This paper investigates properties of a new parametric distribution generated by Marshall and Olkin ( 1997 Marshall , A. W. , Olkin , I. ( 1997 ). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families . Biometrika 84 : 641 – 652 . [Crossref], [Web of Science ®] [Google Scholar] ) 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.

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.

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.

In this paper, we establish the existence and uniqueness of the maximum-likelihood estimates of the parameters of a general class of inverse exponentiated distributions based on complete as well as progressively Type-I and Type-II censored data.

In this paper, we present a formal simple proof for the existence and uniqueness of the maximum likelihood estimates (MLEs) of the parameters of a general class of exponentiated distributions. This class includes the exponentiated (general) exponential, exponentiated Rayleigh (scaled Burr X) and exponentiated Pareto distributions, as special cases, and thus the proof given here establishes the existence and uniqueness of the MLEs for these important special cases as well.