scholarly journals RELATIONS BETWEEN STOCHASTIC ORDERINGS AND GENERALIZED STOCHASTIC PRECEDENCE

2015 ◽  
Vol 29 (3) ◽  
pp. 329-343 ◽  
Author(s):  
Emilio De Santis ◽  
Fabio Fantozzi ◽  
Fabio Spizzichino
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Stochastic Independence ◽  
Stochastic Orderings ◽  
Stochastic Dependence ◽  
Natural Concept ◽  
Decisions Under Risk ◽  
Stochastic Precedence ◽  
Special Classes

The concept of stochastic precedence between two real-valued random variables has often emerged in different applied frameworks. In this paper, we analyze several aspects of a more general, and completely natural, concept of stochastic precedence that also had appeared in the literature. In particular, we study the relations with the notions of stochastic ordering. Such a study leads us to introducing some special classes of bivariate copulas. Motivations for our study can arise from different fields. In particular, we consider the frame of Target-Based Approach in decisions under risk. This approach has been mainly developed under the assumption of stochastic independence between “Prospects” and “Targets”. Our analysis concerns the case of stochastic dependence.

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.

scholarly journals ITERATED FAILURE RATE MONOTONICITY AND ORDERING RELATIONS WITHIN GAMMA AND WEIBULL DISTRIBUTIONS

2018 ◽  
Vol 33 (1) ◽  
pp. 64-80 ◽  
Author(s):  
Idir Arab ◽  
Paulo Eduardo Oliveira
Keyword(s):  
Explicit Expression ◽  
Failure Rate ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Higher Order ◽  
Distribution Functions ◽  
Weibull Distributions ◽  
Stochastic Orderings ◽  
Definition Of ◽  
Relative Convexity

Stochastic ordering of random variables may be defined by the relative convexity of the tail functions. This has been extended to higher order stochastic orderings, by iteratively reassigning tail-weights. The actual verification of stochastic orderings is not simple, as this depends on inverting distribution functions for which there may be no explicit expression. The iterative definition of distributions, of course, contributes to make that verification even harder. We have a look at the stochastic ordering, introducing a method that allows for explicit usage, applying it to the Gamma and Weibull distributions, giving a complete description of the order of relations within each of these families.

scholarly journals GENERALIZATION OF THE PAIRWISE STOCHASTIC PRECEDENCE ORDER TO THE SEQUENCE OF RANDOM VARIABLES

2020 ◽  
pp. 1-9
Author(s):  
Maxim Finkelstein ◽  
Nil Kamal Hazra
Keyword(s):  
Random Variables ◽  
Stochastic Ordering ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Sufficient Condition ◽  
New Approach ◽  
Stochastic Precedence

Abstract We discuss a new stochastic ordering for the sequence of independent random variables. It generalizes the stochastic precedence (SP) order that is defined for two random variables to the case n > 2. All conventional stochastic orders are transitive, whereas the SP order is not. Therefore, a new approach to compare the sequence of random variables had to be developed that resulted in the notion of the sequential precedence order. A sufficient condition for this order is derived and some examples are considered.

The distribution of the maximum of a random number of random variables with applications

1973 ◽  
Vol 10 (01) ◽  
pp. 122-129 ◽  
Author(s):  
Janos Galambos
Keyword(s):  
Service Time ◽  
Joint Distribution ◽  
Random Number ◽  
Random Variables ◽  
Stochastic Independence ◽  
Fixed Integer ◽  
Number Of Components ◽  
Distribution Of The Maximum ◽  
Distribution Of Elements ◽  
Mild Assumption

The asymptotic distribution of the maximum of a random number of random variables taken from the model below is shown to be the same as when their number is a fixed integer. Applications are indicated to determine the service time of a system of a large number of components, when the number of components to be serviced is not known in advance. A much slighter assumption is made than the stochastic independence of the periods of time needed for servicing the different components. In our model we assume that the random variables can be grouped into a number of subcollections with the following properties: (i) the random variables taken from different groups are asymptotically independent, (ii) the largest number of elements in a subgroup is of smaller order than the overall number of random variables. In addition, a very mild assumption is made for the joint distribution of elements from the same group.

The stochastic precedence ordering with applications in sampling and testing

2004 ◽  
Vol 41 (01) ◽  
pp. 73-82 ◽  
Author(s):  
Philip J. Boland ◽  
Harshinder Singh ◽  
Bojan Cukic
Keyword(s):  
Random Sampling ◽  
Stochastic Order ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Simple Random Sampling ◽  
Order Type ◽  
Bernoulli Random Variables ◽  
Binomial Distributions ◽  
Necessary And Sufficient ◽  
Stochastic Precedence

Stratified and simple random sampling (or testing) are two common methods used to investigate the number or proportion of items in a population with a particular attribute. Although it is known that cost factors and information about the strata in the population are often crucial in deciding whether to use stratified or simple random sampling in a given situation, the stochastic precedence ordering for random variables can also provide the basis for an interesting criteria under which these methods may be compared. It may be particularly relevant when we are trying to find as many special items as possible in a population (for example individuals with a disease in a country). Properties of this total stochastic order on the class of random variables are discussed, and necessary and sufficient conditions are established which allow the comparison of the number of items of interest found in stratified random sampling with the number found in simple random sampling in the stochastic precedence order. These conditions are compared with other results established on stratified and simple random sampling (testing) using different stochastic-order-type criteria, and applications are given for the comparison of sums of Bernoulli random variables and binomial distributions.

Stochastic comparisons for queueing models via random sums and intervals

1992 ◽  
Vol 24 (4) ◽  
pp. 960-985 ◽  
Author(s):  
Alain Jean-Marie ◽  
Zhen Liu
Keyword(s):  
Point Process ◽  
Response Times ◽  
Waiting Times ◽  
Random Variables ◽  
Stochastic Ordering ◽  
Partial Sums ◽  
Polling Systems ◽  
Convex Ordering ◽  
Random Intervals ◽  
Increasing Convex Ordering

We consider the relationships among the stochastic ordering of random variables, of their random partial sums, and of the number of events of a point process in random intervals. Two types of result are obtained. Firstly, conditions are given under which a stochastic ordering between sequences of random variables is inherited by (vectors of) random partial sums of these variables. These results extend and generalize theorems known in the literature. Secondly, for the strong, (increasing) convex and (increasing) concave stochastic orderings, conditions are provided under which the numbers of events of a given point process in two ordered random intervals are also ordered.These results are applied to some comparison problems in queueing systems. It is shown that if the service times in two M/GI/1 systems are compared in the sense of the strong stochastic ordering, or the (increasing) convex or (increasing) concave ordering, then the busy periods are compared for the same ordering. Stochastic bounds in the sense of increasing convex ordering on waiting times and on response times are provided for queues with bulk arrivals. The cyclic and Bernoulli policies for customer allocation to parallel queues are compared in the transient regime using the increasing convex ordering. Comparisons for the five above orderings are established for the cycle times in polling systems.

scholarly journals Ordering results between the largest claims arising from two general heterogeneous portfolios

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1315-1332
Author(s):  
Sangita Das ◽  
Suchandan Kayal
Keyword(s):  
Sufficient Conditions ◽  
Random Variables ◽  
Model Parameters ◽  
Absolutely Continuous ◽  
Stochastic Orderings ◽  
General Family ◽  
Insurance Portfolio ◽  
Family Of Distributions

This work is entirely devoted to compare the largest claims from two heterogeneous portfolios. It is assumed that the claim amounts in an insurance portfolio are nonnegative absolutely continuous random variables and belong to a general family of distributions. The largest claims have been compared based on various stochastic orderings. The established sufficient conditions are associated with the matrices and vectors of model parameters. Applications of the results are provided for the purpose of illustration.

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