M. E. Ghitany

Marshall–Olkin Extended Lomax Distribution and Its Application to Censored Data

This paper investigates properties of a new parametric distribution generated by Marshall and Olkin (1997) extended family of distributions based on the Lomax model. We show that the proposed distribution can be expressed as a compound distribution with mixing exponential model. Simple sufficient conditions for the shape behavior of the density and hazard rate functions are given. The limiting distributions of the sample extremes are shown to be of the exponential and Fréchet type. Finally, utilizing maximum likelihood estimation, the proposed distribution is fitted to randomly censored data.

Power Lindley distribution and associated inference

A new two-parameter power Lindley distribution is introduced and its properties are discussed. These include the shapes of the density and hazard rate functions, the moments, skewness and kurtosis measures, the quantile function, and the limiting distributions of order statistics. Maximum likelihood estimation of the parameters and their estimated asymptotic standard errors are derived. Three algorithms are proposed for generating random data from the proposed distribution. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators of the parameters as well as the coverage probability and the width of the confidence interval for each parameter. An application of the model to a real data set is presented finally and compared with the fit attained by some other well-known two-parameter distributions.

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 distributions parameter are compared in small and large samples. Application of the model to real data is given.

Original article: A two-parameter weighted Lindley distribution and its applications to survival data

Abstract: A two-parameter weighted Lindley distribution is proposed for modeling survival data. The proposed distribution has the property that the hazard rate (mean residual life) function exhibits bathtub (upside-down bathtub) or increasing (decreasing) shapes. Simulation studies are conducted to investigate the performance of the maximum likelihood estimators and the asymptotic confidence intervals of the parameters. Applications of the proposed model to real survival data are presented.

Inferences on Stress-Strength Reliability from Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X) when X and Y are independent Lindley random variables with different shape parameters. The uniformly minimum variance unbiased estimator has explicit expression, however, its exact or asymptotic distribution is very difficult to obtain. The maximum likelihood estimator of the unknown parameter can also be obtained in explicit form. We obtain the asymptotic distribution of the maximum likelihood estimator and it can be used to construct confidence interval of R. Different parametric bootstrap confidence intervals are also proposed. Bayes estimator and the associated credible interval based on independent gamma priors on the unknown parameters are obtained using Monte Carlo methods. Different methods are compared using simulations and one data analysis has been performed for illustrative purposes.

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.

ESTIMATION OF RELIABILITY FROM MARSHALL-OLKIN EXTENDED LOMAX DISTRIBUTION

In this paper, we are mainly interested in estimating the reliability R=P(X>Y) in the Marshall–Olkin extended Lomax distribution, recently proposed by Ghitany et al. [Marshall–Olkin extended Lomax distribution and its application, Commun. Statist. Theory Methods 36 (2007), pp. 1855–1866]. The model arises as a proportional odds model where the covariate effect is replaced by an additional parameter. Maximum likelihood estimators of the parameters are developed and an asymptotic confidence interval for R is obtained. Extensive simulation studies are carried out to investigate the performance of these intervals. Using real data we illustrate the procedure.

On a recent generalization of gamma distribution

In this paper, we investigate a generalized gamma distribution recentIy developed by Agarwal and Kalla (1996). Also, we show that such generalized distribution, like the ordinary gamma distribution, has a unique mode and, unlike the ordinary gamma distribution, may have a hazard rate (mean residual life) function which is upside-down bathtub (bathtub) shaped.

Estimation of the Reliability of a Stress-Strength System from Power Lindley Distributions

In this paper, we are interested in the estimation of the reliability parameter R = P(X > Y) where X, a component strength, and Y, a component stress, are independent power Lindley random variables. The point and interval estimation of R, based on maximum likelihood, nonparametric and parametric bootstrap methods, are developed. The performance of the point estimate and confidence interval of R under the considered estimation methods is studied through extensive simulation. A numerical example, based on a real data, is presented to illustrate the proposed procedure.

Comparison of estimation methods for the parameters of the weighted Lindley distribution

The aim of this paper is to compare through Monte Carlo simulations the finite sample properties of the estimates of the parameters of the weighted Lindley distribution obtained by four estimation methods: maximum likelihood, method of moments, ordinary least-squares, and weighted least-squares. The bias and mean-squared error are used as the criterion for comparison. The study reveals that the ordinary and weighted least-squares estimation methods are highly competitive with the maximum likelihood method in small and large samples. Statistical analysis of two real data sets are presented to demonstrate the conclusion of the simulation results.