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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.

All UK insurers exposed to longevity risk need to perform stress tests for their Individual Capital Assessment (ICA). Some have put in place deterministic models which are arguably too simple; others have developed stochastic models that can be demanding and complex.This paper presents a simple model to turn any deterministic mortality scenario into a stochastic model. We propose a simple model of stochastic variation that is easy to explain and to implement, which could be an alternative to and/or complete some of the well known models. The model can be applied to any best estimates of future mortality rates, as it aims to describe how longevity behaves around the projected expected values.The paper proposes a possible calibration on the England and Wales population mortality that produces a minimum indication of possible future variation and uses the results to validate the model's assumptions. Using sample portfolios and the stochastic model, we can simulate cash flows to determine the distribution of the net present values (NPV) of annuity outgo.

This work focuses on Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) for point processes in discrete time. As our main purpose, we show how definitions and results presented in Shaked, Spizzichino and Suter [11] can be unified and generalized in the frame of occupancy models. We first show some general facts about EOM's. Then we introduce a class of EOM's, called M (a) -models, and a concept of generalized Uniform Order Statistics Property in discrete time. For processes with this property, we prove a general characterization result in terms of M (a) -models. Our interest is also focused on properties of closure w.r.t. some natural transformations of EOM's.The so called occupancy distributions give rise, as well known, to a class of multivariate models useful in the description of randomized phenomena. The name "occupancy" comes from the interpretation in terms of particles that are randomly distributed among several cells. In particular, three classical examples, related to as many well-known physical systems, belong to this class: Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac model, see Feller [5]. In the Maxwell-Boltzmann (MB) statistics the capacity of each cell is unlimited, and the particles are distinguishable. In the Bose-Einstein (BE) statistics, the capacity of each cell is unlimited but the particles are indistinguishable. In the Fermi Dirac (FD) statistics, the particles are indistinguishable, but cells can only hold a maximum of one particle. All these three statistics assume that the cells are distinguishable. These models are attractive for many reasons. First of all they have wide range of applications in Sciences, Engineering and also in Statistics, as pointed out by Charalambides [2, Chapters 4 and 5] and Gadrich and Ravid [6]. Moreover Mahmoud [10] provided an interpretation of occupancy distributions in terms of Pólya Urns, which are very flexible and applicable to problems arising in various areas; e.g. Clinical Trials (see Crimaldi and Leisen [4] for some references), Economics (see Aruka [1]) and Computer Science (see Shah, Kothari, Jayadeva andChandra [13]). From a probabilistic and combinatoric point of view, the three fundamental models (MB, BE and FD) have many interesting properties. Indeed they are, in particular, exchangeable and this is a basic remark for our aims. This paper is in fact concentrated on the theme of Exchangeable Occupancy Models (EOM) and their relations with the Uniform Order Statistics Property (UOSP) of counting processes in discrete time. As one main purpose of ours, we show that some notions and results given in Shaked, Spizzichino and Suter [11,12] admit completely natural generalizations in the frame of EOM's.After appropriate preliminaries, we will consider some general properties of the EOM's. For our purposes, we then introduce the notion of M (a) -models, a relevant sub-class of EOM's that turns out to have an important role in our derivations. In the final part of the paper, we will introduce ...

In this paper we recall, extend and compute some information measures for the concomitants of the generalized order statistics (GOS) from the Farlie–Gumbel–Morgenstern (FGM) family. We focus on two types of information measures: some related to Shannon entropy, and some related to Tsallis entropy. Among the information measures considered are residual and past entropies which are important in a reliability context.