NONHOMOGENEOUS BIRTH PROCESSES AND l_∞-SPHERICAL DENSITIES, WITH APPLICATIONS IN RELIABILITY THEORY

Abstract

In this article we characterize l∞-spherical density functions by means of epoch times of nonhomogeneous pure birth processes. Some further properties of l∞-spherical densities, such as Schur-concavity, positive dependence, and stochastic comparisons, are also given. The relationships of l∞-spherical densities to notions of interest in reliability theory are highlighted.

Reliability, signature, and relative quality functions of systems under time‐homogeneous load‐sharing models

“…Such distributions rather arise by adequately extending to the multivariate case the “lack of memory property” and are described by (Schur‐constant) joint densities of the form See, eg, the monograph of Spizzichino for motivations, technical details, discussion, and bibliographic references. A simple connection exists between the densities of form and those related with l ∞ ‐spherical densities (see, in particular, the work of Shaked et al).…”

“…See, eg, the monograph of Spizzichino 26 for motivations, technical details, discussion, and bibliographic references. A simple connection exists between the densities of form (40) and those related with l ∞ -spherical densities (see, in particular, the work of Shaked et al 29 ).…”

“…Exchangeable load‐sharing models are also related with the theme of sequential order statistics (see, in particular, the work of Kamps). Connections with birth processes and l ∞ ‐spherical densities have been analyzed in the work of Shaked et al Several different aspects of the time‐homogeneous linear breakdown case with L ( t ) = θ in have been analyzed in monograph of Spizzichino …”

The de Finetti structure behind some norm-symmetric multivariate densities with exponential decay

“…, Intuitively, between T i− and T i each single component has the same failure intensity r i and the components work independently, so that the (a priori unknown) next component that is hit at time T i has intensity (d − i + ) r i , since the minimum of d − i + independent exponentials with rate r i has this rate. This explains the explicit occurrence of the constant d − i + before r i in our notation, which is slightly di erent to the one in [24] for instance. The density of (X , .…”

Nonnegativity of Covariances Between Functions of Ordered Random Variables

“…0 are constants: It is known from Grandell [16, pp. 67] or Shaked et al [32] that such a process is also a mixed Poisson process with G in (3.4) having G(b, g) distribution, whose density function is given by…”