Increasing Convex Order Research Articles

Ordering properties of generalized aggregation with applications

AbstractThe generalized aggregation arises in many research fields including applied probability, actuarial science, and reliability theory, where is a bivariate kernel function and a is a parameter vector. One of its remarkable features is that both X and W are dependent in many practical situations. Therefore, studying the stochastic properties of generalized aggregations under various dependence structures is an interesting and meaningful problem. In this paper, by using left tail weakly stochastic arrangement increasing, right tail weakly stochastic arrangement increasing, and comonotonicity to characterize the dependent structures among X or W, we establish the increasing convex ordering and the expectation ordering of generalized aggregations to investigate the effects of the arrangement and heterogeneity among ai's. Numerical examples and three practical applications are presented to illustrate our results as well.

On Partial Stochastic Comparisons Based on Tail Values at Risk

The tail value at risk at level p, with p ∈ ( 0 , 1 ) , is a risk measure that captures the tail risk of losses and asset return distributions beyond the p quantile. Given two distributions, it can be used to decide which is riskier. When the tail values at risk of both distributions agree, whenever the probability level p ∈ ( 0 , 1 ) , about which of them is riskier, then the distributions are ordered in terms of the increasing convex order. The price to pay for such a unanimous agreement is that it is possible that two distributions cannot be compared despite our intuition that one is less risky than the other. In this paper, we introduce a family of stochastic orders, indexed by confidence levels p 0 ∈ ( 0 , 1 ) , that require agreement of tail values at risk only for levels p > p 0 . We study its main properties and compare it with other families of stochastic orders that have been proposed in the literature to compare tail risks. We illustrate the results with a real data example.

A note on stochastic dominance, uniform integrability and lattice properties

In this work, we discuss completeness for the lattice orders of first and second order stochastic dominance. The main results state that, both, first and second order stochastic dominance induce Dedekind super complete lattices, i.e.~lattices in which every bounded nonempty subset has a countable subset with identical least upper bound and greatest lower bound. Moreover, we show that, if a suitably bounded set of probability measures is directed (e.g.~a lattice), then the supremum and infimum w.r.t.~first or second order stochastic dominance can be approximated by sequences in the weak topology or in the Wasserstein-$1$ topology, respectively. As a consequence, we are able to prove that a sublattice of probability measures is complete w.r.t.~first order stochastic dominance or second order stochastic dominance and increasing convex order if and only if it is compact in the weak topology or in the Wasserstein-$1$ topology, respectively. This complements a set of characterizations of tightness and uniform integrability, which are discussed in a preliminary section.

Comparison of the multivariate skew-normal random vectors based on the integral stochastic ordering

In this paper, we derive the conditions to compare multivariate skew-normal random vectors X and Y. Then, we present an identity for where the function f fulfills some weak regularity conditions. For comparison purposes, we use six important cases of the integral stochastic orders: the usual order, the convex order, the increasing convex order, the upper orthant order, the directionally convex order, and the supermodular order.

On the increasing convex order of generalized aggregation of dependent random variables

In this article, we study stochastic properties of the generalized sum of right tail weakly stochastic arrangement increasing (RWSAI) nonnegative random variables accompanied with stochastic arrangement increasing (SAI) Bernoulli variables. In terms of monotonicity, supermodularity/submodularity, and convexity of the bivariate kernel function, sufficient conditions are developed for the increasing convex ordering on the generalized aggregation. Applications in actuarial science including the individual risk model and the reserving capital allocation are presented to highlight our results.

A new aging concept derived from the increasing convex ordering

In this paper, a comparison between the life distribution of a new unit with that of a used unit in the increasing convex order is made leading to a new class of life distributions which we call “new better than used in convex ordering of second order”. This class includes as subclasses the NBU and the NBUC and is a subclass of the NBUCA class. Preservation properties under convolution, random maxima, mixing and formation of coherent structures are established. Stochastic comparisons of the excess lifetime when the inter-arrival times belong to the NBUC(2) class are developed. Some applications of Poisson shock models and a test of exponentiality against NBUC(2) alternative are presented.

Preservation of increasing convex/concave order under the formation of parallel/series system of dependent components

This paper investigates sufficient conditions for preservation property of the increasing convex order and the increasing concave order under the taking of maximum and minimum of statistically dependent random variables, respectively. As applications, we develop the preservation of NBUC and NBU(2) aging properties respectively under the parallel and series systems of components with statistically dependent lifetimes. Some copulas are presented as illustrations on statistical dependence structure satisfying the sufficient condition as well.

On stochastic comparisons ofk-out-of-nsystems with Weibull components

AbstractIn this paper we prove that a parallel system consisting of Weibull components with different scale parameters ages faster than a parallel system comprising Weibull components with equal scale parameters in the convex transform order when the lifetimes of components of both systems have different shape parameters satisfying some restriction. Moreover, while comparing these two systems, we show that the dispersive and the usual stochastic orders, and the right-spread order and the increasing convex order are equivalent. Further, some of the known results in the literature concerning comparisons ofk-out-of-nsystems in the exponential model are extended to the Weibull model. We also provide solutions to two open problems mentioned by Balakrishnan and Zhao (2013) and Zhaoet al.(2016).

Ordering results for series and parallel systems comprising heterogeneous exponentiated Weibull components

ABSTRACTIn this paper, we discuss the usual stochastic and reversed hazard rate orders between the series and parallel systems from two sets of independent heterogeneous exponentiated Weibull components. We also obtain the results concerning the convex transform orders between parallel systems and obtain necessary and sufficient conditions under which the dispersive and usual stochastic orders, and the right spread and increasing convex orders between the lifetimes of the two systems are equivalent. Finally, in the multiple-outlier exponentiated Weibull models, based on weak majorization and p-larger orders between the vectors of scale and shape parameters, some characterization results for comparing the lifetimes of parallel and series systems are also established, respectively. The results of this paper can be used in practical situations to find various bounds for the important aging characteristics of these systems.

Stochastic orders and co-risk measures under positive dependence

Conditional risk measures (or co-risk measures) and risk contribution measures are increasingly used in actuarial portfolio analysis to evaluate the systemic risk, which is related to the risk that the failure or loss of a component spreads to another component or even to the whole portfolio: while co-risk measures are risk-adjusted versions of measures usually employed to assess isolate risks, risk contribution measures quantify how a stress situation for a component affects another one. In this paper, we provide sufficient conditions under which two random vectors could be compared in terms of CoVaR (conditional value-at-risk), CoES (conditional expected shortfall) and different risk contribution measures. Conditions are given in terms of the increasing convex order, the dispersive order and the excess wealth order of the marginals under some assumptions of positive dependence.

Sampling of Probability Measures in the Convex Order and Approximation of Martingale Optimal Transport Problems

In this paper, motivated by the approximation of Martingale Optimal Transport problems, we are interested in sampling methods preserving the convex order for two probability measures µ and ν on ℝd, with ν dominating µ. When (Xi)1≤i≤I (resp. (Yj)1≤j≤J ) are independent and identically distributed according µ (resp. ν), in general µI = 1/1 ΣIi=1δXi and νJ = 1/J ΣJj=1δYj are not rankable for the convex order. We investigate modifications of µI (resp. νJ) smaller than νJ (resp. greater than µI) in the convex order and weakly converging to µ (resp. ν) as I, J → ∞. We first consider the one dimensional case d = 1, where, according to Kertz and Rosler, the set of probability measures with a finite first order moment is a lattice for the increasing and the decreasing convex orders. Given µ and ν in this set, we define µ ∨ ν (resp. µ ∧ v) as the supremum (resp. infimum) of µ and ν for the decreasing convex order when ∫ℝ xµ(dx) ≤ ∫ℝ xν(dx) and for the increasing convex order otherwise. This way, µ∨v (resp. µ∧ν) is greater than µ (resp. smaller than ν) in the convex order. We give efficient algorithms permitting to compute µ ∨ ν and µ ∧ ν (and therefore µI ∨ νJ and µI ∧ νJ ) when µ and ν are convex combinations of Dirac masses. In general dimension, when µ and ν have finite moments of order ϱ≥ 1, we define the projection µ ⋏ϱν (resp. µ ⋎ϱν) of µ (resp. ν) on the set of probability measures dominated by ν (resp. larger than µ) in the convex order for the Wasserstein distance with index ϱ. When ϱ= 2, µI ⋏2 νJ can be computed efficiently by solving a quadratic optimization problem with linear constraints. It turns out that, in dimension d = 1, the projections do not depend on ϱ and their quantile functions are explicit in terms of those of µ and ν, which leads to efficient algorithms for convex combinations of Dirac masses. Last, we illustrate by numerical experiments the resulting sampling methods that preserve the convex order and their application to approximate Martingale Optimal Transport problems.

Increasing convex order on generalized aggregation of SAI random variables with applications

Abstract In this paper we study general aggregation of stochastic arrangement increasing random variables, including both the generalized linear combination and the standard aggregation as special cases. In terms of monotonicity, supermodularity, and convexity of the kernel function, we develop several sufficient conditions for the increasing convex order on the generalized aggregations. Some applications in reliability and risks are also presented.

Stochastic comparisons of interfailure times under a relevation replacement policy

AbstractWe 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 capital allocation for stochastic arrangement increasing actuarial risks

Abstract This paper studies the increasing convex ordering of the optimal discounted capital allocations for stochastic arrangement increasing risks with stochastic arrangement decreasing occurrence times. The application to optimal allocation of policy limits is presented as an illustration as well.

A multivariate extension of the increasing convex order to compare risks

In this paper, we propose a generalization of the increasing convex order to the multivariate setting to compare vectors of risks that accounts for both the marginal impacts and the dependence structures of the vectors. This generalization is suitable for comparing vectors with heterogeneous components and extends some well-known properties of the univariate increasing convex order. For example, comparisons of vectors with the same copula can be characterized in terms of the multivariate tail conditional expectations introduced by Cousin and Di Bernardino (2014). Moreover, if the copula reflects a particular positive dependence structure, the order among the vectors can be easily verified simply by checking the univariate increasing convex order of the marginals.

Stochastic orderings for elliptical random vectors

The authors provide sufficient and/or necessary conditions for classifying multivariate elliptical random vectors according to the convex ordering and the increasing convex ordering. Their results generalize the corresponding ones for multivariate normal random vectors in the literature.

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

Abstract 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.

Uncertainty orders on the sublinear expectation space

Abstract In this paper, we introduce some definitions of uncertainty orders for random vectors in a sublinear expectation space. We all know that, under some continuity conditions, each sublinear expectation 𝔼 has a robust representation as the supremum of a family of probability measures. We describe uncertainty orders from two different viewpoints. One is from sublinear operator viewpoint. After giving definitions such as monotonic orders, convex orders and increasing convex orders, we use these uncertainty orders to derive characterizations for maximal distributions, G-normal distributions and G-distributions, which are the most important random vectors in the sublinear expectation space theory. On the other hand, we also establish some uncertainty orders’ characterizations from the viewpoint of probability measures and build some connections with the theory of risk measures.

On the Fast Fading Gaussian Wiretap Channel With Statistical Channel State Information at the Transmitter

In this paper, we investigate the ergodic secrecy capacity of the fast fading Gaussian wiretap channel when only the statistics of the channel state information are known at the transmitter. We derive conditions for the existence of degradedness and a positive ergodic secrecy capacity under the usual stochastic order, the convex order, and the increasing convex order between the legitimate and eavesdropper channels. For more general orders, we prove the secrecy capacity of layered erasure wiretap channels and propose a layered signaling for the achievable scheme, and we derive an upper bound on the capacity for fast fading Gaussian wiretap channels. Finally, the numerical results show that under Nakagami-m fast fading channels, the proposed layered signaling outperforms the Gaussian codebook in several cases. In particular, in certain cases, the Gaussian codebook can achieve only a zero secrecy rate, whereas the proposed scheme achieves positive secrecy rates. Therefore, the connectivity of wireless networks can be significantly improved by the proposed scheme.

The Increasing Convex Order and the Trade–off of Size for Risk

ABSTRACTOne random variable is larger than another in the increasing convex order if that random variable is preferred or indifferent to the other by all decision makers with increasing and convex utility functions. Decision makers in this set prefer larger random variables and are risk loving. When a decision maker whose utility function is increasing and concave is indifferent between such a pair of random variables, a trade‐off of size for risk is revealed, and this information can be used to make comparative static predictions concerning the choices of other decision makers. Specifically, the choices of all those who are strongly more (or less) risk averse than the reference decision maker can be predicted. Thus, the increasing convex order, together with Ross's (1981) definition of strongly more risk averse, can provide additional comparative static findings in a variety of decision problems. The analysis here discusses the decisions to self‐protect and to purchase insurance.