Dhaifalla K. Al-Mutairi

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 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.

Estimation of reliability from a bivariate log-normal data

It has been established that the bivariate log-normal distribution is appropriate for modelling certain paired observations. In this paper, we have developed large-sample confidence intervals of the dependence and reliability R=P(X>Y) parameters from a bivariate log-normal distribution with equal log-normal means. The parameter R provides a general measure of difference between the two populations and has applications in many areas. The performance of these confidence intervals has been examined by extensive simulation studies. The results are illustrated with an example dealing with a quantitative assay problem.

An adaptive concatenated failure rate model for software reliability

Abstract This article introduces a software reliability model whose concatenated failure rate function is motivated via considerations that reflect an engineers knowledge about the stochastic nature of software failures. The model is adaptive (in a sense explained), has two parameters, and has characteristics that generalize those of existing models. A Bayesian approach for estimating the model parameters and for testing hypotheses about reliability growth is proposed. The prior distributions reflect structural considerations, and Markov chain Monte Carlo techniques are used to implement the approach.

Inferences on Stress-Strength Reliability from Weighted Lindley Distributions

This article deals with the estimation of the stress-strength parameter R = P(Y < X), when X and Y are two independent weighted Lindley random variables with a common shape parameter. The MLEs can be obtained by maximizing the profile log-likelihood function in one dimension. The asymptotic distribution of the MLEs are also obtained, and they have been used to construct the asymptotic confidence interval of R. Bootstrap confidence intervals are also proposed. Monte Carlo simulations are performed to verify the effectiveness of the different estimation methods, and data analysis has been performed for illustrative purposes.

A large deviation principle for bootstrapped sample means

A large deviation principle for bootstrapped sample means is established. It relies on the Bolthausen large deviation principle for sums of i.i.d. Banach space valued random variables. The rate function of the large deviation principle for bootstrapped sample means is the same as the classical one.