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.

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.

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.

Dynamic multivariate mean residual life functions

1991 ◽  
Vol 28 (03) ◽  
pp. 613-629 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Partial Ordering ◽  
Mean Residual Life ◽  
Positive Dependence ◽  
Basic Properties ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Weak Notion

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied. A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given. Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

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.

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.

Dynamic multivariate mean residual life functions

1991 ◽  
Vol 28 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Moshe Shaked ◽  
J. George Shanthikumar
Keyword(s):  
Hazard Rate ◽  
Residual Life ◽  
Stochastic Ordering ◽  
Partial Ordering ◽  
Mean Residual Life ◽  
Positive Dependence ◽  
Basic Properties ◽  
Rate Functions ◽  
Hazard Rate Ordering ◽  
Weak Notion

In this paper we introduce and study a dynamic notion of mean residual life (mrl) functions in the context of multivariate reliability theory. Basic properties of these functions are derived and their relationship to the multivariate conditional hazard rate functions is studied.A partial ordering, called the mrl ordering, of non-negative random vectors is introduced and its basic properties are presented. Its relationship to stochastic ordering and to other related orderings (such as hazard rate ordering) is pointed out. Using this ordering it is possible to introduce a weak notion of positive dependence of random lifetimes. Some properties of this positive dependence notion are given.Finally, using the mrl ordering, a dynamic notion of multivariate DMRL (decreasing mean residual life) is introduced and studied. The relationship of this multivariate DMRL notion to other notions of dynamic multivariate aging is highlighted in this paper.

scholarly journals Stochastic Comparisons of the Smallest Claim Amounts from Two Sets of Independent Portfolios

2021 ◽  
Vol 50 (3) ◽  
pp. 54-65
Author(s):  
Hossein Nadeb ◽  
Hamzeh Torabi
Keyword(s):  
Hazard Rate ◽  
General Theorem ◽  
Sufficient Conditions ◽  
Scale Model ◽  
Stochastic Comparisons ◽  
Hazard Rate Ordering

The aim of this paper is detecting the ordering properties of the smallest claim amounts arising from two sets of independent heterogeneous portfolios in insurance. First, we prove a general theorem that it presents some sufficient conditions in the sense of the hazard rate ordering to compare the smallest claim amounts from two batches of independent heterogeneous portfolios. Then, we show that the exponentiated scale model as a famous model and the Harris family satisfy the sufficient conditions of the proven general theorem. Also, to illustrate our results, some used models in actuary are numerically applied.

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.

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