Mixed discrete distributions are primarily used for modeling over-dispersed count data. The construction of mixed models, such as the mixed Poisson model, is based on the assumption that the distribution parameter of interest is a random variable that follows a specified distribution. In this framework, the marginal distribution of a discrete random variable forms a mixed distribution. This paper introduces a novel discrete distribution derived from the geometric distribution, assuming that the model's parameter follows the log-Lindley distribution. This approach is motivated by situations where the parameter of the geometric distribution is not constant across populations, as is often the case in insurance data, where the probability of a claim varies between different portfolios. This distribution is particularly well-suited for modeling over-dispersed discrete data. The statistical properties of the proposed distribution are examined, and the parameters of the resulting model are estimated. To evaluate the accuracy of the estimates, a simulation study is conducted, and the model's performance is demonstrated using real data.

In this paper, we introduce a three-parameter generalization of the Lindley distribution. This includes as special cases the exponential and gamma distributions. The distribution exhibits decreasing, increasing and bathtub hazard rate depending on its parameters. We study various properties of the new distribution and provide numerical examples to show the flexibility of the model. We also derive a bivariate version of the proposed distribution.

A new continuous distribution called Lindley-Lindley distribution is defined and studied. Relevant mathematical properties are derived. We present three characterizations of the new distribution based on the truncated moments of certain functions of the random variable,the hazard function and in terms of the conditional expectation of a function of the random variable. Some new bivariate type distributions using Farlie Gumbel Morgenstern copula, modified Farlie Gumbel Morgenstern copula and Clayton copula are introduced. The main justification of this paper is to show how different frequentist estimators of the new model perform for different sample sizes and different parameter values and to provide a guideline for choosing the best estimation method for the parameters of the proposed model. The unknown parameters of the new distribution are estimated using the maximum likelihood, ordinary least squares, Cramer-Von-Mises, weighted least squares and Bayesian methods. The obtained estimators are compared using Markov Chain Monte Carlo simulations and observed that Bayesian estimators are generally more efficient than the other estimators.

In this study, we introduce a new model called the Extended Exponentiated PowerLindley distribution which extends the Lindley distribution and has increasing, bathtub andupside down shapes for the hazard rate function. It also includes the power Lindley distributionas a special case. Several statistical properties of the distribution are explored, such as thedensity, hazard rate, survival, quantile functions, and moments. Estimation using the maximumlikelihood method and inference on a random sample from this distribution are investigated. Asimulation study is performed to compare the performance of the di® erent parameter estimatesin terms of bias and mean square error. We apply a real data set to illustrate the applicabilityof the new model. Empirical ¯ ndings show that proposed model provides better ¯ ts than otherwell-known extensions of Lindley distributions.

In certain statistical process control applications, quality of a process or product can be characterized by a function commonly referred to as profile. Some of the potential applications of profile monitoring are cases where quality characteristic of interest is modelled using binary, multinomial or ordinal variables. In this paper, profiles with multinomial response are studied. For this purpose, multinomial log it regression (MLR) is considered as the basis. Then, the MLR is converted to Poisson GLM with log link. Two methods including Multivariate exponentially weighted moving average (MEWMA) statistics, and Likelihood ratio test (LRT) statistics are proposed to monitor MLR profiles in phase II. Performances of these three methods are evaluated by average run length criterion (ARL). A case study from alloy fasteners manufacturing process is used to illustrate the implementation of the proposed approach. Results indicate satisfactory performance for the proposed method.

In this paper, a new mixture modelling using the normal meanvariance mixture of Lindley (NMVL) distribution has been considered. The proposed model is heavy-tailed and multimodal and can be used in dealing with asymmetric data in various theoretic and applied problems. We present a feasible computationally analytical EM algorithm for computing the maximum likelihood estimates. The behavior of the obtained maximum likelihood estimators is studied with respect to bias and mean squared errors through conducting a simulation study. Two examples with flow cytometry data are used to illustrate the applicability of the proposed model.

In this paper‎, ‎order statistics and associated inferences are considered from Lindley distribution‎. ‎We derive the exact forms of means‎, ‎variances and covariances as well as the moment generating functions of order statistics‎. ‎These obtained forms allow us to compute the means‎, ‎variances‎, ‎and covariances of the order statistics for various values of the shape parameter‎. ‎These values are then used to compute the coefficients of the best linear unbiased estimators‎, ‎the best linear invariant estimators‎, ‎and the least square estimators of the location and scale parameters‎. ‎The variances and covariances of these estimators are also presented‎. ‎Using the best linear unbiased estimators and best linear invariant estimators we construct confidence intervals for the location and scale parameters through Monte Carlo simulations‎. ‎In addition‎, ‎based on the ordered data‎, ‎we investigate how to obtain the best linear unbiased predictor and the best linear invariant predictor for future order statistics‎. ‎Finally‎, ‎data analysis and Monte Carlo simulation have been performed for illustrative purposes and comparative studies‎, ‎respectively.