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.     

Monotonie des Hill-Schätzers in endlichen Stichproben

If the Pareto parameter Alpha of a Pareto-distributed random variable is calculated as Maximum-Likelihood estimator (Hill-estimator) from a random sample, the expected value of the estimator is a decreasing function of the sample size. In the present paper we show this property also to be valid, if a finite collection of positive real numbers is approximated by a Pareto-distribution, the Pareto parameter of which is calculated as Hill-estimator from samples of the collection. Using the Hill-estimator for Alpha, the expected value of the collection can be estimated as the expected value of the corresponding Pareto-distribution. Generally, the expected value of a Pareto-distributed random variable is a decreasing function of Alpha. Paradoxically however, if the mean value of the Pareto-distribution is used as an estimator for the expected value of the above collection, this estimator also proves to be a decreasing function of the sample size. This property is relevant for the quotation of reinsurance contracts.     

Asymptotic behavior of the expected optimal value of the multidimensional assignment problem

The Multidimensional Assignment Problem (MAP) is a higher-dimensional version of the Linear Assignment Problem that arises in the areas of data association, target tracking, resource allocation, etc. This paper elucidates the question of asymptotical behavior of the expected optimal value of the large-scale MAP whose assignment costs are independent identically distributed random variables with a prescribed probability distribution. We demonstrate that for a broad class of continuous distributions the limiting value of the expected optimal cost of the MAP is determined by the location of the left endpoint of the support set of the distribution, and construct asymptotical bounds for the expected optimal cost.     

关于图值随机元序列的强极限定理

本文研究在局部连通图中取值的随机元的r-阶均值集、r-阶广义样本均值集的基本性质及其极限性质.利用关于随机元的分离引理以及随机游动的常返性,得到了关于图值随机元序列广义强大数定理.推广了已有的结果.     

Derived random measures

A derived random measure is constructed by integration of a random process with respect to a random measure independent of that process. Basic distributional properties, a continuity theorem, sample path properties, a strong law of large numbers, and a central limit theorem for derived random measures are established. Applications are given to compounding and thinning of point processes and the measure of a random set.     

Expected utility for nonstochastic risk

Stochastic random phenomena considered in von Neumann–Morgenstern utility theory constitute only a part of all possible random phenomena (Kolmogorov, 1986). We show that any sequence of observed consequences generates a corresponding sequence of frequency distributions, which in general does not have a single limit point but a non-empty closed limit set in the space of finitely additive probabilities. This approach to randomness allows to generalize the expected utility theory in order to cover decision problems under nonstochastic random events. We derive the maxmin expected utility representation for preferences over closed sets of probability measures. The derivation is based on the axiom of preference for stochastic risk, i.e. the decision maker wishes to reduce a set of probability distributions to a single one. This complements Gilboa and Schmeidler’s (1989) consideration of the maxmin expected utility rule with objective treatment of multiple priors.     

基于样本均值的最优无重叠k-序对排序集抽样

本文提出了无重叠κ-序对排序集抽样方法,即在每个排序集中对κ-序对个体进行观测,并且不同的排序集的κ-序对之间没有任何重复.我们首先探究了此抽样方法得到的样本均值的有效性随每个排序集中κ-序对个体间的相关性变化的趋势.κ-序对个体间的相关性越强,样本均值的有效性损失越大.本文的目的是找到无重叠κ-序对排序集抽样方法中κ-序对分配的最优方案从而使样本均值的有效性损失最小,并证明了最优的无重叠κ-序对排序集抽样比广义排序集抽样以及简单随机抽样更有效.尽管无重叠κ-序对排序集抽样方法的统计效率低于经典的排序集抽样,但是在成本模型下,最优的无重叠κ-序对排序集抽样方法可以比经典的排序集抽样更有效.     

Estimating the Mean of Heavy-Tailed Distributions

An asymptotically normally distributed estimate for the expected value of a positive random variable with infinite variance is introduced. Its behavior relative to estimation using the sample mean is investigated by simulations. An example of how to apply the estimate to file-size measurements on Internet traffic is also shown.     

A note on uniform exponential convergence of sample average approximation of random functions

Shapiro and Xu (2008) [17] investigated uniform large deviation of a class of Hölder continuous random functions. It is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size, the sample average approximation of the random function converges to its expected value uniformly over a compact set. This note extends the result to a class of discontinuous functions whose expected values are continuous and the Hölder continuity may be violated for some negligible random realizations. The extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization.     

Stochastic Differential Equations with Nonlocal Sample Dependence

Stochastic ordinary differential equations are investigated for which the coefficients depend on nonlocal properties of the current random variable in the sample space such as the expected value or the second moment. The approach here covers a broad class of functional dependence of the right-hand side on the current random state and is not restricted to pathwise relations. Existence and uniqueness of solutions is obtained as a limiting process by freezing the coefficients over short time intervals and applying existence and uniqueness results and appropriate estimates for stochastic ordinary differential equations.     

A fuzzy bootstrap test for the mean with <Emphasis Type="Italic">D</Emphasis><Subscript><Emphasis Type="Italic">p,q</Emphasis></Subscript>-distance

In this paper, we consider the problem of testing a simple hypothesis about the mean of a fuzzy random variable. For this purpose, we take a distance between the sample mean and the mean in the null hypothesis as a test statistic. An asymptotic test about the fuzzy mean is obtained by using a central limit theorem. The asymptotical distribution is ω ₂-distribution. The ω ₂-distribution is only known for special cases, thus we have considered random LR-fuzzy numbers. In the fuzzy concept, in addition to the existence of several versions of the central limit theorem, there is another practical disadvantage: The limit law is, in most cases, difficult to handle. Therefore, the central limit theorem for fuzzy random variable does not seem to be a very useful tool to make inferences on the mean of fuzzy random variable. Thus we use the bootstrap technique. Finally, by means of a simulation study, we show that the bootstrap method is a powerful tool in the statistical hypothesis testing about the mean of fuzzy random variables.     

Stochastic convergence of random search methods to fixed size Pareto front approximations

In this paper we investigate to what extent random search methods, equipped with an archive of bounded size to store a limited amount of solutions and other data, are able to obtain good Pareto front approximations. We propose and analyze two archiving schemes that allow for maintaining a sequence of solution sets of given cardinality that converge with probability one to an ?-Pareto set of a certain quality, under very mild assumptions on the process used to sample new solutions. The first algorithm uses a hierarchical grid to define a family of approximate dominance relations to compare solutions and solution sets. Acceptance of a new solution is based on a potential function that counts the number of occupied boxes (on various levels) and thus maintains a strictly monotonous progress to a limit set that covers the Pareto front with non-overlapping boxes at finest resolution possible. The second algorithm uses an adaptation scheme to modify the current value of ? based on the information gathered during the run. This way it will be possible to achieve convergence to the best (smallest) ? value, and to a corresponding solution set of k solutions that ?-dominate all other solutions, which is probably the best possible result regarding the limit behavior of random search methods or metaheuristics for obtaining Pareto front approximations.     

随机环境中伴有移民两性分枝过程的极限性质

本文在独立同分布的随机环境下,建立带有移民的两性分枝过程{Zn}n≥0,且移民人口数依赖当前人口数,证得{Zn}n≥0和{(Fn,Mn)}n≥1是随机环境中的马氏链,并得到第n代每个配对单元平均增长率{rk}k≥0的极限性质,从而推广了经典两性分枝过程的相关理论.     

Approximation to the mean curve in the LCS problem

The problem of sequence comparison via optimal alignments occurs naturally in many areas of applications. The simplest such technique is based on evaluating a score given by the length of a longest common subsequence divided by the average length of the original sequences. In this paper we investigate the expected value of this score when the input sequences are random and their length tends to infinity. The corresponding limit exists but is not known precisely. We derive a theoretical large deviation, convex analysis and Monte Carlo based method to compute a consistent sequence of upper bounds on the unknown limit. An empirical practical version of our method produces promising numerical results.     

树上路径过程的随机路径条件概率的极限性质

本文研究了树上路径过程的极限性质.利用构造鞅的方法得到了树上路径过程的条件概率调和平均的极限性质.所得结果推广了树上非齐次马氏链随机转移概率和任意随机变量序列随机条件概率的调和平均极限性质.     

The total median in statistical quality control

In industry, most of the process observations are assumed to come from a normal population, but usually we merely want to control the process mean value. It is thus sensible to find control statistics, which are ‘robust’ to monitor the process mean, giving the expected rate of false alarms whenever that mean is close to the target value, although not under a normal regime. Simulation studies for a few symmetric and asymmetric distributions allow us to suggest the total median as a robust median estimator. We shall here analyse such a robustness, as well as the robustness of the total median chart comparatively to the sample mean chart, whenever we want to control the mean value of a symmetric underlying parent. Some indication is also provided on the comparative out‐of‐control behaviour of the two charts. Copyright © 2004 John Wiley & Sons, Ltd.     

Asymptotic sequential Rademacher complexity of a finite function class

For a finite function class, we describe the large sample limit of the sequential Rademacher complexity in terms of the viscosity solution of a G-heat equation. In the language of Peng’s sublinear expectation theory, the same quantity equals to the expected value of the largest order statistics of a multidimensional G-normal random variable. We illustrate this result by deriving upper and lower bounds for the asymptotic sequential Rademacher complexity.     

多元随机序列泛函的强偏差定理

利用熵密度和样本偏差率的概念,建立了多元随机序列泛函关于条件期望的用不等式表示的强极限性质(称之为强偏差定理),在推论部分得到了非齐次马氏链的强偏差定理和随机条件概率的调和平均值的极限性质等相关结论.证明中给出了将条件矩母函数应用于研究多元随机序列泛函的强极限性质的一种途径.     

Calculating the mean growth rate of the vector of states of a stochastic system with synchronization of events

A stochastic dynamical system with synchronization is considered. The dynamics of the system is described by a linear vector equation with a second-order matrix in an idempotent semiring with the operations of taking maximum and addition. It is assumed that one diagonal entry of the matrix is an exponentially distributed random variable, whereas all other entries are equal to some nonnegative constant. To solve the problem of calculating the mean rate of growth of the state vector of the system, we make a change of variables: instead of the random coordinates of the state vector of the system we introduce new random variables which are more convenient to analyze. After that, the corresponding sequences of one-dimensional distribution functions are constructed and examined for convergence. The mean rate of growth is calculated as the mean value of the limit distribution. In addition, expressions for the limit probabilities of some events in the systems are derived.