Poisson-Lindley Distribution Research Articles

The Discrete Poisson-Amarendra Distribution

In this paper, a Poisson mixture of the Amarendra distribution, introduced by Shanker (2016 c), is proposed, and called the, “Poisson-Amarendra distribution”. The first four raw moments (about the origin) and central moments (about the mean) are obtained. The expression for coefficient of variation, skewness and kurtosis are also given. For the estimation of its parameter, the maximum likelihood estimation and the method of moments are discussed. Moreover, the distribution is fitted using maximum likelihood estimate to certain data sets to test its goodness of fit over Poisson, Poisson-Lindley and Poisson-Sujatha distributions. The corresponding fitting are found to be quite satisfactory in almost all data sets.

Estimation of Parameters of Inflated Poisson-Lindley Distribution for Adult Out-migration

The aim of this paper is to study the distribution of adult migrants from the household. The inflated Poisson-Lindley distribution is found to be a suitable model to describe this distribution. The parameters of this distribution have been estimated by two methods and suitability of model is checked by using real data sets. Findings may help planners and policy makers for designing more effective and reasonable policies.Journal of Institute of Science and Technology, 2015, 20(2): 6-10

On Zero-Truncation of Poisson and Poisson-Lindley Distributions and Their Applications

In this paper, a general expression for ther th factorial moment of zero-truncated Poisson-Lindley distribution (ZTPLD) has been obtained and hence the first four moments about origin has been given. A very simple and alternative method for finding moments of ZTPLD has also been suggested. The expression for the moment generating function of ZTPLD has been obtained. Both ZTPD (Zero- truncated Poisson distribution) and ZTPLD have been fitted using maximum likelihood estimate to a number of data- sets from demography, biological sciences and social sciences and it has been observed that in most cases ZTPLD gives much closer fits than ZTPD while in some cases ZTPD gives much closer fits than ZTPLD.

The complementary Poisson-Lindley class of distributions

This paper proposed a new general class of continuous lifetime distributions, which is a complementary to the Poisson-Lindley family proposed by Asgharzadeh et al. [3]. The new class is derived by compounding the maximum of a random number of independent and identically continuous distributed random variables, and Poisson-Lindley distribution. Several properties of the proposed class are discussed, including a formal proof of probability density, cumulative distribution, and reliability and hazard rate functions. The unknown parameters are estimated by the maximum likelihood method and the Fisher’s information matrix elements are determined. Some sub-models of this class are investigated and studied in some details. Finally, a real data set is analyzed to illustrate the performance of new distributions.

Approximate tolerance intervals for the discrete Poisson–Lindley distribution

The Poisson–Lindley distribution is a compound discrete distribution that can be used as an alternative to other discrete distributions, like the negative binomial. This paper develops approximate one-sided and equal-tailed two-sided tolerance intervals for the Poisson–Lindley distribution. Practical applications of the Poisson–Lindley distribution frequently involve large samples, thus we utilize large-sample Wald confidence intervals in the construction of our tolerance intervals. A coverage study is presented to demonstrate the efficacy of the proposed tolerance intervals. The tolerance intervals are also demonstrated using two real data sets. The R code developed for our discussion is briefly highlighted and included in the tolerance package.

On Poisson-Lindley Distribution and Its Applications to Biological Sciences

A general expression for the r th factorial moment of Poisson-Lindley distribution has been obtained and hence its first four moments about origin has been obtained. The distribution has been fitted to some data-sets relating to ecology and genetics to test its goodness of fit and the fit shows that it can be an important tool for modeling biological science data.

Bivariate Poisson-Lindley Distribution with Application

This study applies a Bivariate Poisson-Lindley (BPL) distribution for modeling dependent and over-dispersed count data. The advantage of using this form of BPL distribution is that the correlation coefficient can be positive, zero or negative, depending on the multiplicative factor parameter. Several properties such as mean, variance and correlation coefficient of the BPL distribution are discussed. A numerical example is given and the BPL distribution is compared to Bivariate Poisson (BP) and Bivariate Negative Binomial (BNB) distributions which also allow the correlation coefficient to be positive, zero or negative. The results show that BPL distribution provides the smallest Akaike Information Criterion (AIC), indicating that the distribution can be used as an alternative for fitting dependent and over-dispersed count data, with either negative or positive correlation.

A new family of compound lifetime distributions

In this paper, we introduce a general family of continuous lifetime distributions by compounding any continuous distribution and the Poisson-Lindley distribution. It is more flexible than several recently introduced lifetime distributions. The failure rate functions of our family can be increasing, decreasing, bathtub shaped and unimodal shaped. Several properties of this family are investigated including shape characteristics of the probability density, moments, order statistics, (reversed) residual lifetime moments, conditional moments and Renyi entropy. The parameters are estimated by the maximum likelihood method and the Fisher's information matrix is determined. Several special cases of this family are studied in some detail. An application to a real data set illustrates the performance of the family of distributions.

Pareto Poisson–Lindley distribution with applications

A new lifetime distribution is introduced based on compounding Pareto and Poisson–Lindley distributions. Several statistical properties of the distribution are established, including behavior of the probability density function and the failure rate function, heavy- and long-right tailedness, moments, the Laplace transform, quantiles, order statistics, moments of residual lifetime, conditional moments, conditional moment generating function, stress–strength parameter, Rényi entropy and Song's measure. We get maximum-likelihood estimators of the distribution parameters and investigate the asymptotic distribution of the estimators via Fisher's information matrix. Applications of the distribution using three real data sets are presented and it is shown that the distribution fits better than other related distributions in practical uses.

The Bayes Premium in an Aggregate Loss Poisson-Lindley Model with Structure Function STSP

Many premium calculating problems in actuarial science consider the number of claims, denoted as K, as the variable risk. Traditionally, this random variable is modelled by the Poisson distribution. However, it is well known that automobile insurance portfolios are characterized by zero-inflation (high percentage of zero values in the sample) and overdispersion (the variance is greater than the mean), and the Poisson distribution does not properly reflect the last phenomenon. In this paper we determine the Bayes premium considering that K follows a Poisson-Lindley distribution, with parameter $\theta_{1}$ in $[0,1]$, which is a potential alternative to describe these situations. As the structure function for $\theta_{1}$ we elicit the standardized two-sided power distribution, which is a reasonable alternative to the usual beta distribution. In addition, an aggregate loss model is considered with primary distribution given by the Poisson-Lindley distribution. A Bayesian analysis is developed to obtain the Bayes premium. The conclusion is that the STSP is not an adequate alternative in the problem in question because it is more informative and less dispersed than the Beta distribution.

An application of the Morgenstern family with standard two‐sided power and gamma marginal distributions to the Bayes premium in the collective risk model

The Bayes premium is a quantity of interest in the actuarial collective risk model, under which certain hypotheses are assumed. The usual assumption of independence among risk profiles is very convenient from a computational point of view but is not always realistic. Recently, several authors in the field of actuarial and operational risks have examined the incorporation of some dependence in their models. In this paper, we approach this topic by using and developing a Farlie–Gumbel–Morgenstern (FGM) family of prior distributions with specified marginals given by standard two‐sided power and gamma distributions. An alternative Poisson–Lindley distribution is also used to model the count data as the number of claims. For the model considered, closed expressions of the main quantities of interest are obtained, which permit us to investigate the behavior of the Bayes premium under the dependence structure adopted (Farlie–Gumbel–Morgenstern) when the independence case is included. Copyright © 2012 John Wiley & Sons, Ltd.

On the independence between risk profiles in the compound collective risk actuarial model

This paper examines a compound collective risk model in which the primary distribution comprised the Poisson–Lindley distribution with a λ parameter, and where the secondary distribution is an exponential one with a θ parameter. We consider the case of dependence between risk profiles (i.e., the parameters λ and θ), where the dependence is modelled by a Farlie–Gumbel–Morgenstern family. We analyze the consequences of the dependence on the Bayes premium. We conclude that the consequences of the dependence on the Bayes premium may vary considerably.

Computer generation of random variables with Lindley or Poisson–Lindley distribution via the Lambert W function

We provide procedures to generate random variables with Lindley distribution, and also with Poisson–Lindley or zero-truncated Poisson–Lindley distribution, as simple alternatives to the existing algorithms. Our procedures are based on the fact that the quantile functions of these probability distributions can be expressed in closed form in terms of the Lambert W function. As a consequence, the extreme order statistics from the above distributions can also be computer generated in a straightforward manner.

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.

Estimation methods for the discrete Poisson–Lindley distribution

In this paper, we show that the method of moments and maximum likelihood estimators of the parameter of the discrete Poisson–Lindley distribution are consistent and asymptotically normal. The asymptotic variances of the two estimators are almost equal, indicating that the two estimators are almost equally efficient. Also, a simulation study is presented to compare the two estimators. Finally, the two estimators, their standard errors, and the confidence intervals are compared for two published data sets.

Zero-truncated Poisson–Lindley distribution and its application

The zero-truncated Poisson–Lindley distribution is introduced and investigated. In particular, the method of moments and maximum likelihood estimators of the distribution’s parameter are compared in small and large samples. Application of the model to real data is given.