In this paper, we define weighted failure rate and their means from the stand point of an application. We begin by emphasizing that the formation of n independent component series system having weighted failure rates with sum of weight functions being unity is same as a mixture of n distributions. We derive some parametric and non-parametric characterization results. We discuss on the form invariance property of baseline failure rate for a specific choice of weight function. Some bounds on means of aging functions are obtained. Here, we establish that weighted increasing failure rate average (IFRA) class is not closed under formation of coherent systems unlike the IFRA class. An interesting application of the present work is credited to the fact that the quantile version of means of failure rate is obtained as a special case of weighted means of failure rate.

The theory of utility is a well-known method of constructing insurance premiums (see e.g., Newton et al. (1986) Actuarial Mathematics. Itasca, Illinois: The Society of Actuaries.). Furman and Zitikis ((2008) Insurance: Mathematics and Economics, 42, 459–465.) proposed an alternative method using the mean value of a weighted random variable. According to this approach, for various choices of weighting, popular premiums such as net premium, modified variance premium, Esscher premium, and Kamps premium are obtained. On the other hand, some premiums cannot be obtained with this method, such as the premium of the exponential principle. In this paper, we provide a complementary theory by introducing a family of unimodal weighted distributions for which the mode is a premium principle.

A method for the construction of Stein-type covariance identities for a nonnegative continuous random variable is proposed, using a probabilistic analogue of the mean value theorem and weighted distributions. A generalized covariance identity is obtained, and applications focused on actuarial and financial science are provided. Some characterization results for gamma and Pareto distributions are also given. Identities for risk measures which have a covariance representation are obtained; these measures are connected with the Bonferroni, De Vergottini, Gini, and Wang indices. Moreover, under some assumptions, an identity for the variance of a function of a random variable is derived, and its performance is discussed with respect to well-known upper and lower bounds.

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.

Bounds of the quantile entropy in the past lifetime of some ageing classes are explored firstly. The quantile entropy in the past lifetime of a random variable is shown to be increasing if its expected inactivity time is increasing. Some closure properties of the less quantile entropy in the past lifetime order are obtained under the model of generalized order statistics. Moreover, sufficient conditions are given for a function of a random variable and for a weighted random variable to have more quantile entropy in the past lifetime than original random variable.