ORDERING PROPERTIES OF EXTREME CLAIM AMOUNTS FROM HETEROGENEOUS PORTFOLIOS

2019 ◽  
Vol 49 (2) ◽  
pp. 525-554 ◽  
Author(s):  
Yiying Zhang ◽  
Xiong Cai ◽  
Peng Zhao
Keyword(s):  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Stochastic Orders ◽  
Reliability Engineering ◽  
Research Fields ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering ◽  
Arrangement Increasing ◽  
The Right ◽  
Stochastic Properties

AbstractIn the context of insurance, the smallest and largest claim amounts turn out to be crucial to insurance analysis since they provide useful information for determining annual premium. In this paper, we establish sufficient conditions for comparing extreme claim amounts arising from two sets of heterogeneous insurance portfolios according to various stochastic orders. It is firstly shown that the weak supermajorization order between the transformed vectors of occurrence probabilities implies the usual stochastic ordering between the largest claim amounts when the claim severities are weakly stochastic arrangement increasing. Secondly, sufficient conditions are established for the right-spread ordering and the convex transform ordering of the smallest claim amounts arising from heterogeneous dependent insurance portfolios with possibly different number of claims. In the setting of independent multiple-outlier claims, we study the effects of heterogeneity among sample sizes on the stochastic properties of the largest and smallest claim amounts in the sense of the hazard rate ordering and the likelihood ratio ordering. Numerical examples are provided to highlight these theoretical results as well. Not only can our results be applied in the area of actuarial science, but also they can be used in other research fields including reliability engineering and auction theory.

COMPARISONS ON LARGEST ORDER STATISTICS FROM HETEROGENEOUS GAMMA SAMPLES

2020 ◽  
pp. 1-20
Author(s):  
Ting Zhang ◽  
Yiying Zhang ◽  
Peng Zhao
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Shape Parameters ◽  
Reliability Engineering ◽  
Stochastic Comparisons ◽  
Maximum Order ◽  
Scale Parameters ◽  
Hazard Rate Ordering

This paper deals with stochastic comparisons of the largest order statistics arising from two sets of independent and heterogeneous gamma samples. It is shown that the weak supermajorization order between the vectors of scale parameters together with the weak submajorization order between the vectors of shape parameters imply the reversed hazard rate ordering between the corresponding maximum order statistics. We also establish sufficient conditions of the usual stochastic ordering in terms of the p-larger order between the vectors of scale parameters and the weak submajorization order between the vectors of shape parameters. Numerical examples and applications in auction theory and reliability engineering are provided to illustrate these results.

Laplace Transform Characterization of Probabilistic Orderings

1994 ◽  
Vol 8 (1) ◽  
pp. 69-77 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Laplace Transform ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
The Laplace Transform ◽  
Life Distributions ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering ◽  
Necessary And Sufficient

Using the Laplace transform, we characterize, by means of necessary and sufficient conditions, the property that two life distributions are ordered in the sense of stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Order-Preserving Shock Models

1994 ◽  
Vol 8 (1) ◽  
pp. 125-134 ◽  
Author(s):  
Y. Kebir
Keyword(s):  
Likelihood Ratio ◽  
Hazard Rate ◽  
Stochastic Ordering ◽  
Shock Models ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering

To date, research in shock models has been primarily concerned with the classpreserving properties of certain shock and damage kernels. This article focuses on the order-preserving properties of those kernels. We show that they preserve stochastic ordering, hazard rate ordering, backward hazard rate ordering, and likelihood ratio ordering.

Further monotonicity properties of renewal processes

1992 ◽  
Vol 24 (03) ◽  
pp. 575-588 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Continuous Time ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Recurrence Time ◽  
Renewal Time ◽  
Hazard Rate Ordering ◽  
Forward Recurrence Time ◽  
Decreasing Failure Rate ◽  
Monotonicity Results

In a discrete-time renewal process {Nk, k= 0, 1, ·· ·}, letZkandAkbe the forward recurrence time and the renewal age, respectively, at timek.In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k= 0, 1, ·· ·} and {Zk, k= 0, 1, ·· ·} are monotonically non-decreasing inkin hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m– Nk, k= 0, 1, ·· ·} to be non-increasing inkin hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

scholarly journals On hazard rate ordering of dependent variables

1993 ◽  
Vol 25 (02) ◽  
pp. 477-482 ◽  
Author(s):  
Emad-Eldin A. A. Aly ◽  
Subhash C. Kochar
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Scale Model ◽  
Stochastic Orders ◽  
Marginal Distributions ◽  
New Concepts ◽  
Dependent Variables ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering ◽  
Stochastic Order Relations

Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X 1, X 2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.

scholarly journals On hazard rate ordering of dependent variables

1993 ◽  
Vol 25 (2) ◽  
pp. 477-482 ◽  
Author(s):  
Emad-Eldin A. A. Aly ◽  
Subhash C. Kochar
Keyword(s):  
Hazard Rate ◽  
Stochastic Order ◽  
Scale Model ◽  
Stochastic Orders ◽  
Marginal Distributions ◽  
New Concepts ◽  
Dependent Variables ◽  
Likelihood Ratio Ordering ◽  
Hazard Rate Ordering ◽  
Stochastic Order Relations

Shanthikumar and Yao (1991) introduced some new stochastic order relations to compare the components of a bivariate random vector (X1, X2). As they point out in their paper, even if according to their hazard rate (or likelihood ratio) ordering, the marginal distributions may not be ordered accordingly. We introduce some new concepts where the marginal distributions preserve the corresponding stochastic orders. Also a relation between the bivariate scale model and the introduced bivariate hazard rate ordering is established.

Further monotonicity properties of renewal processes

1992 ◽  
Vol 24 (3) ◽  
pp. 575-588 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Continuous Time ◽  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Ordering ◽  
Recurrence Time ◽  
Renewal Time ◽  
Hazard Rate Ordering ◽  
Forward Recurrence Time ◽  
Decreasing Failure Rate ◽  
Monotonicity Results

In a discrete-time renewal process {Nk, k = 0, 1, ·· ·}, let Zk and Ak be the forward recurrence time and the renewal age, respectively, at time k. In this paper, we prove that if the inter-renewal time distribution is discrete DFR (decreasing failure rate) then both {Ak, k = 0, 1, ·· ·} and {Zk, k = 0, 1, ·· ·} are monotonically non-decreasing in k in hazard rate ordering. Since the results can be transferred to the continuous-time case, and since the hazard rate ordering is stronger than the ordinary stochastic ordering, our results strengthen the corresponding results of Brown (1980). A sufficient condition for {Nk+m – Nk, k = 0, 1, ·· ·} to be non-increasing in k in hazard rate ordering as well as some sufficient conditions for the opposite monotonicity results are given. Finally, Brown's conjecture that DFR is necessary for concavity of the renewal function in the continuous-time case is discussed.

scholarly journals Stochastic Comparisons of Some Distances between Random Variables

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 981
Author(s):  
Patricia Ortega-Jiménez ◽  
Miguel A. Sordo ◽  
Alfonso Suárez-Llorens
Keyword(s):  
Financial Risk ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Random Variable ◽  
Distribution Functions ◽  
Stochastic Comparisons ◽  
Stochastic Orderings ◽  
Absolute Value ◽  
The Difference

The aim of this paper is twofold. First, we show that the expectation of the absolute value of the difference between two copies, not necessarily independent, of a random variable is a measure of its variability in the sense of Bickel and Lehmann (1979). Moreover, if the two copies are negatively dependent through stochastic ordering, this measure is subadditive. The second purpose of this paper is to provide sufficient conditions for comparing several distances between pairs of random variables (with possibly different distribution functions) in terms of various stochastic orderings. Applications in actuarial and financial risk management are given.

On stochastic comparisons of k-out-of-n systems with Weibull components

2018 ◽  
Vol 55 (1) ◽  
pp. 216-232 ◽  
Author(s):  
Narayanaswamy Balakrishnan ◽  
Ghobad Barmalzan ◽  
Abedin Haidari
Keyword(s):  
Weibull Model ◽  
Exponential Model ◽  
Parallel System ◽  
Stochastic Orders ◽  
Open Problems ◽  
Increasing Convex Order ◽  
Scale Parameters ◽  
Convex Transform Order ◽  
Right Spread Order ◽  
The Right

Abstract In 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 of k-out-of-n systems 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 Zhao et al. (2016).

scholarly journals Potential Operators on Cones of Nonincreasing Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Alexander Meskhi ◽  
Ghulam Murtaza
Keyword(s):  
Mixed Type ◽  
Sufficient Conditions ◽  
Product Type ◽  
Necessary And Sufficient Conditions ◽  
Lebesgue Spaces ◽  
Potential Operators ◽  
Right Hand ◽  
The Right ◽  
Necessary And Sufficient

Necessary and sufficient conditions on weight pairs guaranteeing the two-weight inequalities for the potential operators(Iαf)(x)=∫0∞(f(t)/|x−t|1−α)dtand(ℐα1,α2f)(x,y)=∫0∞∫0∞(f(t,τ)/|x−t|1−α1|y−τ|1−α2)dtdτon the cone of nonincreasing functions are derived. In the case ofℐα1,α2, we assume that the right-hand side weight is of product type. The same problem for other mixed-type double potential operators is also studied. Exponents of the Lebesgue spaces are assumed to be between 1 and ∞.

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