Excess Wealth Order Research Articles

Characterizations of classes of risk measures by dispersive orders

In this paper, a class C 1 of risk measures, which generalizes the class of risk measures for the right-tail deviation suggested by Wang [Wang, S., 1998. An actuarial index of the right-tail risk. North Amer. Actuarial J. 2, 88–101], is characterized in terms of dispersive order. If dispersive order does not hold, unanimous comparisons are still possible by restricting our attention to a subclass C 2 ⊂ C 1 and then the criterion is the excess-wealth order. Sufficient conditions for stochastic equivalence of excess-wealth ordered random variables are derived in terms of some particular measures of C 2 .

A Kolmogorov–Smirnov-Type Test for Shortfall Dominance Against Parametric Alternatives

This article introduces a Kolmogorov-type test for the shortfall order (also known in the literature as the right-spread or excess-wealth order) against parametric alternatives. In the case of the null hypothesis corresponding to the negative exponential distribution, this provides a test for the new-better-than-used-in-expectation and for the new-worse-than-used-in-expectation properties. Such a test is particularly useful in reliability applications, as well as duration and income distribution analysis. The theoretical properties of the testing procedure are established first for uncensored data, and then for censored and truncated data. Simulation studies reveal that the test based on a bootstrap procedure performs well, even with moderate sample sizes. Applications to real data (chief executive officer compensation data, flight delay data, and throttle failure data) illustrate its empirical relevance.

Excess wealth order and sample spacings

In this note, we further study the properties of excess wealth (or right spread) order and the location independent riskier order. It is proved that if X is less variable than Y according to excess wealth order, then X n : n − X k : n ≤ icx Y n : n − Y k : n for k = 0 , 1 , … , n − 1 , where X 0 : n = Y 0 : n ≡ 0 . Similar results are obtained for location independent riskier order. An application in k -price business auction models is presented as well.

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.

Preservation of the location independent risk order under convolution

The location independent risk order has been used to compare different random assets in risk analysis without the requirement of equal means. Let ( X i , Y i ) , i = 1 , 2 , … , n , be independent pairs of random assets. It is shown that if X i is less than Y i in the location independent risk order for each i and the X i and Y i have log-concave density or probability functions, or if X i is less than Y i in the dispersive order and the X i and Y i have log-concave distribution functions, then ∑ i = 1 n X i is less than ∑ i = 1 n Y i in the location independent risk order. Similar results also hold for the excess wealth order.

A Kolmogorov-Smirnov Type Test for Shortfall Dominance Against Parametric Alternatives

This paper proposes a Kolmogorov-type test for the shortfall order (also known in the literature as the right-spread or excess-wealth order) against parametric alternatives. In the case of the null hypothesis corresponding to the Negative Exponential distribution, this provides a test for the new better than used in expectation (NBUE) property. Such a test is particularly useful in reliability applications as well as duration and income distribution analysis. The theoretical properties of the testing procedure are established. Simulation studies reveal that the test proposed in this paper performs well, even with moderate sample sizes. Applications to real data, namely chief executive officer (CEO) compensation data and flight delay data, illustrate the empirical relevance of the techniques described in this paper.

The total time on test transform and the excess wealth stochastic orders of distributions

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

The total time on test transform and the excess wealth stochastic orders of distributions

For nonnegative random variables X and Y we write X ≤TTTY if ∫0F-1(p)(1-F(x))dx ≤ ∫0G-1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

Lorenz and Excess Wealth Orders, with Applications in Reinsurance Theory

The purpose of this paper is to study the stochastic orderings defined by means of pointwise comparison of the Lorenz curves, or of the excess wealth transforms, of two risks. The resulting order relations are presented in an actuarial context and put in relation with classical stochastic orderings, namely the stochastic dominance, the stop-loss order, the convex order and the dispersive order. Several relevant applications in reinsurance theory are provided.

Two Variability Orders

In this paper we study a new variability order that is denoted by ≤st:icx. This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤st:icxY implies that Var[h(X)] ≤ Var[h (Y)] for all increasing convex functions h. The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤st:icxY are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤st:icx are derived, and some typical applications in queueing theory are described.