Increasing Convex Order Research Articles

Positive Dependence of Signals

In this paper we further investigate the problem considered by Mizuno (2006) in the special case of identically distributed signals. Specifically, we first propose an alternative sufficient condition of crossing type for the convex order to hold between the conditional expectations given signal. Then, we prove that the bivariate (2,1)-increasing convex order ensures that the conditional expectations are ordered in the convex sense. Finally, the L 2 distance between the quantity of interest and its conditional expectation given signal (or expected conditional variance) is shown to decrease when the strength of the dependence increases (as measured by the (2,1)-increasing convex order).

Bounds for the bias of the empirical CTE

The Conditional Tail Expectation (CTE) is gaining an increasing level of attention as a measure of risk. It is known that nonparametric unbiased estimators of the CTE do not exist, and that CTE n α , the empirical α -level CTE (the average of the n ( 1 − α ) largest order statistics in a random sample of size n ), is negatively biased. In this article, we show that increasing convex order among distributions is preserved by E ( CTE n α ) . From this result it is possible to identify the specific distributions, within some large classes of distributions, that maximize the bias of CTE n α . This in turn leads to best possible bounds on the bias under various sets of conditions on the sampling distribution F . In particular, we show that when the α -level quantile is an isolated point in the support of a non-degenerate distribution (for example, a lattice distribution) then the bias is either of the order 1 / n or vanishes exponentially fast. This is intriguing as the bias of CTE n α vanishes at the in-between rate of 1 / n when F possesses a positive derivative at the α th quantile.

Upper comonotonicity and convex upper bounds for sums of random variables

It is well-known that if a random vector with given marginal distributions is comonotonic, it has the largest sum with respect to convex order. However, replacing the (unknown) copula by the comonotonic copula will in most cases not reflect reality well. For instance, in an insurance context we may have partial information about the dependence structure of different risks in the lower tail. In this paper, we extend the aforementioned result, using the concept of upper comonotonicity, to the case where the dependence structure of a random vector in the lower tail is already known. Since upper comonotonic random vectors have comonotonic behavior in the upper tail, we are able to extend several well-known results of comonotonicity to upper comonotonicity. As an application, we construct different increasing convex upper bounds for sums of random variables and compare these bounds in terms of increasing convex order.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Unreliable M/G/1 retrial queue: monotonicity and comparability

In this paper we investigate the monotonicity properties of an unreliable M/G/1 retrial queue using the general theory of stochastic ordering. We show the monotonicity of the transition operator of the embedded Markov chain relative to the strong stochastic ordering and increasing convex ordering. We obtain conditions of comparability of two transition operators and we obtain comparability conditions of the number of customers in the system. Inequalities are derived for the mean characteristics of the busy period, number of customers served during a busy period, number of orbit busy periods and waiting times. Inequalities are also obtained for some probabilities of the steady-state distribution of the server state. An illustrative numerical example is presented.

STOCHASTIC ORDERING OF A CLASS OF SYMMETRIC DISTRIBUTIONS

In this article, we investigate the sufficient and/or necessary conditions in order to stochastically compare the order statistics and their spacing vectors of two random vectors X and Y with special symmetric distributions. The conditions are imposed on the sample ranges Xn:n–X1:n and Yn:n–Y1:n or on (X1:n, Xn:n–X1:n) and (Y1:n, Yn:n–Y1:n). In particular, we consider the multivariate usual stochastic order, the convex order, the increasing convex order, and the directionally convex order. Several examples are also given to illustrate the power of the main results.

Nonparametric two-sample tests for increasing convex order

Given two independent samples of non-negative random variables with unknown distribution functions $F$ and $G$, respectively, we introduce and discuss two tests for the hypothesis that $F$ is less than or equal to $G$ in increasing convex order. The test statistics are based on the empirical stop-loss transform, critical values are obtained by a bootstrap procedure. It turns out that for the resampling a size switching is necessary. We show that the resulting tests are consistent against all alternatives and that they are asymptotically of the given size $α$. A specific feature of the problem is the behavior of the tests ‘inside’ the hypothesis, where $F≠G$. We also investigate and compare this aspect for the two tests.

Stochastic comparisons of spacings of record values from one or two sample sequences

This paper investigates the stochastic comparison of spacings of record values. The likelihood ratio order, the hazard rate order and the mean residual life order between spacings of record values from one sample sequence are developed. Furthermore, the increasing convex order and the mean residual life order between simple spacings of record values from two sample sequences are built as well.

Stochastic orders of scalar products with applications

In this paper, we study stochastic orders of scalar products of random vectors. Based on the study of Ma [Ma, C., 2000. Convex orders for linear combinations of random variables. J. Statist. Plann. Inference 84, 11–25], we first obtain more general conditions under which linear combinations of random variables can be ordered in the increasing convex order. As an application of this result, we consider the scalar product of two random vectors which separates the severity effect and the frequency effect in the study of the optimal allocation of policy limits and deductibles. Finally, we obtain the ordering of the optimal allocation of policy limits and deductibles when the dependence structure of the losses is unknown. This application is a further study of Cheung [Cheung, K.C., 2007. Optimal allocation of policy limits and deductibles. Insurance: Math. Econom. 41, 382–391].

Characterization of stochastic orders by L-functionals

Random variables may be compared with respect to their location by comparing certain functionals ad hoc, such as the mean or median, or by means of stochastic ordering based directly on the properties of the corresponding distribution functions. These alternative approaches are brought together in this paper. We focus on the class of L-functionals discussed by Bickel and Lehmann (1975) and characterize the comparison of random variables in terms of these measures by means of several stochastic orders based on iterated integrals, including the increasing convex order.

Increasing convex ordering of queue length in bulk queues

This paper considers single-server bulk queues M ( X ) / G ( Y ) / 1 and G ( X ) / M ( Y ) / 1 . In the former queue, service times and service capacity are dependent, while in the latter queue, inter-arriving times and arriving group size are dependent. We show that stronger dependence between those leads to shorter queue lengths in the increasing convex ordering sense.

Test for Increasing Convex Order in Multivariate Response

Trend test based on cross-classified data in dose-response has been a central problem in medicine. Most of existing test methods are known to only fit to binary response variables. However, the approaches for binary response tables may suffer from the lack of a clear choice for dichotomization. For multivariate response with ordered categories, some studies have been done for simple stochastic order, likelihood ratio order and so on. However, methods of statistical inference on increasing convex order for more than two multinomial populations have not been fully developed. For testing the increasing convex order alternative, this article provides a model-free test method which can be used in the case of two-way tables and stratified data. Two real examples will be used to illustrate how to apply our test method.

On risk dependence and mrl ordering

Recently, it was shown that as dependence among random variables increases their sum increases in the sense of increasing convex ordering. The present article assumes strong dependence among Bernoulli random variables. This dependence implies that the sum of the random variables increases in the mean residual life (mrl) order. This result is stronger than the previous result, since the mrl ordering implies the increasing convex ordering. We thus extend a result presented in Bäuerle and Müller [1998. Modelling and comparing dependencies in multivariable risk portfolios. ASTIN Bull. 28, 59–76].

FURTHER RESULTS INVOLVING THE MIT ORDER AND THE IMIT CLASS

The purpose of this article is to study several preservation properties of the mean inactivity time order under the reliability operations of convolution, mixture, and shock models. In that context, the increasing mean inactivity time class of lifetime distributions is characterized by means of right spread order and increasing convex order. Some applications in reliability theory are described. Finally, a new test of such a class is discussed.

An aging notion derived from the increasing convex ordering: the NBUCA class

In this paper, comparison between the life distribution of a new unit with that of the remaining life or a used unit in the increasing convex order leads us to introduce a new class of life distributions which we call new better than used in increasing convex average order and denote by NBUCA. This class includes as subclasses the new better than used, NBU and the new better than used in increasing convex ordering, NBUC. Several properties of this class are presented, including the preservation under convolution, random maxima, mixing and formation of coherent structures. Stochastic comparisons of the excess lifetime at different times of a renewal process when the interarrival times belong to the NBUCA class are established. We provide also some applications of Poisson shock models. Finally testing exponentially against NBUCA is presented.

A NOTE ON BOUNDS AND MONOTONICITY OF SPATIAL STATIONARY COX SHOT NOISES

We consider shot-noise and max-shot-noise processes driven by spatial stationary Cox (doubly stochastic Poisson) processes. We derive their upper and lower bounds in terms of the increasing convex order, which is known as the order relation to compare the variability of random variables. Furthermore, under some regularity assumption of the random intensity fields of Cox processes, we show the monotonicity result which implies that more variable shot patterns lead to more variable shot noises. These are direct applications of the results obtained for so-called Ross-type conjectures in queuing theory.

Preservation of stochastic orders for random minima and maxima, with applications

AbstractIt is shown, in this note, that the right spread order and the increasing convex order are both preserved under the taking of random maxima, and the total time on test transform order and the increasing concave order are preserved under the taking of random minima. Some inequalities and preservation properties in reliability and economics are given as applications. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004.

Testing for increasing convex order in several populations

Increasing convex order is one of important stochastic orderings. It is very often used in queueing theory, reliability, operations research and economics. This paper is devoted to studying the likelihood ratio test for increasing convex order in several populations against an unrestricted alternative. We derive the null asympotic distribution of the likelihood ratio test statistic, which is precisely the chi-bar-squared distribution. The methodology for computing critical values for the test is also discussed. The test is applied to an example involving data for survival time for carcinoma of the oropharynx.

Stochastic comparisons of mixtures of convexly ordered distributions with applications in reliability theory

In this paper we first obtain several results which compare mixtures of distributions in the (increasing) convex and concave stochastic orders. We employ these results to derive relatively weak conditions on the intensity functions of a pair of nonhomogeneous Poisson processes (in fact, on the distribution functions that are associated with these intensity functions) under which the corresponding epoch times of the two nonhomogeneous Poisson processes are ordered in the increasing convex stochastic order. Applications include bounds on the epoch times of a nonhomogeneous Poisson process whose intensity function is the hazard rate function of a new better than used in expectation (new worse than used in expectation) random variable, and the increasing convex ordering of times to the first perfect repair in a Bayesian imperfect repair model.