Stochastic Comparisons of Weighted Distributions and Their Mixtures

In this paper, various stochastic ordering properties of a parametric family of weighted distributions and the associated mixture model are developed. The effect of stochastic variation of the output random variable with respect to the parameter and/or the underlying random variable is specifically investigated. Special weighted distributions are considered to scrutinize the consistency as well as the usefulness of the results. Stochastic comparisons of coherent systems made of identical but dependent components are made and also a result for comparison of Shannon entropies of weighted distributions is developed.

PRESERVATION OF LOG-CONCAVITY AND LOG-CONVEXITY UNDER OPERATORS

Log-concavity [log-convexity] and their various properties play an increasingly important role in probability, statistics, operations research and other fields. In this paper, we first establish general preservation theorems of log-concavity and log-convexity under operator $\phi \longmapsto T(\phi , \theta )=\mathbb {E}[\phi (X_\theta )]$ , θ ∈ Θ, where Θ is an interval of real numbers or an interval of integers, and the random variable $X_\theta$ has a distribution function belonging to the family $\{F_\theta , \theta \in \Theta \}$ possessing the semi-group property. The proofs are based on the theory of stochastic comparisons and weighted distributions. The main results are applied to some special operators, for example, operators occurring in reliability, Bernstein-type operators and Beta-type operators. Several known results in the literature are recovered.

Connection of Generalized Failure Rate and Generalized Reversed Failure Rate with Inequality Curves

The purpose of this paper is to introduce some properties of generalized failure rate (GFR) and generalized reversed failure rate (GRFR), study their connection with inequality measures and then apply it to make comparisons of some reliability models. In the second part of our paper, we obtain the properties for the GRFR based on the properties of the GFR and weighted distributions. The importance of such a study lies in the usefulness of using. Finally, we present two examples to illustrate the applications of the relationships outlined in this paper. A real dataset is used for clarifying presented concepts.

Form-Invariance of the Non-Regular Exponential Family of Distributions

The weighted distributions are used when the sampling mechanism records observations according to a nonnegative weight function. Sometimes the form of the weighted distribution is the same as the original distribution except possibly for a change in the parameters that is called the form-invariant weighted distribution. In this paper, by identifying a general class of weight functions, we introduce an extended class of form-invariant weighted distributions belonging to the non-regular exponential family which included two common families of distribution: exponential family and non-regular family as special cases. Some properties of this class of distributions such as the sufficient and minimal sufficient statistics, maximum likelihood estimation and the Fisher information matrix are studied.