A quasi Lindley distribution

In survival analysis interest lies in modeling data that describe the time to a particular event. Informative functions, namely the hazard function and mean residual life function, can be obtained from the model's distribution function. We focus on the mean residual life function which provides the expected remaining life given that the subject has survived (i.e., is event-free) up to a particular time. This function is of direct interest in reliability, medical, and actuarial fields. In addition to its practical interpretation, the mean residual life function characterizes the survival distribution. In terms of mean residual life function inference, there are two shortcomings present in the current literature. First off, the shape of the functional is often restricted, which forces the researcher to make an assumption that may not be appropriate. Secondly, in cases where the shape of the functional is not parametrically specified, full inference is not obtained. The aim of our research is to provide a modeling approach that yields full inference for the mean residual life function, and is not restrictive on the shape of the functional. In particular, we develop general Bayesian nonparametric modeling methods for inference for mean residual life functions built from a mixture model for the associated survival distribution. Although the prior model is not placed on the mean residual life function directly, our methods offer rich inference for the desired functional. We place a Dirichlet process mixture model on the survival function, and discuss the importance of careful kernel selection to ensure desirable properties for the mean residual life function. We advocate for a mixture model with a gamma kernel and dependent baseline distribution for the Dirichlet process prior. We extend our model to the regression setting by modeling the joint distribution for the survival response and random covariates. This approach provides a flexible method for obtaining inference for the regression functionals when the number of random covariates is small to moderate. We further extend our methods to the scenario where interest lies in comparison of survival between two experimental groups. Typically, we expect the range of survival in the two groups to be the same, but exhibiting different characteristics over that range. Here, we develop a dependent Dirichlet process prior for the mixing distributions having shared locations across the two groups and varying weights to incorporate dependency between populations and achieve richer inferential results. The final scenario we consider is the case in which the researcher believes two populations have ordered mean residual life functions. For such applications, a prior model that incorporates an ordering constraint on the mean residual life functions is attractive. We introduce a mixture of Erlang distributions with weights constructed using Dirichlet process priors that provides the mean residual life ordering result. We demonstrate the utility of our modeling methods through simulation and real data examples. In addition, we draw comparisons with both parametric and semiparametric models.

In this paper, we introduce a new continuous probability distribution which is based on the product of linear and exponential functions having a single parameter and hence, we named it ‘One-parameter Linear-exponential distribution’. Different structural properties such as probability density function, probability distribution function and moment generating function of this distribution have been derived. Moments about origin and hence, the first four central moments of the proposed distribution have been obtained. The hazard rate function and the mean residual life function of the proposed distribution have been discussed, and show its flexibility over Lindley distribution. Estimation of parameters has been discussed by the method of moments as well as the method of maximum likelihood. Goodness of fit has been applied to different data-sets which were used by others. It has been observed that the proposed distribution gives better fit to the most of data-sets of same nature, having variance greater than the mean, than Lindley distribution.

A two-parameter generalization of Sujatha distribution (AGSD), which includes Lindley distribution and Sujatha distribution as particular cases, has been proposed. It's important mathematical and statistical properties including its shape for varying values of parameters, moments, coefficient of variation, skewness, kurtosis, index of dispersion, hazard rate function, mean residual life function, stochastic ordering, mean deviations, Bonferroni and Lorenz curves, and stress-strength reliability have been discussed. Maximum likelihood estimation method has been discussed for estimating its parameters. AGSD provides better fit than Sujatha, Aradhana, Lindley and exponential distributions for modeling real lifetime data.Journal of Institute of Science and TechnologyVolume 22, Issue 1, July 2017, Page: 66-83

In the present paper, a new one parameter lifetime distribution named, “Adya distribution’ has been proposed for modeling lifetime data from engineering. Its various statistical properties including moments and moments based measures, hazard rate function, mean residual life function, stochastic ordering, deviations from the mean and the median, Bonferroni and Lorenz curves, and stress-strength reliability have been studied. Both the method of moment and the maximum likelihood estimation have been discussed for estimating the parameter of the proposed distribution. A numerical example has been presented to test the goodness of fit of the proposed distribution over other one parameter lifetime distributions available in statistical literature.

In this paper, we introduce a new distribution named as the Pseudo Lindley Distribution (PsLD) as a generalization of the Lindley distribution (LD). A full and detailed description are provided in terms of moments, cumulates, characteristic function, failure, rate function, stochastic ordering, distributions of sums, and parameters estimation. Simulations studies and data driven applications are also reported. Keywords: Lindley distribution; Exponential distribution; Gamma distribution; Stochastic ordering; Maximum-likelihood estimation.

This article introduces a two-parameter exponentiated Teissier distribution. It is the main advantage of the distribution to have increasing, decreasing and bathtub shapes for its hazard rate function. The expressions of the ordinary moments, identifiability, quantiles, moments of order statistics, mean residual life function and entropy measure are derived. The skewness and kurtosis of the distribution are explored using the quantiles. In order to study two independent random variables, stress–strength reliability and stochastic orderings are discussed. Estimators based on likelihood, least squares, weighted least squares and product spacings are constructed for estimating the unknown parameters of the distribution. An algorithm is presented for random sample generation from the distribution. Simulation experiments are conducted to compare the performances of the considered estimators of the parameters and percentiles. Three sets of real data are fitted by using the proposed distribution over the competing distributions.

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.

In reliability studies, the expected additional life time given that a component has survived until time t is called the mean residual life function (MRLF). This MRLF determines the distribution function uniquely. In this paper an interpretation of MRLF in renewal theory is presented and some characterizations of the exponential distribution are obtained. Finally, considering the general MRLF, a method is developed for obtaining the mixing distribution when the original distribution is exponential. Some examples are discussed, in one of which Morrison’s (1978) result is obtained as a special case.

Statistical dependence and stochastic orders are two common tools used in distribution theory or, broadly speaking, reliability theory. In the context of the present work, we draw attention to the fact that reliability practitioners usually assume that the components constituting a system work independently. However, there are instances where the components are found to be stochastically dependent following a certain multivariate distribution. This gives rise to some error in computation of aging functions. In this paper, we propose a method based on statistical dependence and stochastic orders to compute the error incurred due to the faulty assumption that components are independently working. Our method makes the task of error analysis readily comprehensible. Earlier, the authors working in this direction considered some bivariate distributions and inferred about over/under assessment (OA/UA) of aging functions in silos, viz., survival function (SF), hazard rate, mean residual life function. However, in our work, we propose that OA/UA in the SF is closely related to statistical dependence, and OA/UA in some specific aging functions can be inferred with the help of stochastic orders, assuming that the orders for some other class of aging functions are known.

پدیده یخبندان دیررس رخدادی فرین و زیانبخش جوی بوده که در شاخه های مختلف مهندسی مانند کشاورزی، باغبانی و غیره بررسی می شود. تحلیل احتمالی این رخداد و به کارگیری نتایج آن می تواند آسیب های وارده به بخش کشاورزی، باغبانی و غیره را کاهش دهد. موضوع این تحقیق تحلیل آخرین روز یخبندان کلیه ایستگاه های همدید خراسان رضوی است. هشت توزیع احتمالی نرمال، گامبل نوع 1، گامای2 پارامتری، لوگ نرمال 2 و 3 پارامتری، پارتوی تعمیم یافته، حدی تعمیم یافته و پیرسون نوع 3 بر داده ها برازش و پارامترها با هفت روش (5 روش گشتاوری، حداکثر درست نمائی و آنتروپی) برآورد شد. بهترین تابع برازشی برای هر ایستگاه با آزمون K-S انتخاب و آخرین روز یخبندان در دوره های بازگشت مختلف با توزیع برتر پیش بینی شد. بهترین تابع برازشی در ایستگاه ها متفاوت و عمدتا شامل پارتوی تعمیم یافته، پیرسون نوع 3 و لوگ نرمال 3 پارامتری است. بیشترین کاربرد روش برآورد پارامترها مربوط به انواع گشتاورها است. بنابراین نمی توان یک تابع برازشی با یک روش خاص برآورد پارامترها را در بررسی این پدیده پیشنهاد داد. همچنین بررسی روابط رگرسیونی بین دوره های بازگشت مختلف آخرین روز یخبندان و شاخص های دمائی سالانه هر ایستگاه نشان داد که می توان آخرین روز یخبندان را براساس متوسط دمای کمینه هر ایستگاه با دقت مناسب برآورد کرد.

The skew-normal distribution proposed by Azzalini [1] is suitable for the analysis of data exhibiting a unimodal empirical distribution function but having some skewness present, a structure often occurring in data analysis. In this paper we study the skew-normal distribution from a reliability point of view. More specifically, we obtain the failure rate, the mean residual life function (MRLF) and the reliability function of the skew-normal distribution. We also compare it with the normal distribution with respect to certain stochastic orderings and prove that the failure rate of the skew-normal distribution is increasing, a property enjoyed by the normal distribution. Appropriate machinery is developed to obtain the reliability in the strength-stress model where the variables involved follow the skew-normal distribution. Finally, data from Roberts [2] is analyzed to illustrate the results.

The purpose of this article is to study the bivariate distributions with second kind of beta conditionals, presented by Castillo and Sarabia, from a reliability point of view. The reliability functions, viz the survival function, the failure rate and mean residual life function, are derived and their monotonicity is discussed for the univariate model as well as the conditional model. Finally, Clayton’s association measure is obtained in terms of the hazard gradient and its value, in the case of our model, is derived.

To represent lifetime data, a new one-parameter lifetime distribution referred to as ''Sujit distribution'' has been developed. This model is a mixture of the gamma and exponential distributions. It has been discussed how the recommended distribution's moments, kurtosis, shape, hazard rate function, skewness, as well as mean residual functions are among its key characteristics. Maximum Likelihood Valuation has been utilized to evaluate the parameters. A real-world data set was used in the last session to determine the distribution's best fit.. KEYWORDS :Lifetime distribution, Mean residual life function, Hazard rate function, Moments.

An overview is presented of some theoretical results concerning the mean residual life function used in reliability theory. An extension of the concept to the bivariate case is introduced, and the relationship between the reliability and mean residual life function is derived. The properties of the function and conditions for asymptotic exponentiality of component life lengths are discussed. >

This study introduces the Lindley Stochastic Frontier Analysis—LSFA model, a novel approach that incorporates the Lindley distribution to enhance the flexibility and accuracy of efficiency estimation. The LSFA model is compared against traditional SFA models, including the half-normal, exponential, and gamma models, using advanced numerical methods such as the Gauss–Hermite Quadrature, Monte Carlo Integration, and Simulated Maximum Likelihood Estimation for parameter estimation. Simulation studies revealed that the LSFA model outperforms in scenarios involving small sample sizes and complex, skewed distributions, particularly those characterized by gamma distributions. In contrast, traditional models such as the half-normal model perform better in larger samples and simpler settings, while the gamma model is particularly effective under exponential inefficiency distributions. Among the numerical techniques, the Gauss–Hermite Quadrature demonstrates a strong performance for half-normal distributions, the Monte Carlo Integration offers consistent results across models, and the Simulated Maximum Likelihood Estimation shows robustness in handling gamma and Lindley distributions despite higher errors in simpler cases. The application to a banking dataset assessed the performance of 12 commercial banks pre-COVID-19 and during COVID-19, demonstrating LSFA’s superior ability to handle skewed and intricate data structures. LSFA achieved the best overall reliability in terms of the root mean square error and bias, while the gamma model emerged as the most accurate for minimizing absolute and percentage errors. These results highlight LSFA’s potential for evaluating efficiency during economic shocks, such as the COVID-19 pandemic, where data patterns may deviate from standard assumptions. This study highlights the advantages of the Lindley distribution in capturing non-standard inefficiency patterns, offering a valuable alternative to simpler distributions like the exponential and half-normal models. However, the LSFA model’s increased computational complexity highlights the need for advanced numerical techniques. Future research may explore the integration of generalized Lindley distributions to enhance model adaptability with enriched numerical optimization to establish its effectiveness across diverse datasets.