Increasing Convex Order Research Articles

Some properties of convex and increasing convex orders under Archimedean copula

Abstract In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

Increasing convex order of capital allocation with dependent assets under threshold model

In this manuscript, we consider a risk-preference investor allocating some amount of capital to the dependent risky asset, where the responding asset will occur default if the stochastic return is less than some predetermined threshold. Then, we present sufficient conditions of the increasing convex order on capital allocation with dependent risky assets when the stochastic return is right tail weakly stochastic arrangement increasing. Finally, some numerical examples are given as illustrations.

Stochastic ordering of variability measure estimators

Variability measures, such as the variance or the Gini mean difference, are widely used to summarize the dispersion of random variables. In the statistical setting, it is quite natural to assume that if a random quantity has more variability, then the estimators of its variability measures should be greater in some stochastic sense. Stochastic orders can be used to give inequalities in this regard, confirming, or not, this suitable property. This paper is devoted to the stochastic comparison of some variability measure estimators; conditions such that some of these estimators are comparable in the usual stochastic order and in the increasing convex order whenever the involved random variables have different variability are provided. Special attention is devoted to the cases of sample variance and Gini mean differences, and to the case of simple random samples.

Adjusted higher-order expected shortfall

How to detect different tail behaviors of two risk random variables with the same mean is an important task. In this paper, motivated by Burzoni et al. (2022), a class of convex risk measures, referred to as adjusted higher-order Expected Shortfall (ES), is introduced and studied. The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its higher-order ES does not exceed a pre-specified threshold for every probability level. This new risk measure is intimately linked to dual higher-order increasing convex order by choosing the risk threshold to be the higher-order ES of a special benchmark random loss. The dual representation for (adjusted) higher-order Expected Shortfall is also given.

Ordering Results of Aggregate Claim Amounts from Two Heterogeneous Portfolios

In this paper, we discuss the stochastic comparison of two classical surplus processes in an one-year insurance period. Under the Marshall-Olkin extended Weibull random aggregate claim amounts, we establish some sufficient conditions for the comparison of aggregate claim amounts in the sense of the usual stochastic order. Applications of our results to the Value-at-Risk and ruin probability are also given. The obtained results show that the heterogeneity of the risks in a given insurance portfolio tends to make the portfolio volatile, which in turn leads to requiring more capital. We also obtain some sufficient conditions for comparing aggregate non-random claim amounts with different occurrence frequency vectors in terms of increasing convex order.

The NBRULC Reliability Class: Mathematical Theory and Goodness-of-Fit Testing with Applications to Asymmetric Censored and Uncensored Data

The majority of approaches proposed in the past few decades to solve life test problems have differed markedly from those used for closely related, yet broader, issues. Due to the complexity of data that are generated each day in many practical domains, as a result of the development of scales for rating the success or failure of reliability, a new domain of reliability has been created. This domain is referred to as life classes, where specific probability distributions are presented. In this study, it is shown that the use of the quality-of-fit technique to solve problems involving life testing makes sense, and produces simpler processes that are roughly equivalent or superior to those used in traditional procedures. They may also behave better in limited samples. This work investigates a novel quality-of-fit test statistic; it is based on an exponential transform and is compared to the best renewal used Laplace test in increasing convex ordering (NBRULC). Evidence for approach normality is provided. The calculated variables include powers, Pitman asymptotic effectiveness, and critical points. Methods on how to handle censored data were also studied. Our experiments have real-world applications in the fields of medicine and engineering.

Generalized location-scale mixtures of elliptical distributions: Definitions and stochastic comparisons

This article proposes a unified class of generalized location-scale mixture of multivariate elliptical distributions and studies integral stochastic orderings of random vectors following such distributions. Given a random vector Z , independent of X and Y , the scale parameter of this class of distributions is mixed with a function α ( Z ) and its skew parameter is mixed with another function β ( Z ) . Sufficient (and necessary) conditions are established for stochastically comparing different random vectors stemming from this class of distributions by means of several stochastic orders including the usual stochastic order, convex order, increasing convex order, supermodular order, and some related linear orders. Two insightful assumptions for the density generators of elliptical distributions, aiming to control the generators’ tail, are provided to make stochastic comparisons among mixed-elliptical vectors. Some applications in applied probability and actuarial science are also provided as illustrations on the main findings.

A Moment Inequality for the NBRULC Class: Statistical Properties with Applications to Model Asymmetric Data

In this paper, the moment inequalities for some aging distributions are derived based on a mathematical class entitled “a new better than renewal used in Laplace transform order in increasing convex order class (NBRULC)”. The introduced inequalities can be utilized as a new mathematical test for the exponentiality property versus NBRULC. If the mean life is finitely based on these inequalities, then all higher-order moments exist. Pitman’s asymptotic efficiency of the new mathematical test is derived and studied in detail for some asymmetric probability models. The new mathematical test’s power is estimated in reliability studies for a few well-known alternative asymmetric models. The problem in the case of right-censored data is also handled. After that, applying the suggested test to practical issues is demonstrated using asymmetric, real datasets.

Some Results on Measures of Interaction among Risks

It has become a common understanding that financial risk can spread rapidly from one institution to another, and the stressful status of one institution may finally result in a systemic crisis. One popular method to assess and quantify the risk of contagion is employing the co-risk measures and risk contribution measures. It is interesting and important to understand how the underlining dependence structure and magnitude of random risks jointly affect systemic risk measures. In this paper, we mainly focus on the conditional value-at-risk, conditional expected shortfall, the delta conditional value-at-risk, and the delta conditional expected shortfall. Existing studies mainly focus on the situation with two random risks, and this paper makes some contributions by considering the scenario with possibly more than two random risks. By employing the tools of stochastic order, positive dependence concepts and arrangement monotonicity, several results concerning the usual stochastic order, increasing convex order, dispersive order and excess wealth order are presented. Concisely speaking, it is found that for a large enough stress level, a larger random risk tends to lead to a more severe systemic risk. We also performed some Monte Carlo experiments as illustrations for the theoretical findings.

Baire category results for stochastic orders

In the sense of Baire categories, we prove that the elements of a typical pair of univariate distribution functions (defined on a bounded subset of {mathbb {R}}) cannot be compared in the sense of the usual stochastic order, the increasing convex order and the mean residual lifetime order. A similar result is also proved in the class of copulas, i.e. multivariate distribution functions with standard uniform marginals, equipped with the orthant order.

An Identity for Expectations and Characteristic Function of Matrix Variate Skew-normal Distribution with Applications to Associated Stochastic Orderings

We establish an identity for $$Ef\left( \varvec{Y}\right) - Ef\left( \varvec{X}\right) $$ , when $$\varvec{X}$$ and $$\varvec{Y}$$ both have matrix variate skew-normal distributions and the function f satisfies some weak conditions. The characteristic function of matrix variate skew normal distribution is then derived. We then make use of it to derive some necessary and sufficient conditions for the comparison of matrix variate skew-normal distributions under six different orders, such as usual stochastic order, convex order, increasing convex order, upper orthant order, directionally convex order and supermodular order.

Consistent Estimation of Distribution Functions under Increasing Concave and Convex Stochastic Ordering

A random variable Y 1 is said to be smaller than Y 2 in the increasing concave stochastic order if for all increasing concave functions for which the expected values exist, and smaller than Y 2 in the increasing convex order if for all increasing convex ψ. This article develops nonparametric estimators for the conditional cumulative distribution functions of a response variable Y given a covariate X, solely under the assumption that the conditional distributions are increasing in x in the increasing concave or increasing convex order. Uniform consistency and rates of convergence are established both for the K-sample case and for continuously distributed X.

Adjusted Rényi entropic Value-at-Risk

Entropy is a measure of self information or uncertainty. Using different concepts of entropy, we may get different risk measures by dual representation. In this paper, we introduce and study the main properties of a class of convex risk measures, called as adjusted Rényi entropic Value-at-Risk (VaR). The adjusted risk measure quantifies risk as the minimum amount of capital that has to be raised and injected into a financial position to ensure that its Rényi entropic-VaR does not exceed a pre-specified threshold for every probability level. When p∈[1,∞), the adjusted Rényi entropic-VaR of order p is intimately linked to the (p+1)-increasing convex order by choosing the risk threshold to be the Rényi entropic-VaR of a benchmark random loss.

Some stochastic comparisons of lower records and lower record spacings

We obtain here sufficient conditions for increasing concave order and location independent more riskier order of lower record values based on stochastic comparisons of minimum order statistics. We further discuss stochastic orderings of lower record spacings. In particular, we show that increasing convex order of adjacent spacings between minimum order statistics is a sufficient condition for increasing convex order of adjacent spacings of their lower records.

Diffusion-limited annihilating systems and the increasing convex order

We consider diffusion-limited annihilating systems with mobile A-particles and stationary B-particles placed throughout a graph. Mutual annihilation occurs whenever an A-particle meets a B-particle. Such systems, when ran in discrete time, are also referred to as parking processes. We show for a broad family of graphs and random walk kernels that augmenting either the size or variability of the initial placements of particles increases the total occupation time by A-particles of a given subset of the graph. A corollary is that the same phenomenon occurs with the total lifespan of all particles in internal diffusion-limited aggregation.

Stochastic comparisons on extreme order statistics from observations associated by FGM copula

This study deals with the stochastic comparison of extreme order statistics from random samples with FGM copula or FGM survival copula. For general marginal distributions, several sufficient conditions on the usual stochastic order, the reversed hazard rate order, the hazard rate order, the increasing convex order and the increasing concave order on sample maximums or minimums are derived. Then, we apply these conditions to samples following proportional hazard rate model, proportional reversed hazard rate model, Weibull distributions and negative binomial distributions, and generalize some ordering results about samples having independent observations to the case with FGM copula or FGM survival copula. Two comparing results on the second-order statistics from proportional hazard samples are presented. Some numerical examples are provided to illustrate the theoretical results as well.

Stochastic Order and Generalized Weighted Mean Invariance.

In this paper, we present order invariance theoretical results for weighted quasi-arithmetic means of a monotonic series of numbers. The quasi-arithmetic mean, or Kolmogorov–Nagumo mean, generalizes the classical mean and appears in many disciplines, from information theory to physics, from economics to traffic flow. Stochastic orders are defined on weights (or equivalently, discrete probability distributions). They were introduced to study risk in economics and decision theory, and recently have found utility in Monte Carlo techniques and in image processing. We show in this paper that, if two distributions of weights are ordered under first stochastic order, then for any monotonic series of numbers their weighted quasi-arithmetic means share the same order. This means for instance that arithmetic and harmonic mean for two different distributions of weights always have to be aligned if the weights are stochastically ordered, this is, either both means increase or both decrease. We explore the invariance properties when convex (concave) functions define both the quasi-arithmetic mean and the series of numbers, we show its relationship with increasing concave order and increasing convex order, and we observe the important role played by a new defined mirror property of stochastic orders. We also give some applications to entropy and cross-entropy and present an example of multiple importance sampling Monte Carlo technique that illustrates the usefulness and transversality of our approach. Invariance theorems are useful when a system is represented by a set of quasi-arithmetic means and we want to change the distribution of weights so that all means evolve in the same direction.

Weekly scheduling of emergency department physicians to cope with time-varying demand

Overcrowding, long waiting times and delays frequently occur in hospital Emergency Departments (EDs). The main causes are the stochastic and strongly time-varying demands of patient arrivals at an ED and the temporary overloading of EDs. Motivated by collaboration with large EDs, we investigate the physicians scheduling problem in the ED for a weekly planning horizon to address the stochastic and time-varying demands. The patient–physician service system is modeled as a time-varying and temporarily overloaded queueing system without abandonments. We employ a continuous-time Markov chain and uniformization method for the analytical evaluation of waiting times of patients. Based on an increasing convex order property, patient waiting times are proven to be convex in a system state. Based on this convexity, an approximation technique is established to model the physician scheduling problem as a mixed-integer program to decide the start and end working times of physicians. We also obtain a tight lower bound of the optimal solution to this scheduling problem. A local search-based algorithm is designed to solve this scheduling problem. Our method improves the physician schedule obtained via the approaches from the literature, significantly improves actual hospital scheduling, and simultaneously reduces physician working times and patient waiting times.

On the Increasing Convex Order of Relative Spacings of Order Statistics

Relative spacings are relative differences between order statistics. In this context, we extend previous results concerning the increasing convex order of relative spacings of two distributions from the case of consecutive spacings to general spacings. The sufficient conditions are given in terms of the expected proportional shortfall order. As an application, we compare relative deprivation within some parametric families of income distributions.

Stochastic dominance relations for generalised parametric distributions obtained through composition

Investigating stochastic dominance within flexible multi-parametric families of distributions is often complicated, owing to the high number of parameters or non-closed functional forms. To simplify the problem, we use the T–X method, making it possible to obtain generalised models through the composition of cumulative distributions and quantile functions. We derive conditions for the second-order stochastic dominance and for the increasing convex order within multi-parametric families in two steps, namely: (i) breaking them down via the T–X approach and (ii) checking dominance conditions of the (more) manageable distributions composing the model. We apply our method to some special distributions and focus on the beta-generated family, which enables the comparisons of order statistics of i.i.d. samples from (possibly) different random variables.