Model Selection Using Conditional Densities

In this article we propose a new method to select a discrete model f(x; θ), based on the conditional density of a sample given the value of a sufficient statistic for θ. The main idea is to work with a broad family of discrete distributions, called the family of power series distribution, for which there is a common sufficient statistic for the parameter of interest. The proposed method uses the maximum conditional density in order to select the best model. We compare our proposal with the usual methodology based on Bayes factors. We provide several examples that show that our proposal works fine in most instances. Bayes factors are strongly dependent on the prior information about the parameters. Since our method does not require the specification of a prior distribution, it provides a useful alternative to Bayes factors.

Parametric bootstrap tests for continuous and discrete distributions

Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.

Zero biasing and a discrete central limit theorem

We introduce a new family of distributions to approximate $\mathbb {P}(W\in A)$ for $A\subset\{...,-2,-1,0,1,2,...\}$ and $W$ a sum of independent integer-valued random variables $\xi_1$, $\xi_2$, $...,$ $\xi_n$ with finite second moments, where, with large probability, $W$ is not concentrated on a lattice of span greater than 1. The well-known Berry--Esseen theorem states that, for $Z$ a normal random variable with mean $\mathbb {E}(W)$ and variance $\operatorname {Var}(W)$, $\mathbb {P}(Z\in A)$ provides a good approximation to $\mathbb {P}(W\in A)$ for $A$ of the form $(-\infty,x]$. However, for more general $A$, such as the set of all even numbers, the normal approximation becomes unsatisfactory and it is desirable to have an appropriate discrete, nonnormal distribution which approximates $W$ in total variation, and a discrete version of the Berry--Esseen theorem to bound the error. In this paper, using the concept of zero biasing for discrete random variables (cf. Goldstein and Reinert [J. Theoret. Probab. 18 (2005) 237--260]), we introduce a new family of discrete distributions and provide a discrete version of the Berry--Esseen theorem showing how members of the family approximate the distribution of a sum $W$ of integer-valued variables in total variation.

Computational techniques in the study of discrete distributions generated by the function 3 F 2

In this paper, we present the use of computational aspects in the study of the family of discrete distributions generated by the hypergeometric function 3 F 2, which is a univariate extension of the Gaussian hypergeometric function. These computational techniques allow us to obtain the probability mass function, the mean, the mode in an explicit form as well as the knowledge of the most important properties. We can also obtain a summation result and implement different methods of estimation. Finally, we present an example of an application to real data already fitted by other discrete distributions.

Extremal Limit Laws for Discrete Random Variables

If X1, X2, . . . , Xn are independent and identically distributed discrete random variables andMn = max(X1, . . . , Xn) we examine the limiting behavior of (Mn−b(n))/a(n) as n→∞. It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n → ∞, it is possible to obtain non-degenerate limits for (Mn − b(n))/a(n). We consider five families of discrete distributions and show how this can be done.