Stochastic equivalence of ordered random variables with applications in reliability theory

This paper presents some conditions for stochastic equality of (univariate or multivariate) random variables under various stochastic orderings. The main results provide generalizations of several known results. Applications to stochastic equivalence and characterization problems associated with multivariate NBU (NBUE) distributions are also included.     

Stochastic programming: Potential hazards when random variables reflect market interaction

There are two types of random phenomena modeled in stochastic programs. One type is what we may term “external” or “natural” random variables, such as temperature or the roll of a dice. But in many other cases, random variables are used to reflect the behavior of other market participants. This is the case for such as price and demand of a product. Using simple game theoretic models, we demonstrate that stochastic programming may not be appropriate in these cases, as there may be no feasible way to replace the decisions of others by a random variable, and arrive at the correct decision. Hence, this simple note is a warning against certain types of stochastic programming models. Stochastic programming is unproblematic in pure forms of monopoly and perfect competition, and also with respect to external random phenomena. But if market power is involved, such as in oligopolies, the modeling may not be appropriate.     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

Characterization of conditional covariance and unified theory in the problem of ordering random variables

Under the assumption that a (p+q)-dimensional row vector (Y, X) is elliptically contoured distributed, the conditional covariance of Y given X=x is characterized in the context of correctly ordering the coordinates Y _k 's of Y based on X. This is an answer to a conjecture implicit in Portnoy (1982). Moreover some unified theory is presented for the problem of ordering Y _k 's based on X. An essential tool is the decreasing in transposition (D. T.) function theory of Hollander et al. (1977, Ann. Statist., 5(4), 722–733).     

On the linear combination of normal and Laplace random variables

Summary  The exact distribution of the linear combination αX+βY is derived when X and Y are normal and Laplace random variables distributed independently of each other. A program in MAPLE is provided to compute the associated percentage points.     

关于Tsallis熵与随机序的一些结果

本文讨论了具有相同Tsallis熵或相同Tsallis相对熵的两个连续随机变量随机等价的一些条件,以及随机序与Tsallis熵序的一些关系.     

Correlations of functions of normal variables

Let (X₁, X₂,…, X_k, Y₁, Y₂,…, Y_k) be multivariate normal and define a matrix C by C_ij = cov(X_i, Y_j). If (i) (X₁,…, X_k) = (Y₁,…, Y_k) and (ii) C is symmetric positive definite, then 0 < varf(X₁,…, X_k) < ∞ corr(f(X₁,…, X_k),f(Y₁,…, Y_k)) > 0. Condition (i) is necessary for the conclusion. The sufficiency of (i) and (ii) follows from an infinite-dimensional version, which can also be applied to a pair of jointly normal Brownian motions.     

On hazard rate ordering of the sums of heterogeneous geometric random variables

In this paper, we treat convolutions of heterogeneous geometric random variables with respect to the p-larger order and the hazard rate order. It is shown that the p-larger order between two parameter vectors implies the hazard rate order between convolutions of two heterogeneous geometric sequences. Specially in the two-dimensional case, we present an equivalent characterization. The case when one convolution involves identically distributed variables is discussed, and we reveal the link between the hazard rate order of convolutions and the geometric mean of parameters. Finally, we drive the “best negative binomial bounds” for the hazard rate function of any convolution of geometric sequence under this setup.     

Stochastic programming with random processes

Three possible approaches to stochastic programming problems defined in time (so that they contain random processes) are described in this paper: (1) an application of the extremal theory of random processes; (2) an exponential penalty model approach related to scenario analysis; (3) a modification of the entropic penalty approach. Explicit results are derived for some special cases.     

Stochastic ordering of medians in exchangeable trivariate normal vectors

In this paper, we establish the stochastic ordering of median from an exchangeable trivariate normal vector based on the strength of the correlation coefficient. Specifically, by considering two exchangeable trivariate normal vectors with different correlation coefficients, we show that the absolute value of the median in the vector with smaller correlation coefficient is stochastically smaller than the absolute value of the median in the vector with larger correlation coefficient. We prove this result by utilizing skew-normal distributions.     

Approximate analysis and optimization of batch ordering policies in capacitated supply chains

Devising manufacturing/distribution strategies for supply chains and determining their parameter values have been challenging problems. Linking production management to stock keeping processes improves the planning of the supply chain activities, including material management, culminating in improved customer service levels. In this study, we investigate a multi-echelon supply chain consisting of a supplier, a plant, a distribution center and a retailer. Material flow between stages is driven by reorder point/order quantity inventory control policies. We develop a model to analyze supply chain behavior using some key performance metrics such as the time averages of inventory and backorder levels, as well as customer service levels at each echelon. The model is validated against simulation, yielding good agreement of robust performance metrics. The metrics are then used within an optimization framework to design the supply chain so as to minimize expected total system costs. The outcome of the optimization framework specifies how to move inventory throughout the supply chain and how to set inventory control parameters, i.e., reorder levels and replenishment batch sizes.     

Stochastic approximation of Banach-valued random variables with smooth distributions

A random variablef taking values in a Banach spaceE is estimated from another Banach-valued variableg. The best (with respect to theL _p-metrix) estimator is proved to exist in the case of Bochnerp-integrable variables. For a Hilbert spaceE andp=2, the best estimator is expressed in terms of the conditional expectation and, in the case of jointly Gaussian variables, in terms of the orthoprojection on a certain subspace ofE. More explicit expressions in terms of surface measures are given for the case in which the underlying probability space is a Hilbert space with a smooth probability measure. The results are applied to the Wiener process to improve earlier estimates given by K. Ritter [4]. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 425–444, September, 1995. The author is grateful to B. S. Kashin for posing the problem and helping to write the article, and to the reviewer for a number of useful remarks.     

A partially observable decision problem under a shifted likelihood ratio ordering

In this paper, we discuss a partially observable sequential decision problem under a shifted likelihood ratio ordering. Since we employ the Bayes' theorem for the learning procedure, we treat this problem under several assumptions. Under these assumptions, we obtain some fundamental results about the relation between prior and posterior information. We also consider an optimal stopping problem for this partially observable Markov decision process.     

Rook numbers and the normal ordering problem

For an element w in the Weyl algebra generated by D and U with relation DU=UD+1, the normally ordered form is w=∑c_i,jUⁱD^j. We demonstrate that the normal order coefficients c_i,j of a word w are rook numbers on a Ferrers board. We use this interpretation to give a new proof of the rook factorization theorem, which we use to provide an explicit formula for the coefficients c_i,j. We calculate the Weyl binomial coefficients: normal order coefficients of the element (D+U)ⁿ in the Weyl algebra. We extend these results to the q-analogue of the Weyl algebra. We discuss further generalizations using i-rook numbers.     

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.     

Convexity of chance constraints with independent random variables

We investigate the convexity of chance constraints with independent random variables. It will be shown, how concavity properties of the mapping related to the decision vector have to be combined with a suitable property of decrease for the marginal densities in order to arrive at convexity of the feasible set for large enough probability levels. It turns out that the required decrease can be verified for most prominent density functions. The results are applied then, to derive convexity of linear chance constraints with normally distributed stochastic coefficients when assuming independence of the rows of the coefficient matrix.