Dispersive order comparisons on extreme order statistics from homogeneous dependent random vectors

In this paper, we investigate sufficient conditions for preservation property of the dispersive order for the smallest and largest order statistics of homogeneous dependent random vectors. Moreover, we establish sufficient conditions for ordering with the dispersive order the largest order statistics from dependent homogeneous samples of different sizes.

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

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.

A class of location-independent variability orders, with applications

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real functionhthat can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate functionh. In this paper we study these orders whenhis nondecreasing. We note that all these orders are location independent, and we characterize the dispersive order, and the location-independent riskier order, by means of the generalized TTT orders with nondecreasingh. Further properties, which strengthen known properties of the dispersive order, are given. A useful nontrivial closure property of the generalized TTT orders with nondecreasinghis obtained. Applications in poverty comparisons, risk management, and reliability theory are described.

A class of location-independent variability orders, with applications

Li and Shaked (2007) introduced the family of generalized total time on test transform (TTT) stochastic orders, which is parameterized by a real function h that can be used to capture the preferences of a decision maker. It is natural to look for properties of these orders when there is an uncertainty in determining the appropriate function h. In this paper we study these orders when h is nondecreasing. We note that all these orders are location independent, and we characterize the dispersive order, and the location-independent riskier order, by means of the generalized TTT orders with nondecreasing h. Further properties, which strengthen known properties of the dispersive order, are given. A useful nontrivial closure property of the generalized TTT orders with nondecreasing h is obtained. Applications in poverty comparisons, risk management, and reliability theory are described.

A relation between positive dependence of signal and the variability of conditional expectation given signal

Let S 1 and S 2 be two signals of a random variable X, where G 1(s 1 ∣ x) and G 2(s 2 ∣ x) are their conditional distributions given X = x. If, for all s 1 and s 2, G 1(s 1 ∣ x) - G 2(s 2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S 1 is greater than the conditional expectation of X given S 2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S 1 and S 2 have the same marginal distribution and, when S 1 and S 2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

A relation between positive dependence of signal and the variability of conditional expectation given signal

Let S1 and S2 be two signals of a random variable X, where G1(s1 ∣ x) and G2(s2 ∣ x) are their conditional distributions given X = x. If, for all s1 and s2, G1(s1 ∣ x) - G2(s2 ∣ x) changes sign at most once from negative to positive as x increases, then the conditional expectation of X given S1 is greater than the conditional expectation of X given S2 in the convex order, provided that both conditional expectations are increasing. The stochastic order of the sufficient condition is equivalent to the more stochastically increasing order when S1 and S2 have the same marginal distribution and, when S1 and S2 are sums of X and independent noises, it is equivalent to the dispersive order of the noises.

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.

DISPERSION-TYPE VARIABILITY ORDERS

Dispersion-type orders are introduced and studied. The new orders can be used to compare the variability of the underlying random variables, among which are the usual dispersive order and the right spread order. Connections among the new orders and other common stochastic orders are examined and investigated. Some closure properties of the new orders under the operation of order statistics, transformations, and mixtures are derived. Finally, several applications of the new orders are given.