Stochastic Order Relations Research Articles

Uniqueness of the Inverse First-Passage Time Problem and the Shape of the Shiryaev Boundary

Given a distribution on the positive extended real line, the two-sided inverse first-passage time problem for Brownian motion asks for a function such that the first passage time of this function by a reflected Brownian motion has the given distribution. We combine the ideas of Ekström and Janson, which were developed within the scope of the one-sided inverse first-passage time problem, with the methods of De Masi et al., which were used in the context of free boundary problems, in order to give a different proof for the uniqueness for the two-sided inverse first-passage time problem by using a stochastic order relation. We provide criteria for qualitative properties of solutions of the inverse first-passage problem, which apply to the boundary corresponding to the exponential distribution.

A model of discrete choice based on reinforcement learning under short-term memory

A family of models of individual discrete choice are constructed by means of statistical averaging of choices made by a subject in a reinforcement learning process, where the subject has a short, k-term memory span. The choice probabilities in these models combine in a non-trivial, non-linear way the initial learning bias and the experience gained through learning. The properties of such models are discussed and, in particular, it is shown that probabilities deviate from Luce’s Choice Axiom. Moreover, it is shown that the latter property is recovered as the memory span increases.Two different applications in utility theory are considered. In the first, we use the discrete choice model to generate a binary preference (weak stochastic order) relation on simple lotteries. We show that the relation violate the transitivity and independence axioms of expected utility theory. Furthermore, we establish the dependence of the preferences on frames, with risk aversion for gains, and risk seeking for losses. Based on these findings, we consider a parametric model of choice based on the probability maximization principle, as a model for deviations from the expected utility principle. To illustrate the approach, we apply this model to the classical problem of demand for insurance.

Dependency in multisensory integration: a copula-based analysis.

The notion of copula has attracted attention from the field of contextuality and probability. A copula is a function that joins a multivariate distribution to its one-dimensional marginal distributions. Thereby, it allows characterizing the multivariate dependency separately from the specific choice of margins. Here, we demonstrate the use of copulas by investigating the structure of dependency between processing stages in a stochastic model of multisensory integration, which describes the effect of stimulation by several sensory modalities on human reaction times. We derive explicit terms for the covariance and Kendall's tau between the processing stages and point out the specific role played by two stochastic order relations, the usual stochastic order and the likelihood ratio order, in determining the sign of dependency. This article is part of the theme issue 'Contextuality and probability in quantum mechanics and beyond'.

Two-stage stochastic programming under multivariate risk constraints with an application to humanitarian relief network design

In this study, we consider two classes of multicriteria two-stage stochastic programs in finite probability spaces with multivariate risk constraints. The first-stage problem features a multivariate stochastic benchmarking constraint based on a vector-valued random variable representing multiple and possibly conflicting stochastic performance measures associated with the second-stage decisions. In particular, the aim is to ensure that the associated random outcome vector of interest is preferable to a specified benchmark with respect to the multivariate polyhedral conditional value-at-risk (CVaR) or a multivariate stochastic order relation. In this case, the classical decomposition methods cannot be used directly due to the complicating multivariate stochastic benchmarking constraints. We propose an exact unified decomposition framework for solving these two classes of optimization problems and show its finite convergence. We apply the proposed approach to a stochastic network design problem in a pre-disaster humanitarian logistics context and conduct a computational study concerning the threat of hurricanes in the Southeastern part of the United States. Our numerical results on these large-scale problems show that our proposed algorithm is computationally scalable.

A New Compound Gamma and Lindley Distribution with Application to Failure Data

In this paper, we propose a new lifetime distribution by compounding the gamma and Lindley distributions. Construction of it can be interpreted in the viewpoint of the reliability analysis and Bayesian inference. Moreover, the distribution has decreasing and unimodal hazard rate shapes. Several properties of the distribution are obtained, involving characteristics of the (reverse) hazard rate function, quantiles, moments, extreme order statistics and some stochastic order relations. Estimating the distribution parameters is discussed by some estimation methods and their performance is evaluated by a simulation study. Also, estimation of the stress-strength parameter is investigated. Usefulness of the distribution among other models is illustrated by fitting two failure data sets and using some goodness-of-fit measures.

Systemic risk and copula models

Systemic risk describes the phenomenon that dependency adds a specific component of risk to a system or network of (financial) institutions as a whole, which would not be present if the institutions were independent from each other. This paper introduces the concept of systemic risk measures. We describe and study its behavior as a function of the copula, which represents the loss variables of the institutions in the network. Further, we define stochastic order relations on copulas and relate them with systemic risk measures.

Muirhead inequality for convex orders and a problem of I. Raşa on Bernstein polynomials

We present a new, very short proof of a conjecture by I. Raşa, which is an inequality involving basic Bernstein polynomials and convex functions. It was affirmed positively very recently by J. Mrowiec, T. Rajba and S. Wąsowicz (2017) by the use of stochastic convex orders, as well as by Abel (2017) who simplified their proof. We give a useful sufficient condition for the verification of some stochastic convex order relations, which in the case of binomial distributions are equivalent to the I. Raşa inequality. We give also the corresponding inequalities for other distributions. Our methods allow us to give some extended versions of stochastic convex orderings as well as the I. Raşa type inequalities. In particular, we prove the Muirhead type inequality for convex orders for convolution polynomials of probability distributions.

On extreme order statistics from heterogeneous Weibull variables

ABSTRACTIn this paper, we focus on stochastic comparisons of extreme order statistics from heterogeneous independent/interdependent Weibull samples. Specifically, we study extreme order statistics from Weibull distributions with (i) common shape parameter but different scale parameters, and (ii) common scale parameter but different shape parameters. Several new comparison results in terms of the likelihood ratio order, reversed hazard rate order and usual stochastic order are studied in those scenarios. The results established here strengthen and generalize some of the results known in the literature including Khaledi and Kochar [Weibull distribution: some stochastic comparisons. J Statist Plann Inference. 2006;136:3121–3129], Fang and Zhang [Stochastic comparisons of series systems with heterogeneous Weibull components. Statist Probab Lett. 2013;83:1649–1653], Torrado [Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters. J Korean Statist Soc. 2015;44:68–76] and Torrado and Kochar [Stochastic order relations among parallel systems from Weibull distributions. J Appl Probab. 2015;52:102–116]. Some numerical examples are also provided for illustration.

A contamination model for the stochastic order

Stochastic ordering among distributions has been considered in a variety of scenarios. However, it is often a restrictive model, not supported by the data even in cases in which the researcher tends to believe that a certain variable is somehow smaller than other. Alternatively, we propose to look at a more flexible version in which two distributions satisfy an approximate stochastic order relation if they are slightly contaminated versions of distributions for which stochastic order holds. The minimal level of contamination required for stochastic order to hold is used as a measure of deviation from exact stochastic order model. Our approach is based on the use of trimmings of probabilities. We discuss their connection to approximate stochastic order and provide theoretical support for its use in data analysis, proving uniform consistency and giving non-asymptotic bounds for the error probabilities of our tests. We provide simulation results and a case study for illustration.

Distribution free two-sample methods for judgment post-stratified data

A distribution-free procedure is developed to test a stochastic order relation between two distributions based on judgment post-stratied (JPS) data. The proposed inference relies on Mann-Whitney rank sum statistics. Two classes of test statistics are provided. The

Minimal representation of insurance prices

This paper prices insurance contracts by employing law invariant, coherent risk measures from mathematical finance. We demonstrate that the corresponding premium principle enjoys a minimal representation. Uniqueness–in a sense specified in the paper–of this premium principle is derived from this initial result. The representations are derived from a result by Kusuoka, which is usually given for nonatomic probability spaces. We extend this setting to premium principles for spaces with atoms, as this is of particular importance for insurance.Further, stochastic order relations are employed to identify the minimal representation. It is shown that the premium principles in the minimal representation are extremal with respect to the order relations. The tools are finally employed to explicitly provide the minimal representation for premium principles, which are important in actuarial practice.

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let Xλ1, Xλ2, …, Xλn be independent Weibull random variables with Xλi ∼ W(α, λi), where λi > 0 for i = 1, …, n. Let Xn:nλ denote the lifetime of the parallel system formed from Xλ1, Xλ2, …, Xλn. We investigate the effect of the changes in the scale parameters (λ1, …, λn) on the magnitude of Xn:nλ according to reverse hazard rate and likelihood ratio orderings.

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

On the Harris extended family of distributions

A new method for generating new classes of distributions based on the probability-generating function is presented in Aly and Benkherouf [A new family of distributions based on probability generating functions. Sankhya B. 2011;73:70–80]. In particular, they focused their interest to the so-called Harris extended family of distributions. In this paper, we provide several general results regarding the Harris extended models such as the general behaviour of the failure rate function. We also derive a very useful representation for the Harris extended density function as an absolutely convergent power series of the survival function of the baseline distribution. Additionally, some stochastic order relations are established and limiting distributions of sample extremes are also considered for this model. These general results are illustrated in several special Harris extended models. Finally, we discuss estimation of the model parameters by the method of maximum likelihood and provide an application to real data for illustrative purposes.

Queues and Risk Processes with Dependencies

We study the generalization of the G/G/1 queue obtained by relaxing the assumption of independence between inter-arrival times and service requirements. The analysis is carried out for the class of multivariate matrix exponential distributions introduced in Ref.[13]. In this setting, we obtain the steady-state waiting time distribution, and we show that the classical relation between the steady-state waiting time and workload distributions remains valid when the independence assumption is relaxed. We also prove duality results with the ruin functions in an ordinary and a delayed ruin process. These extend several known dualities between queueing and risk models in the independent case. Finally, we show that there exist stochastic order relations between the waiting times under various instances of correlation.

Hazard Rate Properties of Systems with Two Components

In this article, we study the hazard rate ordering of lifetimes of two-component systems (series and parallel) by considering some bivariate distributions for the joint distribution of component lifetimes. Such that, we aim to investigate the lifetimes of systems with stochastically dependent and with stochastically independent components, the lifetimes of the components, and a stochastic ordering relation between these lifetimes. In addition to these, the mononotonicity of the hazard rates of the parallel and the series systems for the bivariate Farlie–Gumbel–Morgenstern (FGM) family is studied.

Comparison and bounds for functionals of future lifetimes consistent with life tables

We derive a new crossing criterion of hazard rates to identify a stochastic order relation between two random variables. We apply this crossing criterion in the context of life tables to derive stochastic ordering results among three families of fractional age assumptions: the family of linear force of mortality functions, the family of quadratic survival functions and the power family. Further, this criterion is used to derive tight bounds for functionals of future lifetimes that exhibit an increasing force of mortality with given one-year survival probabilities. Numerical examples illustrate our findings.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

Generalized Increasing Convex and Directionally Convex Orders

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.