A simple formula based on quantiles for the moments of beta generalized distributions

In this article, we derive explicit expansions for the moments of beta generalized distributions from power series expansions for the quantile functions of the baseline distributions. We apply our formula to the beta normal, beta Student t, beta gamma and beta beta generalized distributions. We propose a simple way to express the quantile function of any beta generalized distribution as a power series expansion with known coefficients.     

Multivariate discrete exponential family of distributions and their properties

The paper generalizes the univariate discrete exponential family of distributions to the multivariate situation, and this generalization includes the multivariate power series distributions, the multivariate Lagrangian distributions, and the modified multivariate power-series distributions. This provides a unified approach for the study of these three classes of distributions. We obtain recurrence relations for moments and cumulants, and the maximum likelihood estimation for the discrete exponential family. These results are applied to some multivariate discrete distributions like the Lagrangian Poisson, Lagrangian (negative) multinomial, logarithmic series distributions and multivariate Lagrangian negative binomial distribution.     

A new family of generalized distributions

Kumaraswamy [Generalized probability density-function for double-bounded random-processes, J. Hydrol. 462 (1980), pp. 79–88] introduced a distribution for double-bounded random processes with hydrological applications. For the first time, based on this distribution, we describe a new family of generalized distributions (denoted with the prefix ‘Kw’) to extend the normal, Weibull, gamma, Gumbel, inverse Gaussian distributions, among several well-known distributions. Some special distributions in the new family such as the Kw-normal, Kw-Weibull, Kw-gamma, Kw-Gumbel and Kw-inverse Gaussian distribution are discussed. We express the ordinary moments of any Kw generalized distribution as linear functions of probability weighted moments (PWMs) of the parent distribution. We also obtain the ordinary moments of order statistics as functions of PWMs of the baseline distribution. We use the method of maximum likelihood to fit the distributions in the new class and illustrate the potentiality of the new model with an application to real data.     

Geeta distribution and its properties

A new discrete distribution defined over all the positive integers and with the name of Geeta distribution is described. It is L-shaped like the logarithmic series distribution, Yule distribution and the discrete Pareto distribution but is far more versatile than them as it has two parameters. It belongs to the classes of location parameter distributions, modified power series distributions, Lagrange series distributions and exponential distributions. Its mean fi, variance a2 and two recurrence formulae for higher central moments are obtained. Convolution theorem and variations in the model with changes in the parameters have been considered. ML estimators, MVU estimators and estimators based of mean and variance and on mean and first frequency have been derived.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.     

Gompertz-power series distributions

ABSTRACT

In this article, we introduce the Gompertz power series (GPS) class of distributions which is obtained by compounding Gompertz and power series distributions. This distribution contains several lifetime models such as Gompertz-geometric (GG), Gompertz-Poisson (GP), Gompertz-binomial (GB), and Gompertz-logarithmic (GL) distributions as special cases. Sub-models of the GPS distribution are studied in details. The hazard rate function of the GPS distribution can be increasing, decreasing, and bathtub-shaped. We obtain several properties of the GPS distribution such as its probability density function, and failure rate function, Shannon entropy, mean residual life function, quantiles, and moments. The maximum likelihood estimation procedure via a EM-algorithm is presented, and simulation studies are performed for evaluation of this estimation for complete data, and the MLE of parameters for censored data. At the end, a real example is given.     

Bayesian Estimation for Deformed Modified Power Series Distribution

In this article, posterior distribution, posterior moments, and predictive distribution for the modified power series distributions deformed at any of a support point under linex and generalized entropy loss function are derived. It is assumed that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma, or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution deformed at a given point.     

Incomplete moments of modified power series districutions with applications

The recurrence relations between the incomplete moments and the factorial incomplete moments of the modified power series distributions (MPSD) are derived. These relations are employed to obtain the experessions for the incomplete moments and the incomplete factorial moments of some particular members of the MPSD class such as the generalized negative binomial, the generalized Poisson, the generalized logrithmic series, the lost game distribution and the distribution of the number of customers served in a busy period. An application of the incomplete moments of the generalized Poisson distribution is provided in the economic selection of a manufactured product. A numerical example is provided using the Poisson distribution and the Generalized Poisson distribution. The example illustrates the difference in results using the two models     

Non–zero inflated modified power series distributions

In this paper, we study the class of inflated modified power series distributions (IMPSD) where inflation occurs at any of the support points. This class include among other the generalized Poisson, the generalized negative binomial, the generalized logarithmic series and the lost games distributions. We give expressions for the moments, factorial moments and central moments of the IMPSD. The maximum likelihood estimation of the parameters of the IMPSD and the variance – covariance matrix of the estimators is obtained. We derive these estimators and their information matrices for mentioned above particular members of IMPSD class. The second part of this paper deals with the distribution of sum of independent and identically distributed random variables taking values s, s+1. s + 2, …, s ≥ 0, with modified power series distributions inflated at the point s.     

The compound class of linear failure rate-power series distributions: Model,properties, and applications

In this article, a new class of distributions is introduced, which generalizes the linear failure rate distribution and is obtained by compounding this distribution and power series class of distributions. This new class of distributions is called the linear failure rate-power series distributions and contains some new distributions such as linear failure rate-geometric, linear failure rate-Poisson, linear failure rate-logarithmic, linear failure rate-binomial distributions, and Rayleigh-power series class of distributions. Some former works such as exponential-power series class of distributions, exponential-geometric, exponential-Poisson, and exponential-logarithmic distributions are special cases of the new proposed model. The ability of the linear failure rate-power series class of distributions is in covering five possible hazard rate function, that is, increasing, decreasing, upside-down bathtub (unimodal), bathtub and increasing-decreasing-increasing shaped. Several properties of this class of distributions such as moments, maximum likelihood estimation procedure via an EM-algorithm and inference for a large sample, are discussed in this article. In order to show the flexibility and potentiality, the fitted results of the new class of distributions and some of its submodels are compared using two real datasets.     

Inverse Weibull power series distributions: properties and applications

Inverse Weibull (IW) distribution is one of the widely used probability distributions for nonnegative data modelling, specifically, for describing degradation phenomena of mechanical components. In this paper, by compounding IW and power series distributions we introduce a new lifetime distribution. The compounding procedure follows the same set-up carried out by Adamidis and Loukas [A lifetime distribution with decreasing failure rate. Stat Probab Lett. 1998;39:35–42]. We provide mathematical properties of this new distribution such as moments, estimation by maximum likelihood with censored data, inference for a large sample and the EM algorithm to determine the maximum likelihood estimates of the parameters. Furthermore, we characterize the proposed distributions using a simple relationship between two truncated moments and maximum entropy principle under suitable constraints. Finally, to show the flexibility of this type of distributions, we demonstrate applications of two real data sets.     

A target distribution model for nonparametric density estimation

In this paper a model is proposed which represents a wide class of continuous distributions. It is shown how the parameters of this model can be estimated leading to a distribution estimator and a corresponding density estimator. An important property of this estimator is that it can be structured to reflect a priori knowledge of the unknown distribution.

Finally, some examples are shown and some comparisons made with kernel and orthogonal series estimators.     

The use of Fourier series in the evaluation of probability distribution functions

In statistics, Fourier series have been used extensively in such areas as time series and stochastic processes. These series; however, to a large degree have been neglected with regard to their use in statistical distribution theory. This omission appears quite striking when one considers that, after the elementary functions, the trigonometric functions are the most important functions in applied mathematics. In this paper a procedure is developed for utilizing Fourier series to represent distribution functions of finite range random variables as Fourier series with coefficients easily expressible (using Chebyshev polynomials) In terms of the moments of the distribution. This method allows the evaluation of probabilities for a wide class of distributions. It is applied to the     

Approximating Matrix-Exponential Distributions by Global Randomization

Abstract

Based on the general concept of randomization, we develop linear-algebraic approximations for continuous probability distributions that involve the exponential of a matrix in their definitions, such as phase types and matrix-exponential distributions. The approximations themselves result in proper probability distributions. For such a global randomization with the Erlang-k distribution, we show that the sequences of true and consistent distribution and density functions converge uniformly on [0, ∞). Furthermore, we study the approximation errors in terms of the power moments and the coefficients of the Taylor series, from which the accuracy of the approximations can be determined apriori. Numerical experiments demonstrate the feasibility of the presented randomization technique – also in comparison with uniformization.     

Some Properties and Generalizations of Non-negative Bayesian Time Series Models

We study the most basic Bayesian forecasting model for exponential family time series, the power steady model (PSM) of Smith, in terms of observable properties of one-step forecast distributions and sample paths. The PSM implies a constraint between location and spread of the forecast distribution. Including a scale parameter in the models does not always give an exact solution free of this problem, but it does suggest how to define related models free of the constraint. We define such a class of models which contains the PSM. We concentrate on the case where observations are non-negative. Probability theory and simulation show that under very mild conditions almost all sample paths of these models converge to some constant, making them unsuitable for modelling in many situations. The results apply more generally to non-negative models defined in terms of exponentially weighted moving averages. We use these and related results to motivate, define and apply very simple models based on directly specifying the forecast distributions.     

Lifetime distributions motivated by series and parallel structures

According to Ross, any system can be represented either as a series arrangement of parallel structures or as a parallel arrangement of series structures. Motivated by this, we propose new three-parameter lifetime distributions by compounding geometric, power series, and exponential distributions. The distributions can allow for decreasing, increasing, bathtub-shaped, and upside down bathtub-shaped hazard rates. A mathematical treatment of the new distributions is provided including expressions for their density functions, Shannon and Rényi entropies, mean residual life functions, hazard rate functions, quantiles, and moments. The method of maximum likelihood is used for estimating parameters. Five of the new distributions are studied in detail. Finally, two illustrative data examples and a sensitivity analysis are presented.     

A class of regression models for parallel and series systems with a random number of components

We extend the Weibull power series (WPS) class of distributions to the new class of extended Weibull power series (EWPS) class of distributions. The EWPS distributions are related to series and parallel systems with a random number of components, whereas the WPS distributions [Morais AL, Barreto-Souza W. A compound class of Weibull and power series distributions. Computational Statistics and Data Analysis. 2011;55:1410–1425] are related to series systems only. Unlike the WPS distributions, for which the Weibull is a limiting special case, the Weibull law is a particular case of the EWPS distributions. We prove that the distributions in this class are identifiable under a simple assumption. We also prove stochastic and hazard rate order results and highlight that the shapes of the EWPS distributions are markedly more flexible than the shapes of the WPS distributions. We define a regression model for the EWPS response random variable to model a scale parameter and its quantiles. We present the maximum likelihood estimator and prove its consistency and asymptotic normal distribution. Although series and parallel systems motivated the construction of this class, the EWPS distributions are suitable for modelling a wide range of positive data sets. To illustrate potential uses of this model, we apply it to a real data set on the tensile strength of coconut fibres and present a simple device for diagnostic purposes.     

Properties of Coherent Systems with Dependent Components

We introduce two new families of univariate distributions that we call hyperminimal and hypermaximal distributions. These families have interesting applications in the context of reliability theory in that they contain that of coherent system lifetime distributions. For these families, we obtain distributions, bounds, and moments. We also define the minimal and maximal signatures of a coherent system with exchangeable components which allow us to represent the system distribution as generalized mixtures (i.e., mixtures with possibly negative weights) of series and parallel systems. These results can also be applied to order statistics (k-out-of-n systems). Finally, we give some applications studying coherent systems with different multivariate exponential joint distributions.     

Structural properties and estimation in m|EK| 1 queue

In this paper we study the distribution of the number of customers served in a busy period in the framework of modified power series distribution introduced by Gupta (197U) and obtain the moments and probability generating function of this distribution. We also study the maximum likelihood estimation of the parameter θand the variance and the asymptotic bias of the MLE are also obtained. The minimum variance unbiased estimate of θ^ris investigated and an estimate of the probabilities is given.     

The Extended Skew-Exponential Power Distribution and Its Derivation

We consider an extended family of asymmetric univariate distributions generated using a symmetric density, f, and the cumulative distribution function, G, of a symmetric distribution, which depends on two real-valued parameters λ and β and is such that when β = 0 it includes the entire class of distributions with densities of the form g(z | λ) = 2 G(λ z) f(z). A key element in the construction of random variables distributed according to the family is that they can be represented stochastically as the product of two random variables. From this representation we can readily derive theoretical properties, easy-to-implement simulation schemes, as well as extensions to the multivariate case and an explicit procedure for obtaining the moments. We give special attention to the extended skew-exponential power distribution. We derive its information matrix in order to obtain the asymptotic covariance matrix of the maximum likelihood estimators. Finally, an application to a real data set is reported, which shows that the extended skew-exponential power model can provide a better fit than the skew-exponential power distribution.