Poisson-Lindley Distribution Research Articles

Generalised Poisson-new linear-exponential distribution

It is a compound discrete probability distribution which has two parameters and whose particular case is Poisson-New Linear-Exponential Distribution (PNLED) [1]. The statistical characteristics required for this distribution such as probability mass function (pmf), statistical moments, estimation of parameters and goodness of fit have been derived and explained nicely. To test theoretical reliability of this distribution, goodness of fit has been applied to some over-dispersed data which were used by other researchers and further, we found that this distribution looks more appropriate for statistical modelling than PNLED, generalised Poisson-Lindley distribution (GPLD) [1, 2] and Poisson-Lindley distribution (PLD) [3].

Modified Bivariate Poisson-Lindley Model: Properties and Applications in Soccer

Abstract This paper presents the bivariate Poisson-new XLindley distribution (BPNXLD), which may be used to represent dependent and over-dispersed countdata. Among the characteristics considered are the correlation coefficient, mean, and variance of the distribution. Acomparison with several Bivariate distributions is included. The goodness of fit of this novel model iscompared with the bivariate Poisson, bivariate negative binomial and bivariate Poisson-Lindley distributions using two data sets from a German Bundesliga season.

Poisson-New Quadratic-Exponential Distribution

This proposed distribution is a discrete compound probability distribution with only one parameter. To get this distribution, Poisson distribution has been mixed with the New Quadratic-Exponential distribution of Sah (2022). Hence, it is named as “Poisson-New Quadratic-Exponential Exponential Distribution (PNLED)”. The important statistical characteristics needed to check the validity of this distribution have been derived and clearly explained. To check the validity of the theoretical works of this distribution, while using goodness of fit on some over-dispersed count data, what we have been found that this distribution seems a better alternative of Poisson-Lindley distribution (PLD) of Sankaran (1970), Poisson Mishra distribution (PMD) of Sah (2017) and Poisson-Modified Mishra distribution (PMMD) of Sah and Sahani (2023).

Generalised Poisson-Modified Mishra distribution

This distribution is a compound probability distribution constructed by compounding generalised Poisson distribution of Consul and Jain (1973) with Modified Mishra distribution of Sah (2022). All the important characteristics required for this distribution have been obtained and explained. In order to check the validity of the theoretical work, goodness of fit has been applied to some secondary data which have been used by others. It has been observed that this distribution gives a better fit than Poisson-Lindley distribution of Sankaran (1970) as well as Poisson-Modified Mishra distribution of Sah and Sahani (2023).

Premium Linear-Exponential Mixture of Poisson Distribution

The proposed distribution is a discrete probability distribution with only one parameter and it has been derived by compounding the Poisson distribution with the Premium Linear-exponential distribution (PLED) of Sah (2022). So, it is named as Premium Linear-Exponential Mixture of Poisson Distribution (PLEMPD). The required statistical characteristics have been derived and explained to give the theoretical concept of this distribution. The main purpose of proposing this distribution is to provide a better goodness of fit for over-dispersed count data than the previously derived one parameter continuous mixtures of Poisson distribution of same nature by others. Taking some secondary over-dispersed count data and using the theoretical concept of this distribution, it has been observed that this distribution is more suitable for statistical modeling of over-dispersed count data than Poisson-Lindley distribution (PLD) of Sankaran (1970) and Poisson-Mishra distribution (PMD) of Sah (2017).

A Statistical Model for Count Data Analysis and Population Size Estimation: Introducing a Mixed Poisson–Lindley Distribution and Its Zero Truncation

Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the improved second-degree Lindley distribution. This distribution, called the Poisson-improved second-degree Lindley distribution, is capable of effectively modeling standard count data with overdispersion. However, if the frequency of the unobserved events is unknown, the proposed distribution cannot be directly used to describe the events. To address this limitation, we propose a modification by truncating the distribution to zero. This results in a tractable zero-truncated distribution that encompasses all types of dispersions. Due to the unknown frequency of unobserved events, the population size as a whole becomes unknown and requires estimation. To estimate the population size, we develop a Horvitz–Thompson-like estimator utilizing truncated distribution. Both the untruncated and truncated distributions exhibit desirable statistical properties. The estimators for both distributions, as well as the population size, are asymptotically unbiased and consistent. The current study demonstrates that both the truncated and untruncated distributions adequately explain the considered medical datasets, which are the number of dicentric chromosomes after being exposed to different doses of radiation and the number of positive Salmonella. Moreover, the proposed population size estimator yields reliable estimates.

A Generalization of Generalized Poisson-Lindley Distribution and its Applications

In this paper we proposed a generalization of generalized Poisson-Lindley distribution which includes generalized Poisson-Lindley distribution, Poisson-Lindley distribution, Poisson-weighted Lindley distribution, negative binomial distribution and geometric distribution as special cases. Statistical properties based on moments, maximum likelihood estimation and applications of the distribution have been discussed.

Nonparametric Bootstrap Confidence Intervals for the Population Mean of a Zero-Truncated Poisson-Lindley Distribution and Their Application

Recently, the zero-truncated Poisson-Lindley distribution has been proposed for studying count data containing non-zero values. However, the nonparametric bootstrap confidence interval estimation of the population mean has not yet been studied. In this study, confidence interval estimation based on percentile, simple, and biased-corrected bootstrap methods was compared in terms of coverage probability and average interval length via Monte Carlo simulation. The true values of parameter () were set as 0.25, 0.5, 1, 1.5, and 2, and the population means are approximate 7.7586, 4.0909, 2.4000, 1.8817, and 1.6364, respectively. The bootstrap samples (=1,000) of size were generated from the original sample, and each simulation was repeated 1,000 times. The results indicate that attaining the nominal confidence level using the bootstrap confidence intervals was impossible for small sample sizes regardless of the other settings. Moreover, when the sample size was large, the performance of the nonparametric bootstrap confidence intervals was not substantially different. Overall, the bias-corrected bootstrap confidence interval outperformed the others, even for small sample sizes. Last, the nonparametric bootstrap confidence intervals were used to calculate the confidence interval for the population mean of the zero-truncated Poisson-Lindley distribution via two numerical examples, the results of which match those from the simulation study.

A Flexible Dispersed Count Model Based on Bernoulli Poisson–Lindley Convolution and Its Regression Model

Count data are encountered in real-life dealings. More understanding of such data and the extraction of important information about the data require some statistical analysis or modeling. One innovative technique to increase the modeling flexibility of well-known distributions is to use the convolution of random variables. This study examines the distribution that results from adding two independent random variables, one with the Bernoulli distribution and the other with the Poisson–Lindley distribution. The considered distribution is named as the two-parameter Bernoulli–Poisson–Lindley distribution. Many of its statistical properties are investigated, such as moments, survival and hazard rate functions, mode, dispersion behavior, mean deviation about the mean, and parameter inference based on the maximum likelihood method. To evaluate the effectiveness of the bias and mean square error of the produced estimates, a simulation exercise is carried out. Then, applications to two practical data sets are given. Finally, we construct a flexible count data regression model based on the proposed distribution with two practical examples.

The Poisson–Lindley Distribution: Some Characteristics, with Its Application to SPC

Statistical process control (SPC) is a significant method to monitor processes and ensure quality. Control charts are the most important tools in SPC. As production processes and production parts become more complex, there is a need to design control charts using more complex distributions. One of the most important control charts to monitor the number of nonconformities in production processes is the C-chart, which uses the Poisson distribution as a quality characteristic distribution. However, to fit the Poisson distribution to the count data, equality of mean and variance should be satisfied. In some cases, such as biological and medical sciences, count data exhibit overdispersion, which means that the variance of data is greater than the mean. In such cases, we can use the Poisson–Lindley distribution instead of the Poisson distribution to model the count data. In this paper, we first discuss some important characteristics of the Poisson–Lindley distribution. Then, we present parametric and bootstrap control charts when the observations follow the Poisson–Lindley distribution and analyze their performance. Finally, we provide a simulated example and a real-world dataset to demonstrate the implementation of control charts. The results show the good performance of the proposed control charts.

A Pliant Model to Count Data: Nabla Poisson–Lindley Distribution with a Practical Data Example

A Pliant Model to Count Data: Nabla Poisson–Lindley Distribution with a Practical Data Example

BETA TRANSFORMATION OF THE LINDLEY POISSON DISTRIBUTION FOR OVER-DISPERSED COUNT DATA

A new distribution for over-dispersed count data is proposed, and its properties are studied. This is a two-parameter distribution which is obtained by introducing an additional parameter beta into the Poisson-Lindley distribution. The goodness-of-fit of this distribution is compared with other distributions that have been proposed to model overdispersion. Two illustrative examples are presented to show the flexibility of the model.

The Poisson-Adya distribution

In this paper a Poisson mixture of Adya distribution called Poisson-Adya distribution has been suggested. The expressions of statistical constants including coefficients of variation, skewness, kurtosis and index of dispersion have been obtained and their behavior for varying values of parameter has been studied. It is observed that the obtained distribution is unimodal, has increasing hazard rate and over-dispersed. Maximum likelihood estimation and method of moment have been discussed for estimating parameter. Finally, the goodness of fit of the proposed distribution and its comparison with Poisson and Poisson-Lindley distributions has been given.

On statistical properties and applications of PoissonPranav distribution

In this paper, Poisson-Pranav distribution, a Poisson mixture of Pranav distribution, has been proposed. Its moments and moments-based measures including coefficients of variation, skewness, kurtosis, index of dispersion have been obtained and their behavior illustrated graphically. The estimation of parameter of the proposed distribution has been discussed using both the method of moment and the method of maximum likelihood. The simulation study has also been presented in order to illustrate the performance of maximum likelihood estimator. The goodness of fit of the proposed distribution has been explained with two count datasets and its fit was found quite satisfactory over Poisson distribution, Poisson-Lindley distribution, Poisson-Akash distribution and Poisson-Ishita distribution.

Bivariate Discrete Poisson-Lindley Distributions.

Two families of bivariate discrete Poisson–Lindley distributions are introduced. The first is derived by mixing the common parameter in a bivariate Poisson distribution by different models of univariate continuous Lindley distributions. The second is obtained by generalizing a bivariate binomial distribution with respect to its exponent when it follows any of five different univariate discrete Poisson–Lindley distributions with one or two parameters. The use of probability-generating functions is mainly employed to derive some general properties for both families and specific characteristics for each one of their members. We obtain expressions for probabilities, moments, conditional distributions, regression functions, as well as characterizations for certain bivariate models and their marginals. An attractive property of all bivariate individual models is that they contain only two or three parameters, and one of them is readily estimated by simple ratios of their sample means. This feature, and since all marginal distributions are over-dispersed, strongly suggests their potential use to describe bivariate dependent count data in many different areas.

The Poisson-Transmuted Janardan Distribution for Modelling Count Data

In this paper, we introduce a new mixed Poisson distribution, called the Poisson-transmuted Janardan distribution. The Poisson-Janardan and Poisson-Lindley distributions are sub-model of the proposed distribution. Some mathematical properties of the proposed distribution, including the moments, moment generating function, probability generating function and generation of a Poisson-transmuted Janardan random variable, are presented. The parameter estimation is discussed based on the method of moments and the maximum likelihood estimation. In addition, we illustrated the application of the proposed distribution by fitting with 4 real data sets and comparing it with some other distributions based on the Kolmogorov-Smirnov test for criteria.

A New Zero–One-Inflated Poisson–Lindley Distribution for Modelling Overdispersed Count Data

A New Zero–One-Inflated Poisson–Lindley Distribution for Modelling Overdispersed Count Data

A new regression model for Poisson Lindley distribution with application

The main goal of this study is to propose a new regression model using re-parameterization of the Poisson-Lindley distribution for seven parameters. The utility of real-world data is used to assess the accuracy of estimating algorithms. The suggested model is compared to well-known regression models for count data modelling, such as Poisson, on a real data set to demonstrate its utility. While fitting two real data sets, the (GPL7) linear model will be compared to the Poisson for seven parameters and the (GPL4) linear model will be compared to the Poisson for four parameters. The GPL7 linear model was found to be capable of fitting over-dispersed count data and to have the maximum log-likelihood, according to the results.

A Bayes Estimation for the Parameters of Misclassified Size-biased Poisson Lindley Distribution and its Application

This paper introduces a misclassified size-biased Poisson-Lindley distribution by compounding the size-biased Poisson distribution with the size-biased Lindley distribution. The Bayes estimation for the misclassified size-biased Poisson-Lindley distribution is investigated and compared with the Maximum Likelihood estimation. A real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in the modeling count dataset. Finally, a Monte Carlo simulation study of 100000 simulated data is presented to investigate and compare the Bayes estimators and Maximum Likelihood estimators in terms of simulated risk for different sample sizes and varying parameters value

Modelling Neonatal Mortality Rate in Nigeria Using a Continuous Poisson-Lindley Distribution

Nigeria’s effort to reduce under-five mortality has been biased in favour of childhood mortality to the neglect of neonates and as such the literature is short of adequate information on the determinants of neonatal mortality, whereas studies have shown that about half of infant deaths occur in the neonatal period. Knowledge of the determinants of neonatal mortality is essential for the design of intervention programmes that will enhance neonatal survival. Therefore, this study was conducted to investigate the trends in neonatal mortality in Nigeria. It also proposed a Poisson based continuous probability distribution called Poisson-Lindley distribution to neonatal mortality rate in Nigeria. Some properties of the new model and other relevant measures were obtained. The unknown parameters of the model were also estimated using the method of maximum likelihood. The fitness of the proposed model to the neonatal mortality rate was considered using a dataset on neonatal mortality rate from 1967 to 2019.