Tiantian Mao

Equivalent characterizations on orderings of order statistics and sample ranges

The purpose of this article is to present several equivalent characterizations of comparing the largest-order statistics and sample ranges of two sets of n independent exponential random variables with respect to different stochastic orders, where the random variables in one set are heterogeneous and the random variables in the other set are identically distributed. The main results complement and extend several known results in the literature. The geometric distribution can be regarded as the discrete counterpart of the exponential distribution. We also study the orderings of the largest-order statistics from geometric random variables and point out similarities and differences between orderings of the largest-order statistics from geometric variables and from exponential variables.

Properties of second-order regular variation and expansions for risk concentration

The purpose of this study is two-fold. First, we investigate further properties of the second-order regular variation (2RV). These properties include the preservation properties of 2RV under the composition operation and the generalized inverse transform, among others. Second, we derive second-order expansions of the tail probabilities of convolutions of non-independent and identically distributed (i.i.d.) heavy-tail random variables, and establish second-order expansions of risk concentration under mild assumptions. The main results extend some ones in the literature from the i.i.d. case to non-i.i.d. case.

Ordering convolutions of heterogeneous exponential and geometric distributions revisited

Let Sn(a1, …, an) be the sum of n independent exponential random variables with respective hazard rates a1, …, an or the sum of n independent geometric random variables with respective parameters a1, …, an. In this article, we investigate sufficient conditions on parameter vectors (a1, …, an) and

ON ORDERINGS BETWEEN WEIGHTED SUMS OF RANDOM VARIABLES

(a_{1}^{\ast},\ldots,a_{n}^{\ast})

On aggregation sets and lower-convex sets

under which Sn(a1, …, an) and

Closure properties of the second-order regular variation under convolutions

S_{n}(a_{1}^{\ast},\ldots,a_{n}^{\ast})

Risk Aversion in Regulatory Capital Principles

are ordered in terms of the increasing convex and the reversed hazard rate orders for both exponential and geometric random variables and in terms of the mean residual life order for geometric variables. For the bivariate case, all of these sufficient conditions are also necessary. These characterizations are used to compare fail-safe systems with heterogeneous exponential components in the sense of the increasing convex and the reversed hazard rate orders. The main results complement several known ones in the literature.

Second-order regular variation inherited from Laplace–Stieltjes transforms

Linear combinations of independent random variables have been extensively studied in the literature. Xu & Hu [21] and Pan, Xu, & Hu [16] unified the study of linear combinations of independent random variables under the general setup. This paper is a companion one of these two papers. In this paper, we will further study this topic. The results are further generalized to the cases of permutation invariant random variables and of independent but not necessarily identically distributed random variables which are ordered in the likelihood ratio or the hazard ratio order.

PRESERVATION OF LOG-CONCAVITY UNDER CONVOLUTION

It has been a challenge to characterize the set of all possible sums of random variables with given marginal distributions, referred to as an aggregation set in this paper. We study the aggregation set via its connection to the corresponding lower-convex set, which is the set of all sums of random variables that are smaller than the respective marginal distributions in convex order. Theoretical properties of the two sets are discussed, assuming that all marginal distributions have finite mean. In particular, an aggregation set is always a subset of its corresponding lower-convex set, and the two sets are identical in the asymptotic sense after scaling. We also show that a lower-convex set is identical to the set of comonotonic sums with the same marginal constraint. The main theoretical results contribute to the field of multivariate distributions with fixed margins.

A Model-Free Continuum of Degrees of Risk Aversion

ABSTRACT Second-order regular variation (2RV) is a refinement of the concept of RV which appears in a natural way in applied probability, statistics, risk management, telecommunication networks, and other fields. Let X1, …, Xn be independent and non negative random variables with respective survival functions , and assume that is of 2RV with the first-order parameter − α and the second-order parameter ρi for each i and that all the are tail-equivalent. It is shown, in this paper, that the survival function of the sum ∑ni = 1Xi is also of 2RV. The main result is applied to establish the 2RV closure property for the randomly weighted sum ∑ni = 1ΘiXi, where the weights Θ1, …, Θn are independent and non negative random variables, independent of X1, …, Xn, and satisfying certain moment conditions.