Stochastic Order Relations Research Articles

STOCHASTIC COMPARISONS IN REVENUE MANAGEMENT UNDER A DISCRETE CHOICE MODEL OF CONSUMER BEHAVIOR

This study examines the monotonicity properties of expected revenue in a revenue management problem with respect to the consumers' choice behavior and the market size. The consumer behavior is described by a general discrete choice model. The firm decides which subset of fare products to offer at each time period. An example shows that the usual stochastic order relation in consumers' preference over the set of fare products is not sufficient for our naive intuition regarding the monotonicity to hold true. We provide sufficient conditions under which our intuition is valid. These conditions identify desirable changes in consumer behavior for the firm.

Multivariate Stochastic Orders Induced by Case-Control Sampling

It is shown that retrospective sampling induces stochastic order relations in case-control studies. More specifically if the regression function is increasing and the covariates are positively dependent, then the covariates for cases are larger, with respect to some multivariate stochastic order, than the covariates of the controls. Strong dependence concepts yield strong multivariate stochastic orders. Conversely, different multivariate stochastic orders imply different monotonicity properties on the regression function. The results carry over to marginal models, transformed models and to problems involving confounders. The results set forth a new theoretical foundation for the analysis of case-control studies.

The Theory of Assortative Matching Based on Costly Signals

We study two-sided markets with a finite number of agents on each side, and with two-sided incomplete information. Agents are matched assortatively on the basis of costly signals. Asymmetries in signalling activity between the two sides of the market can be explained by asymmetries either in size or in heterogeneity. Our main results identify general conditions under which the potential increase in expected output due to assortative matching (relative to random matching) is offset by the costs of signalling. Finally, we examine the limit model with a continuum of agents and point out differences and similarities to the finite version. Technically, the paper is based on the elegant theory about stochastic order relations among differences of order statistics, pioneered by Barlow and Proschan in 1966 in the framework of reliability theory.

Dependence in failure times due to environmental factors

In many applications, common factors influence a set of failure (or survival) times of interest. This is, for instance, the case in mortality analysis, where mortality is influenced by socio-economic and health factors, or in the analysis of time-to-failure observations, where the items are functioning under similar conditions. This paper aims to answer the following question: is it true that more positively dependent factors generate more positively dependent outcomes? The analysis is conducted with the help of various stochastic order relations.

On Immunisation andS-Convex Extremal Distributions

ABSTRACTThe paper concerns the interest risk management of insurance companies or banks. Classes of stochastic order relations for arbitrary discrete random variables are used to find extremal strategies of immunisation in the context of deterministic immunisation theory. In a special case, the results obtained by Hürlimann (2002) are extended to conditions for immunisation under arbitraryS-convex orS-concave shift factors of the term structure of interest rates. The notion of the Shiu measure is generalised to an immunisation risk measure, accounting for more moments of the asset and liability risks.

ROLE OF EQUILIBRIUM DISTRIBUTION IN RELIABILITY STUDIES

The equilibrium distribution arises as the limiting distribution of the forward recurrence time in a renewal process. The purpose of this article is to study the relationships between the equilibrium distributions (including its higher derivates) and the original distributions. Some stochastic order relations and the relations between their aging properties are investigated and some applications in the field of insurance and financial investments are given. In addition, the relation between the equilibrium distribution of a series system and the series system of equilibrium distribution is investigated. Bivariate equilibrium distribution whose reliability properties are consistent with those of the univariate equilibrium distribution is defined.

PRESERVATION OF STOCHASTIC ORDERS UNDER MIXTURES OF EXPONENTIAL DISTRIBUTIONS

Recently, Bartoszewicz [5,6] considered mixtures of exponential distributions treated as the Laplace transforms of mixing distributions and established some stochastic order relations between them: star order, dispersive order, dilation. In this article the preservation of the likelihood ratio, hazard rate, reversed hazard rate, mean residual life, and excess wealth orders under exponential mixtures is studied. Some new preservation results for the dispersive order are given, as well as the preservation of the convex transform order, and the star one is discussed.

Discrete [formula omitted]-convex extremal distributions: Theory and applications

Given a nondegenerate moment space with s fixed moments, explicit formulas for the discrete s -convex extremal distribution have been derived for s = 1 , 2 , 3 (see [M. Denuit, Cl. Lefèvre, Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences, Insurance Math. Econom. 20 (1997) 197–214]). If s = 4 , only the maximal distribution is known (see [M. Denuit, Cl. Lefèvre, M. Mesfioui, On s -convex stochastic extrema for arithmetic risks, Insurance Math. Econom. 25 (1999) 143–155]). This work goes beyond this limitation and proposes a method for deriving explicit expressions for general nonnegative integer s . In particular, we derive explicitly the discrete 4-convex minimal distribution. For illustration, we show how this theory allows one to bound the probability of extinction in a Galton–Watson branching process. The results are also applied to derive bounds for the probability of ruin in the compound binomial and Poisson insurance risk models.

Stochastic Order Relations and Lattices of Probability Measures

We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with stochastic dominance constraints. In particular we show that the space of probability measures on $\mathbb{R}$ is a lattice under most of the known partial orders, whereas the space of probability measures on $\mathbb{R}^d$ typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions on the dependence structure of the measures, are lattices.

Discrete Stochastic Arithmetic for Validating Results of Numerical Software

The Discrete Stochastic Arithmetic DSA is a probabilistic approach for round-off error propagation. After a brief review of the CESTAC (Controle et Estimation Stochastique des Arrondis de Calculs) method, which is the basis of DSA, the concept of the “informatical zero”, also called “computational zero”, is defined. The stochastic order relations of the DSA are presented. The DSA is the joint use of the synchronous implementation of the CESTAC method and the stochastic order relations. After having summarized the asynchronous implementation of the CESTAC method, which has been used in the Prosolver software, and which has been legitimately criticized, the synchronous implementation is presented. Then the CADNA (Control of Accuracy and Debugging for Numerical Application) library which implements the DSA arithmetic is presented. It is shown that this library is able to dynamically control the validity of the hypotheses which must hold so that results provided by CESTAC method are reliable. If the hypotheses do not hold then warnings are printed in a special file. The user is informed that numerical anomalies have been detected. Depending on these warnings the user may conclude either that the results obtained are not reliable and that they cannot be correctly computed with this computer, or he may try to debug his code. It is shown that the numerical examples that support the criticisms and which make the Prosolver software fail, do not jeopardize the CADNA library.

Conditions for the uniqueness of optimal policies of discounted Markov decision processes

This paper presents three conditions. Each of them guarantees the uniqueness of optimal policies of discounted Markov decision processes. The conditions presented here impose hypotheses specifically on the state space X, the action space A, the admissible action sets A(x),x∈X, the transition probability Q, and on the cost function c. Two of these conditions require mainly convexity assumptions, but the third one does not need this kind of assumptions. However, it needs certain stochastic order relations in Q, and the cost function c to reach its minimum with respect to the actions, just in one action. We illustrate the conditions with several examples including, in particular, discrete models, the linear regulator problem, and also a model of an inventory control system.

Semi-infinite probabilistic optimization: first-order stochastic dominance constrain

We consider stochastic optimization problems involving a continuum of probabilistic constraints. They are equivalent to stochastic dominance constraints of first order, frequently called stochastic ordering constraints. We develop necessary and sufficient conditions of optimality for these models. We show that the Lagrange multipliers corresponding to dominance constraints can be identified with piecewise constant nondecreasing utility functions. We also show that the convexification of stochastic ordering relation is equivalent to second-order stochastic dominance under rather weak assumptions.

Laplace transform ordering of actuarial quantities

This paper aims to study both univariate and multivariate versions of the Laplace transform order. The importance of this stochastic order relation in actuarial sciences is enhanced by outlining several possible applications.

The s-convex orders among real random variables, with applications

In this paper, newclasses of stochastic order relations are introduced. These can be seen as extensions of the usual convex order and are closely related to the orderings discussed inLef`evre and Utev (1996), as well as to the stochastic dominances in economics and stop-loss orders in actuarial sciences. These classes are studied in detail, including properties, characterizations, sufficient conditions, and extrema with respect to these orderings in different sets of distribution functions. Some applications illustrate the theory.

How Does the Value Function of a Markov Decision Process Depend on the Transition Probabilities?

The present work deals with the comparison of (discrete time) Markov decision processes (MDPs), which differ only in their transition probabilities. We show that the optimal value function of an MDP is monotone with respect to appropriately defined stochastic order relations. We also find conditions for continuity with respect to suitable probability metrics. The results are applied to some well-known examples, including inventory control and optimal stopping.

Some new classes of stochastic order relations among arithmetic random variables, with applications in actuarial sciences

Recently, Fishbum and Lavalle (1995) and Lefèvre and Utev (1996) have considered some stochastic order relations specific for arithmetic random variables. The present work is concerned with these orderings, together with two other classes of stochastic order relations closely related. First, attention is paid to characterizations and various properties of all these orderings. Then, sufficient conditions of crossing-type for the two new classes of orderings are derived and extrema among discrete random variables are deduced. This is applied in actuarial sciences to obtain new bounds for the classical single life premiums as well as for the probability of ruin in the compound binomial risk model.

Mixed Poisson approximation in the collective epidemic model

The collective epidemic model is a quite flexible model that describes the spread of an infectious disease of the Susceptible-Infected-Removed type in a closed population. A statistic of great interest is the final number of susceptibles who survive the disease. In the present paper, a necessary and sufficient condition is derived that guarantees the weak convergence of the law of this variable to a mixed Poisson distribution when the initial susceptible population tends to infinity, provided that the outbreak is severe in a certain sense. New ideas in the proof are the exploitation of a stochastic convex order relation and the use of a weak convergence theorem for products of i.i.d. random variables.

Symmetric queues with batch departures and their networks

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

Symmetric queues with batch departures and their networks

Batch departures arise in various applications of queues. In particular, such models have been studied recently in connection with production systems. For the most part, however, these models assume Poisson arrivals and exponential service times; little is known about them under more general settings. We consider how their stationary queue length distributions are affected by the distributions of interarrival times, service times and departing batch sizes of customers. Since this is not an easy problem even for single departure models, we first concentrate on single-node queues with a symmetric service discipline, which is known to have nice properties. We start with pre-emptive LIFO, a special case of the symmetric service discipline, and then consider symmetric queues with Poisson arrivals. Stability conditions and stationary queue length distributions are obtained for them, and several stochastic order relations are considered. For the symmetric queues and Poisson arrivals, we also discuss their network. Stability conditions and the stationary joint queue length distribution are obtained for this network.

Some Results on Interarrival Times of Nonhomogeneous Poisson Processes

It is well known that in the case of a Poisson process with constant intensity function the interarrival times are independent and identically distributed, each having exponential distribution. We study this problem when the intensity function is monotone. In particular, we show that in the case of a nonhomogeneous Poisson process with decreasing (increasing) intensity the interarrival times are increasing (decreasing) in the hazard rate ordering sense and they are also jointly likelihood ratio ordered (cf. Shanthikumar and Yao, 1991, Bivariate characterization of some stochastic order relations, Advances in Applied Probability 23: 642–659). This result is stronger than the usual stochastic ordering between the successive interarrival times. Also in this case, the interarrival times are conditionally increasing in sequence and, as a consequence, they are associated. We also consider the problem of comparing two nonhomogeneous Poisson processes in terms of the ratio of their intensity functions and establish some results on the successive number of events from one process occurring between two consecutive occurrences from the second process.