Inequality for the Moment of a Function of a Random Variable

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Negative Binomial Approximation with Stein's Method

Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discrete random variable written as a sum of indicators, an upper bound of coupling form is expressed as an average of terms each of which measures the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable. When a monotone coupling exists a lower bound can also be shown. Application of these results is illustrated with the example of the Po´lya distribution for which the rate of approach to the negative binomial limit is found.     

Design Adaptive Nearest Neighbor Regression Estimation

This paper deals with nonparametric regression estimation under arbitrary sampling with an unknown distribution. The effect of the distribution of the design, which is a nuisance parameter, can be eliminated by conditioning. An upper bound for the conditional mean squared error of k−NN estimates leads us to consider an optimal number of neighbors, which is a random function of the sampling. The corresponding estimate can be used for nonasymptotic inference and is also consistent under a minimal recurrence condition. Some deterministic equivalents are found for the random rate of convergence of this optimal estimate, for deterministic and random designs with vanishing or diverging densities. The proposed estimate is rate optimal for standard designs.     

On characterizations of distributions by mean absolute deviation and variance bounds

In this paper we present a bound for the mean absolute deviation of an arbitrary real-valued function of a discrete random variable. Using this bound we characterize a mixture of two Waring (hence geometric) distributions by linearity of a function involved in the bound. A double Lomax distribution is characterized by linearity of the same function involved in the analogous bound for a continuous distribution. Finally, we characterize the Pearson system of distributions and the generalized hypergeometric distributions by a quadratic function involved in a similar bound for the variance of a function of a random variable.     

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.     

On nonnegative solutions of random systems of linear inequalities

The set of nonnegative solutions of a system of linear equations or inequalities is a convex polyhedron. If the coefficients of the system are chosen at random, the number of vertices of this polyhedron is a random variable. Its expected value, dependent on the probability distribution of the coefficients, which are assumed to be nonnegative throughout, is investigated, and a distribution-independent upper bound for this expected value is established.     

An upper bound on the expected completion time of PERT networks

In this paper the problem of determining an upper bound on the expected project completion time, described by the PERT network, is considered. It is assumed that activity durations are independent random variables with given means. The exact forms of probability distributions do not have to be known; however, their cumulative distribution functions are expected to belong to the so-called NBUE class. Very simple algorithms for deriving this bound are presented. The computations can even be performed manually for more involved networks. Our approach producing a pessimistic evaluation of the expected value of the project duration, extends considerably the information obtained through the use of the classical PERT that always underestimates this value. The results are illustrated by a simple example, and errors of approximations are discussed.     

带干扰风险的破产模型的研究

本文对古典风险模型中保险公司按单位时间常数率收到保险费的假设做了改进,将每次收到的保险费的次数看作是复合泊松过程,将每次收到的保费和每次的理陪额均看作是服从指数分布的随机变量,并引入带干扰风险的扰动项,从而对古典风险模型进行推广,且给出了相应的破产概率上界,分析了破产概率的上界与准备金,索赔额,净保费和扰动方差之间的关系。     

带干扰风险的破产模型的研究

本文对古典风险模型中保险公司按单位时间常数率收到保险费的假设做了改进,将每次收到的保险费的次数看作是复合泊松过程,将每次收到的保费和每次的理陪额均看作是服从指数分布的随机变量,并引入带干扰风险的扰动项,从而对古典风险模型进行推广,且给出了相应的破产概率上界,分析了破产概率的上界与准备金,索赔额,净保费和扰动方差之间的关系.     

复合二项对偶模型的最优分红问题

研究复合二项对偶模型的最优分红问题,通过分析HJB方程得到了最优分红策略和相应的最优值函数之间的关系以及最优值函数的简单计算方法.通过讨论最优红利策略的一些性质得到了最优值函数的可无限逼近的上界和下界.     

The unsatisfiability threshold revisited

The problem of determining the unsatisfiability threshold for random 3-SAT formulas consists in determining the clause to variable ratio that marks the experimentally observed abrupt change from almost surely satisfiable formulas to almost surely unsatisfiable. Up to now, there have been rigorously established increasingly better lower and upper bounds to the actual threshold value. In this paper, we consider the problem of bounding the threshold value from above using methods that, we believe, are of interest on their own right. More specifically, we show how the method of local maximum satisfying truth assignments can be combined with results for the occupancy problem in schemes of random allocation of balls into bins in order to achieve an upper bound for the unsatisfiability threshold less than 4.571. In order to obtain this value, we establish a bound on the q-binomial coefficients (a generalization of the binomial coefficients). No such bound was previously known, despite the extensive literature on q-binomial coefficients. Finally, to prove our result we had to establish certain relations among the conditional probabilities of an event in various probabilistic models for random formulas. It turned out that these relations were considerably harder to prove than the corresponding ones for unconditional probabilities, which were previously known.     

Characterizing a comonotonic random vector by the distribution of the sum of its components

In this article, we characterize comonotonicity and related dependence structures among several random variables by the distribution of their sum. First we prove that if the sum has the same distribution as the corresponding comonotonic sum, then the underlying random variables must be comonotonic as long as each of them is integrable. In the literature, this result is only known to be true if either each random variable is square integrable or possesses a continuous distribution function. We then study the situation when the distribution of the sum only coincides with the corresponding comonotonic sum in the tail. This leads to the dependence structure known as tail comonotonicity. Finally, by establishing some new results concerning convex order, we show that comonotonicity can also be characterized by expected utility and distortion risk measures.     

Tail behaviour of Gaussian processes with applications to the Brownian pillow

In this paper we investigate the tail behaviour of a random variable S which may be viewed as a functional T of a zero mean Gaussian process X, taking special interest in the situation where X obeys the structure which is typical for limiting processes occurring in nonparametric testing of (multivariate) independency and (multivariate) constancy over time. The tail behaviour of S is described by means of a constant a and a random variable R which is defined on the same probability space as S. The constant a acts as an upper bound, and is relevant for the computation of the efficiency of test statistics converging in distribution to S. The random variable R acts as a lower bound, and is instrumental in deriving approximation for the upper percentage points of S by simulation.     

FEM Shakedown of uncertain structures by chance constrained programming

We present a new primal-dual numerical algorithm of the shakedown problem under uncertainty in which the material strength is considered as a normally distributed random variable. Using chance constrained programming, we obtain deterministic equivalent formulations based on upper bound and lower bound theorems which are dual to each other. The proposed algorithm is tested in numerical examples which are applied to civil engineering structures. The obtained solutions show good performance. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis and optimization of an inventory system with bounded orders

This paper addresses the control of a one-item inventory system subject to random order lead time and random demand. The key parameter of the control policy is the objective inventory. In each period, the order to be placed brings the inventory position as close as possible to the objective inventory. The order of each period is kept between a lower bound and an upper bound. We show that the distribution of the inventory level converges to its stationary distribution provided that the lower bound is smaller than the average demand, the upper bound is greater than the average demand and some regularity conditions hold. The average inventory cost is shown to be a convex function of the objective inventory level. A simulation-based approach is proposed for the determination of the optimal objective inventory. A method of bisection with derivative is then used to determine the optimal objective inventory. The derivatives needed in various iterations of this method are estimated using a single sample path with respect to a given objective inventory. Numerical results are provided.     

Nonlinear integer programming for optimal allocation in stratified sampling

A stratified random sampling plan is one in which the elements of the population are first divided into nonoverlapping groups, and then a simple random sample is selected from each group. In this paper, we focus on determining the optimal sample size of each group. We show that various versions of this problem can be transformed into a particular nonlinear program with a convex objective function, a single linear constraint, and bounded variables. Two branch and bound algorithms are presented for solving the problem. The first algorithm solves the transformed subproblems in the branch and bound tree using a variable pegging procedure. The second algorithm solves the subproblems by performing a search to identify the optimal Lagrange multiplier of the single constraint. We also present linearization and dynamic programming methods that can be used for solving the stratified sampling problem. Computational testing indicates that the pegging branch and bound algorithm is fastest for some classes of problems, and the linearization method is fastest for other classes of problems.     

Green's functions for elliptic and parabolic equations with random coefficients II

This paper is concerned with linear parabolic partial differential equations in divergence form and their discrete analogues. It is assumed that the coefficients of the equation are stationary random variables, random in both space and time. The Green's functions for the equations are then random variables. Regularity properties for expectation values of Green's functions are obtained. In particular, it is shown that the expectation value is a continuously differentiable function in the space variable whose derivatives are bounded by the corresponding derivatives of the Green's function for the heat equation. Similar results are obtained for the related finite difference equations. This paper generalises results of a previous paper which considered the case when the coefficients are constant in time but random in space.

     

The equal flow problem

This paper presents a new algorithm for the solution of a network problem with equal flow side constraints. The solution technique is motivated by the desire to exploit the special structure of the side constraints and to maintain as much of the characteristics of pure network problems as possible. The proposed algorithm makes use of Lagrangean relaxation to obtain a lower bound and decomposition by right-hand-side allocation to obtain upper bounds. The lagrangean dual serves not only to provide a lower bound used to assist in termination criteria for the upper bound, but also allows an initial allocation of equal flows for the upper bound. The algorithm has been tested on problems with up to 1500 nodes and 6000 arcs. Computational experience indicates that solutions whose objective function value is well within 1% of the optimum can be obtained in 1%–65% of the MPSX time depending on the amount of imbalance inherent in the problem. Incumbent integer solutions which are within 99.99% feasible and well within 1% of the proven lower bound are obtained in a straightforward manner requiring, on the average, 30% of the MPSX time required to obtain a linear optimum.     

Characterizations of continuous distributions through inequalities involving the expected values of selected functions

Nanda (2010) and Bhattacharjee et al. (2013) characterized a few distributions with help of the failure rate, mean residual, log-odds rate and aging intensity functions. In this paper, we generalize their results and characterize some distributions through functions used by them and Glaser’s function. Kundu and Ghosh (2016) obtained similar results using reversed hazard rate, expected inactivity time and reversed aging intensity functions. We also, via w(·)-function defined by Cacoullos and Papathanasiou (1989), characterize exponential and logistic distributions, as well as Type 3 extreme value distribution and obtain bounds for the expected values of selected functions in reliability theory. Moreover, a bound for the varentropy of random variable X is provided.     

Continuity and Stability of a Quadratic Mixed-Integer Stochastic Program

For the two-stage quadratic stochastic program where the second-stage problem is a general mixed-integer quadratic program with a random linear term in the objective function and random right-hand sides in constraints, we study continuity properties of the second-stage optimal value as a function of both the first-stage policy and the random parameter vector. We also present sufficient conditions for lower or upper semicontinuity, continuity, and Lipschitz continuity of the second-stage problem's optimal value function and the upper semicontinuity of the optimal solution set mapping with respect to the first-stage variables and/or the random parameter vector. These results then enable us to establish conclusions on the stability of optimal value and optimal solutions when the underlying probability distribution is perturbed with respect to the weak convergence of probability measures.