Lindley Distribution Research Articles

A NEW INVERSE TWO-PARAMETER LINDLEY DISTRIBUTION AND ITS APPLICATION

This paper proposes an inverse two-parameter Lindley distribution by utilizing the one and two parameter Lindley distributions. The key properties of the novel distribution like, its survival function, shape characteristics of the density, entropy measure, hazard rate function, stochastic ordering and stress-strength reliability were examined. Two data sets were employed in the empirical studies. Method of maximum likelihood was used to estimate the parameters. The goodness- of -fit was accessed using the HQIC, BIC, CAIC, and AIC. The proposed distribution was compared with the inverse Lindley and the inverse Akash distributions in order to access its superiority over the two distributions. Ultimately, the new inverse two parameter Lindley distribution was found to be superior by providing a better fit.

Statistical Inferences about Parameters of the Pseudo Lindley Distribution with Acceptance Sampling Plans

Different non-Bayesian and Bayesian techniques were used to estimate the pseudo-Lindley (PsL) distribution’s parameters in this study. To derive Bayesian estimators, one must assume appropriate priors on the parameters and use loss functions such as squared error (SE), general entropy (GE), and linear-exponential (LINEX). Since no closed-form solutions are accessible for Bayes estimates under these loss functions, the Markov Chain Monte Carlo (MCMC) approach was used. Simulation studies were conducted to evaluate the estimators’ performance under the given loss functions. Furthermore, we exhibited the adaptability and practicality of the PsL distribution through real-world data applications, which is essential for evaluating the various estimation techniques. Also, the acceptance sampling plans were developed in this work for items whose lifespans approximate the PsL distribution.

Negative Binomial Three Parameter Lindley Distribution and Its Properties

Many researchers have proposed mixed distributions as one of the most important methods for obtaining new probability distributions. Several studies have shown that mixed Negative Binomial distributions fits count data better than Poisson and Negative Binomial distribution itself. In this paper, we introduce a mixed distribution by mixing the distributions of negative binomial and three Parameter Lindley distribution. This new distribution has a thick tail and may be considered as an alternative for fitting count data with over dispersion. The parameters of the new distribution are estimated using MLE method and properties studied. Special cases of the new distribution and also identified. A simulation study carried out shows that the ML estimators give the parameter estimates close to the parameter when the sample is large, that is, the bias and variance of the parameter estimates decrease with increase in sample size showing the consistent nature of the new compound distribution. The study also compares the performance of the new distribution over distributions of Poisson, Negative Binomial, Negative Binomial oneParameter Lindley Distribution, Negative Binomial two Parameter distribution, three parameter Lindley distribution using a real count over dispersed dataset and the results shows that Negative Binomial three parameter Lindley distribution gave the smallest Kolmogorov Smirnov test statistic, AIC and BIC as compared to other distributions, hence the new distribution provided a better fit compared to other distributions under study for fitting over dispersed count data.

A new parameter estimation method for the extended power Lindley distribution based on order statistics, with application

In this paper, we propose inference procedures for the estimation of parameters by using order statistics. First, we derive some new expressions for single and product moments of the order statistics from the extended power Lindley distribution. We then use these moments to obtain the best linear unbiased estimates (BLUEs) of the location and scale parameters based on Type-II right-censored samples. A real data set is analysed to illustrate the flexibility and importance of the model.

Utilizing Repetitive Sampling in the Construction of a Control Chart for Lindley Distribution with Time Truncation

The monitoring of the process relies on the average product lifetime, which adheres to the Lindley distribution and employs a truncated life test-based attribute control chart. Control limits for these charts are determined through a repetitive sampling procedure. Two inner and outer control limits are established based on the desired average run length (ARL). Tables are generated for varying ARL values by adjusting shift values while keeping the shape parameter of the Lindley distribution constant. The effectiveness of the control charts is assessed through application to real-time data, and the performance of the proposed chart is evaluated using simulation data. A comparative analysis is conducted with single sampling scheme based on ARL values, and the results are presented and discussed.

Determination and economic design of repetitive group sampling plan under two parameter Lindley distribution

ABSTRACT A designing procedure of repetitive group sampling plan for assuring mean life under a two-parameter Lindley distribution is proposed, and the optimization model is developed for finding the optimal plan parameters such that the demands of both supplier and buyer are satisfied simultaneously. A real-time data set of ball bearings is used to explain the application of the proposed plan. A comparison between the proposed plan and the single sampling plan is done to reveal the better performance of the proposed plan based on operating characteristic function and average sample number. Applying the proposed plan to provide lifetime assurance reduces the inspection effort and related costs. We discuss the design of the proposed plan with an economic aspect. In addition, the sensitivity of the optimal sampling plans concerning the failure probability and cost parameters is analyzed. Finally, it is evident from the study that the proposed plan will be preferable to a single sampling plan for providing quality assurance with a minimum cost of the inspection.

Statistical analysis using multiple censoring scheme under partially accelerated life tests for the power Lindley distribution

Statistical analysis using multiple censoring scheme under partially accelerated life tests for the power Lindley distribution

The Efficiency of Hazard Rate Preservation Method for Generating Discrete Rayleigh–Lindley Distribution

In this study, we introduce two novel discrete counterparts for the Rayleigh–Lindley mixture, constructed through the application of survival and hazard rate preservation techniques. These two-parameter discrete models demonstrate exceptional adaptability across various data types, including skewed, symmetric, and monotonic datasets. Statistical analyses were conducted using maximum likelihood estimation and Bayesian approaches to assess these models. The Bayesian analysis, in particular, was implemented with the squared error and LINEX loss functions, incorporating a modified Lwin Prior distribution for parameter estimation. Through simulation studies and numerical methods, we evaluated the estimators’ performance and compared the effectiveness of the two discrete adaptations of the Rayleigh–Lindley distribution. The simulations reveal that Bayesian methods are especially effective in this setting due to their flexibility and adaptability. They provide more precise and dependable estimates for the discrete Rayleigh–Lindley model, especially when using the hazard rate preservation method. This method is a compelling alternative to the traditional survival discretization approach, showcasing its significant potential in enhancing model accuracy and applicability. Furthermore, two real data sets are analyzed to assess the performance of each analog.

Generalization of Power Lindley Distribution: Properties and Applications

Generalization of Power Lindley Distribution: Properties and Applications

A double sampling plan for truncated life tests under two-parameter Lindley distribution

A double sampling plan for truncated life tests under two-parameter Lindley distribution

A New Two-Parameters Lindley-Frailty Model: Censored and Uncensored Schemes under Different Baseline Models: Applications, Assessments, Censored and Uncensored Validation Testing

Classical survival models assume homogeneity among the population of individuals who are susceptible to the event of interest. However, in many practical circumstances, there is a certain amount of unobserved heterogeneity that can be caused by a variety of sources, such as environmental or genetic factors. If the heterogeneity is ignored, many issues could arise, including an overestimation of the hazard rate and inaccurate estimates of the regression coefficients. Frailty models are usually used to model the heterogeneity among individuals. In this paper, we propose a novel univariate frailty model. The frailty variable is assumed to follow the Two Parameter Lindley distribution. The maximum likelihood method is used to estimate the model parameters. The baseline hazard functions are assumed to follow Weibull, Exponential, Gompertz, and Pareto distributions, and a simulation study is performed under this assumption. We examine the characteristics of the distribution and assess its performance compared to other distributions that are frequently applied in frailty modeling by using both Nikulin-Rao-Robson and Bagdonavicius-Nikulin goodness-of-fit tests to determine the adequacy of the model. We analyze a fresh medical dataset collected from an emergency hospital in Algeria to evaluate the effectiveness and applicability of the proposed model.

Extended odd Weibull–Lindley distribution

A novel extension of the Lindley and Weibull distributions is proposed in this article by combining the Lindley distribution with the extended odd Weibull family, termed the extended odd Weibull–Lindley (EOWL) distribution. The statistical properties of the EOWL distribution are derived, including a linear representation for its probability density function, moments, quantiles, and moment generation functions. Three parameter estimation methods (maximum likelihood, maximum spacing product, and least squares estimations) are explored. Monte Carlo simulations are conducted to evaluate the effectiveness of the estimator methods. Finally, in order to illustrate the flexibility and validity of the proposed distribution, three real datasets are employed. The results show that the EOWL distribution demonstrates a superior fitting performance compared to other established distributions.

Classical and Bayesian inference of inverted modified Lindley distribution based on progressive type-II censoring for modeling engineering data

This paper investigates statistical inferences for product lifetimes following the inverted modified Lindley distribution, utilizing progressive Type-II censored data. The estimation of model parameters employs the maximum likelihood method, complemented by the construction of approximate confidence intervals. Bayesian estimates are also explored, incorporating squared error and linear exponential loss functions with noninformative priors. To approximate Bayes estimates, the proposal presents Gibbs sampling based on the MCMC algorithm. This results in the generation of the greatest posterior density credible intervals for the parameters. A real data analysis is conducted to validate the accuracy of all the models and methods discussed. Finally, computational studies using Monte Carlo simulations are presented to compare the suggested estimators.

Fractional Lindley distribution generated by time scale theory, with application to discrete-time lifetime data

ABSTRACT The fractional Lindley distribution is used to model the distribution of perturbations in count data regressions, which allow for dealing with widely dispersed data. It is obtained from the non-fractional Lindley distribution by replacing the support T = R + by T = N ∖ { 0 } and applying time scale theory, whose ambition is to unify the theories of difference equations and differential equations, integral and differential calculus, and the calculus of finite differences. It thus provides a framework for the study of dynamical systems in discrete-continuous time. Delta moments are discrete-time Laplace transforms of the frequency function of the fractional Lindley distribution. The parameter of the fractional Lindley distribution is estimated by least squares, weighted least squares, maximum likelihood, moments, and proportions. The moment estimator always exists, so that delta moments result from the nabla Laplace transform of the frequency function of the fractional Lindley distribution. The maximum likelihood estimates have the least mean-square errors. The proportion method works satisfactorily only when the mode of the distribution is null and the proportion of zeros is high. A simulation allows for quantifying the mean-square errors associated with the estimators. A count regression based on the fractional Lindley distribution with data on the total number of stays after hospital admission among U.S. residents aged 65 and over shows that the Akaike information criteria is significantly lower than with the uniform Poisson and Poisson regressions.

A generalization of the Lindley Distribution

The Lindley distribution is useful in a wide variety of fields, such as biology and astronomy. Many generalizations of the Lindley distribution have been introduced in the literature, with various motivations. Inspired by the concept of superstatistics in nonequilibrium statistical mechanics, we introduce here a novel generalization of the Lindley distribution, by regarding its shape parameter as a stochastic variable. This procedure is particularly well motivated in astronomy applications, since this parameter may fluctuate over a large spatiotemporal scale. By appealing to the central limit theorem, two families of distributions are constructed, associated with additive and multiplicative random processes. We discuss in detail these distributions, we study their asymptotic behavior, and compute their moments. We briefly discuss their possible use in astronomy, biology, and elsewhere.

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 new three-parameter lifetime distribution for environmental data analysis: the Harris extended modified Lindley distribution

A new three-parameter lifetime distribution for environmental data analysis: the Harris extended modified Lindley distribution

Group Acceptance Sampling Plans Based on Truncated Life Tests for Gamma Lindley Distribution with Real Data Application

Group Acceptance Sampling Plans Based on Truncated Life Tests for Gamma Lindley Distribution with Real Data Application

Gini index based goodness-of-fit test for the Lindley distribution

In survival analysis and reliability studies the Lindley distribution has been widely used. In this article, we introduce a new goodness of fit test for the Lindley distribution based on an estimate of the Gini index. The asymptotic distribution of the new test statistic is investigated. The unknown parameter of the Lindley distribution is estimated by the maximum likelihood estimator. The critical points and the actual sizes of the test are obtained by Monte Carlo simulations. A power comparison study of the existing tests is performed. It is shown that the Gini index-based test that we proposed performs well by comparing its powers with that of other tests. A real-life data analysis is finally presented for illustrative purpose.

Bivariate Poisson Generalized Lindley Distributions and the Associated BINAR(1) Processes

This paper proposes new bivariate distributions based on the Poisson generalized Lindley distribution as marginal. These models include the basic bivariate Poisson generalized Lindley (BPGL) and the Sarmanov-based bivariate Poisson generalized Lindley (SPGL) distributions. Subsequently, we introduce the BPGL and SPGL distributions as joint innovation distributions in a novel bivariate first-order integer-valued autoregressive process (BINAR(1)) based on binomial thinning. The model parameters in the BPGL and SPGL distributions are estimated using the method of maximum likelihood (ML) while we apply the conditional maximum likelihood (CML) for the BINAR(1) process. We conduct some simulation experiments to assess the small and large sample performances. Further, we implement the new BINAR(1)s to the Pittsburgh crime series data and they show better fitting criteria than other competing BINAR(1) models in the literature.