The fuzzy hyperbolic inequality index associated with fuzzy random variables

The aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy-valued random variables in general populations. For this purpose, the fuzzy hyperbolic inequality index associated with general fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index (extending those for classical inequality indices) is given. Two examples illustrating the computation of the fuzzy inequality index are also considered. Some comments and suggestions are finally included.     

Fuzzy random renewal process and renewal reward process

So far, there have been several concepts about fuzzy random variables and their expected values in literature. One of the concepts defined by Liu and Liu (2003a) is that the fuzzy random variable is a measurable function from a probability space to a collection of fuzzy variables and its expected value is described as a scalar number. Based on the concepts, this paper addresses two processes—fuzzy random renewal process and fuzzy random renewal reward process. In the fuzzy random renewal process, the interarrival times are characterized as fuzzy random variables and a fuzzy random elementary renewal theorem on the limit value of the expected renewal rate of the process is presented. In the fuzzy random renewal reward process, both the interarrival times and rewards are depicted as fuzzy random variables and a fuzzy random renewal reward theorem on the limit value of the long-run expected reward per unit time is provided. The results obtained in this paper coincide with those in stochastic case or in fuzzy case when the fuzzy random variables degenerate to random variables or to fuzzy variables.     

Some remarks on the moment problem and its relation to the theory of extrapolation of spaces

It is well known that that the coincidence of integer moments (nth-power moments, where n is an integer) of two nonnegative random variables does not imply the coincidence of their distributions. Moreover, we show that, given coinciding integer moments, the ratio of half-integer moments may tend to infinity arbitrarily fast. Also, in this paper, we give a new proof of uniqueness in the continuous moment problem and show that, in that problem, it is impossible to replace the condition of coincidence of all moments by a two-sided inequality between them, while preserving the inequality between the distributions. In conclusion, we study the relationship with the theory of extrapolation of spaces.     

On the convergence of uncertain random sequences

In this paper, a useful inequality for central moment of uncertain random variables is proved. Based on this inequality, a convergence theorem for sum of uncertain random variables is derived. A Borel–Cantelli lemma for chance measure is obtained based on the continuity assumption of uncertain measure. Finally, several convergence theorems for uncertain random sequences are established.     

Laguerre deconvolution with unknown matrix operator

In this paper we consider the convolutionmodel Z = X + Y withX of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent identically distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in [17]. To make the problem identifiable for unknown density of Y, we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, well-suited for nonnegative random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein’s type concentration inequalities developed for random matrices in [23].     

Risk model with fuzzy random individual claim amount

In this paper, we consider a risk model in which individual claim amount is assumed to be a fuzzy random variable and the claim number process is characterized as a Poisson process. The mean chance of the ultimate ruin is researched. Particularly, the expressions of the mean chance of the ultimate ruin are obtained for zero initial surplus and arbitrary initial surplus if individual claim amount is an exponentially distributed fuzzy random variable. The results obtained in this paper coincide with those in stochastic case when the fuzzy random variables degenerate to random variables. Finally, two numerical examples are presented.     

Some inequalities and convergence theorems for Choquet integrals

Fuzzy measure (or non-additive measure), which has been comprehensively investigated, is a generalization of additive probability measure. Several important kinds of non-additive integrals have been built on it. Integral inequalities play important roles in classical probability and measure theory. In this paper, we discuss some of these inequalities for one kind of non-additive integrals—Choquet integral, including Markov type inequality, Jensen type inequality, Hölder type inequality and Minkowski type inequality. As applications of these inequalities, we also present several convergence concepts and convergence theorems as complements to Choquet integral theory.     

Dynamic fuzzy reliability models of degraded hold-down structures for folded solar array

The hold-down structures are of considerable importance to the launch of solar array. Due to the difficulties in obtaining sufficient load specimen, it is imprecise to construct the stress as random variables. Therefore, dynamic fuzzy reliability models are developed in this paper, which resolve the problems in dealing with the interaction between the fuzzy stress process and the stochastic strength process. Even for a deterministic fuzzy stress process, the influences of material statistical properties on reliability can be affected by the level α of fuzzy stress. Meanwhile, the level α relates to investment in the collection of information about the fuzzy stress on hold-down bar. Hence, the models can be used for the economic analysis and optimal design of hold-down bar. Finally, key fuzzy parameters of stress, which have significant influences on both the reliability behavior and the effects of material statistical properties on reliability, are identified and some suggestions for the reliability enhancement of hold-down bar are provided in this paper.     

Asymptotic Property of <Emphasis Type="Italic">M</Emphasis> Estimator in Classical Linear Models Under Dependent Random Errors

In this paper, we first establish a useful result on strong convergence for weighted sums of widely orthant dependent (WOD, in short) random variables. Based on the strong convergence that we established and the Bernstein type inequality, we investigate the strong consistency of M estimators of the regression parameters in linear models based on WOD random errors under some more mild moment conditions. The results obtained in the paper improve and extend the corresponding ones for negatively orthant dependent random variables and negatively superadditive dependent random variables. Finally, the simulation study is provided to illustrate the feasibility of the theoretical result that we established.     

Invariance principles for the law of the iterated logarithm under <Emphasis Type="Italic">G</Emphasis>-framework

We obtain a general invariance principle of G-Brownian motion for the law of the iterated logarithm (LIL for short). For continuous bounded independent and identically distributed random variables in G-expectation space, we also give an invariance principle for LIL. In some sense, this result is an extension of the classical Strassen’s invariance principle to the case where probability measure is no longer additive. Furthermore, we give some examples as applications.     

Some fuzzy stochastic orderings for fuzzy random variables

A new approach to stochastic ordering of fuzzy random variables is investigated in this paper. The traditional definitions of stochastic ordering, hazard rate ordering, and also mean residual life ordering were extended and proposed the unified indexes to ranking fuzzy random variables. Finally, we study the stochastic ordering of fuzzy order statistics by using our proposed approach and established some properties.     

A Bernstein—Chernoff deviation inequality, and geometric properties of random families of operators

In this paper we first describe a new deviation inequality for sums of independent random variables which uses the precise constants appearing in the tails of their distributions, and can reflect in full their concentration properties. In the proof we make use of Chernoff's bounds. We then apply this inequality to prove a global diameter reduction theorem for abstract families of linear operators endowed with a probability measure satisfying some condition. Next we give a local diameter reduction theorem for abstract families of linear operators. We discuss some examples and give one more global result in the reverse direction, and extensions. This research was partially supported by BSF grant 2002-006.     

Rough approximations of vague sets in fuzzy approximation space

The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the β-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on β-(complement) operator. Meantime, λ-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the β-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable.     

An interval approach based on expectation optimization for fuzzy random bilevel linear programming problems

This paper considers a class of bilevel linear programming problems in which the coefficients of both objective functions are fuzzy random variables. The main idea of this paper is to introduce the Pareto optimal solution in a multi-objective bilevel programming problem as a solution for a fuzzy random bilevel programming problem. To this end, a stochastic interval bilevel linear programming problem is first introduced in terms of α-cuts of fuzzy random variables. On the basis of an order relation of interval numbers and the expectation optimization model, the stochastic interval bilevel linear programming problem can be transformed into a multi-objective bilevel programming problem which is solved by means of weighted linear combination technique. In order to compare different optimal solutions depending on different cuts, two criterions are given to provide the preferable optimal solutions for the upper and lower level decision makers respectively. Finally, a production planning problem is given to demonstrate the feasibility of the proposed approach.     

Multidimensional dependency measures

The problem of dependency between two random variables has been studied throughly in the literature. Many dependency measures have been proposed according to concepts such as concordance, quadrant dependency, etc. More recently, the development of the Theory of Copulas has had a great impact in the study of dependence of random variables specially in the case of continuous random variables. In the case of the multivariate setting, the study of the strong mixing conditions has lead to interesting results that extend some results like the central limit theorem to the case of dependent random variables.In this paper, we study the behavior of a multidimensional extension of two well-known dependency measures, finding their basic properties as well as several examples. The main difference between these measures and others previously proposed is that these ones are based on the definition of independence among n random elements or variables, therefore they provide a nice way to measure dependency.The main purpose of this paper is to present a sample version of one of these measures, find its properties, and based on this sample version to propose a test of independence of multivariate observations. We include several references of applications in Statistics.     

Additive fuzzy measures and integrals,III

This paper is the third in a series of works dealing with a class of fuzzy measures and with their corresponding fuzzy integrals. Its aim is to present some important properties of the additive fuzzy integrals introduced in (Butnariu, J. Math. Anal. Appl., in press; J. Math. Anal. Appl. 117 (1986), 385–410) (e.g., the Lebesgue-Beppo-Levi theorem, etc.). These properties are used to explain some applications of the additive fuzzy integrals in proving a Radon-Nikodym representation theorem for additive fuzzy measures and in constructing solutions for fuzzy games.     

Estimations and asymptotic behaviors of coherent entropic risk measure for sums of random variables

In this article, we provide an estimation and several asymptotic behaviors for the coherent entropic risk measure of compound Poisson process. We also establish an estimation for the coherent entropic risk measure of sum of i.i.d. random variables in virtue of Log-Sobolev inequality. As an application, we provide two deviation estimations of the tail probability for compound Poisson process. Finally, several simulation results are given to support our results.     

Actuarial risk measures for financial derivative pricing

We present an axiomatic characterization of price measures that are superadditive and comonotonic additive for normally distributed random variables. The price representation derived involves a probability measure transform that is closely related to the Esscher transform, and we call it the Esscher-Girsanov transform. In a financial market in which the primary asset price is represented by a stochastic differential equation with respect to Brownian motion, the price mechanism based on the Esscher-Girsanov transform can generate approximate-arbitrage-free financial derivative prices.     

马氏环境中马氏链的一个概率不等式及其应用

本文研究了马氏环境中马氏链构成的随机变量之和的概率不等式问题.利用了结尾的方法,获得了马氏环境中马氏链构成的随机变量之和的尾部概率不等式,作为结果的应用,给出了将过程限制在(S,S∩F,PS)上的强大数定律.文中提出的方法和结果对研究独立的随机变量之和的大样本性质是十分有用的.     

Chance-constrained programming with fuzzy stochastic coefficients

We consider fuzzy stochastic programming problems with a crisp objective function and linear constraints whose coefficients are fuzzy random variables, in particular of type L-R. To solve this type of problems, we formulate deterministic counterparts of chance-constrained programming with fuzzy stochastic coefficients, by combining constraints on probability of satisfying constraints, as well as their possibility and necessity. We discuss the possible indices for comparing fuzzy quantities by putting together interval orders and statistical preference. We study the convexity of the set of feasible solutions under various assumptions. We also consider the case where fuzzy intervals are viewed as consonant random intervals. The particular cases of type L-R fuzzy Gaussian and discrete random variables are detailed.