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.

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.

scholarly journals Stochastic Comparisons of Order Statistics and Spacings: A Review

2012 ◽  
Vol 2012 ◽  
pp. 1-47 ◽  
Author(s):  
Subhash Kochar
Keyword(s):  
Order Statistics ◽  
Hazard Rate ◽  
Proportional Hazard ◽  
Rate Model ◽  
Stochastic Comparisons ◽  
Scale Parameters ◽  
Hazard Rate Model ◽  
Recent Developments ◽  
Parent Observations ◽  
Sample Spacings

We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

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 Stochastic Comparisons of Lifetimes of Series and Parallel Systems with Dependent Heterogeneous MOTL-G Components under Random Shocks

Symmetry ◽  
2021 ◽  
Vol 13 (12) ◽  
pp. 2248
Author(s):  
Liang Jiao ◽  
Rongfang Yan
Keyword(s):  
Hazard Rate ◽  
Stochastic Ordering ◽  
Parallel Systems ◽  
Real Data ◽  
Shape Parameters ◽  
Stochastic Comparisons ◽  
Weak Majorization ◽  
Majorization Order ◽  
Distribution Class ◽  
Random Shocks

To measure the magnitude among random variables, we can apply a partial order connection defined on a distribution class, which contains the symmetry. In this paper, based on majorization order and symmetry or asymmetry functions, we carry out stochastic comparisons of lifetimes of two series (parallel) systems with dependent or independent heterogeneous Marshall–Olkin Topp Leone G (MOTL-G) components under random shocks. Further, the effect of heterogeneity of the shape parameters of MOTL-G components and surviving probabilities from random shocks on the reliability of series and parallel systems in the sense of the usual stochastic and hazard rate orderings is investigated. First, we establish the usual stochastic and hazard rate orderings for the lifetimes of series and parallel systems when components are statistically dependent. Second, we also adopt the usual stochastic ordering to compare the lifetimes of the parallel systems under the assumption that components are statistically independent. The theoretical findings show that the weaker heterogeneity of shape parameters in terms of the weak majorization order results in the larger reliability of series and parallel systems and indicate that the more heterogeneity among the transformations of surviving probabilities from random shocks according to the weak majorization order leads to larger lifetimes of the parallel system. Finally, several numerical examples are provided to illustrate the main results, and the reliability of series system is analyzed by the real-data and proposed methods.

scholarly journals ON STOCHASTIC COMPARISONS OF ORDER STATISTICS FROM HETEROGENEOUS EXPONENTIAL SAMPLES

2019 ◽  
pp. 1-6
Author(s):  
Yaming Yu
Keyword(s):  
Order Statistics ◽  
Order Statistic ◽  
Open Problem ◽  
Hazard Rate ◽  
Random Variables ◽  
Stochastic Comparisons ◽  
Heterogeneous Sample ◽  
Exponential Random Variables ◽  
Hazard Rate Ordering

Abstract We show that the kth order statistic from a heterogeneous sample of n ≥ k exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan [14]. As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea [11]. Extensions to general spacings are also presented.

Stochastic comparisons of coherent systems under different random environments

2018 ◽  
Vol 55 (2) ◽  
pp. 459-472 ◽  
Author(s):  
Ebrahim Amini-Seresht ◽  
Yiying Zhang ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Hazard Rate ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Reliability Engineering ◽  
Coherent Systems ◽  
Stochastic Comparisons ◽  
Numerical Examples ◽  
Reversed Hazard Rate ◽  
Collaborative Environment ◽  
Theoretical Results

Abstract For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

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.

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.

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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