Generalized Entropies, Variance and Applications

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

Stochastic comparisons of interfailure times under a relevation replacement policy

We provide some results for the comparison of the failure times and interfailure times of two systems based on a replacement policy proposed by Kapodistria and Psarrakos (2012). In particular, we show that when the first failure times are ordered in terms of the dispersive order (or, the excess wealth order), then the successive interfailure times are ordered in terms of the usual stochastic order (respectively, the increasing convex order). As a consequence, we provide comparison results for the cumulative residual entropies of the systems and their dynamic versions.

On sufficient conditions for the comparison in the excess wealth order and spacings

The purpose of this paper is twofold. On the one hand, we provide sufficient conditions for the excess wealth order. These conditions are based on properties of the quantile functions which are useful when the dispersive order does not hold. On the other hand, we study sufficient conditions for the comparison in the increasing convex order of spacings of generalized order statistics. These results will be combined to show how we can provide comparisons of quantities of interest in reliability and insurance.

Two Variability Orders

In this paper we study a new variability order that is denoted by ≤st:icx. This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤st:icxY implies that Var[h(X)] ≤ Var[h (Y)] for all increasing convex functions h. The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤st:icxY are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤st:icx are derived, and some typical applications in queueing theory are described.