Stability of Order Statistics under Dependence

We precisely evaluate the upper and lower deviations of the expectation of every order statistic from an i.i.d. sample under arbitrary violations of the independence assumption, measured in scale units generated by various central absolute moments of the parent distribution of a single observation. We also determine the distributions for which the bounds are attained. The proof is based on combining the Moriguti monotone approximation of functions with the Hölder inequality applied for proper integral representations of expected order statistics in the independent and dependent cases. The method allows us to derive analogous bounds for arbitrary linear combinations of order statistics.     

Unimodality of uniform generalized order statistics,with applications to mean bounds

We prove that uniform generalized order statistics are unimodal for an arbitrary choice of model parameters. The result is applied to establish optimal lower and upper bounds on the expectations of generalized order statistics based on nonnegative samples in the population mean unit of measurement. The bounds are attained by two-point distributions.     

最大均方差异统计量的一般界

经典的最大均方差异统计量MMDb(F,X,Y)和MMD2u(F,X,Y)基于等量假设(即m=n)来检验两组样本X={x1,x2,…,xm}和Y={y1,y2,…,yn}是否来自不同的分布.本文对样本等量假设进行了放松,推广了经典的最大均方差异统计量界,推导出当m≠n时统计量MMDb(F,X,Y)和MMD2u(F,X,Y...     

Error bounds for set inclusions

A variant of Robinson-Ursescu Theorem is given in normed spaces. Several error bound theorems for convex inclusions are proved and in particular a positive answer to Li and Singer's conjecture is given under weaker assumption than the assumption required in their conjecture. Perturbation error bounds are also studied. As applications, we study error bounds for convex inequality systems.     

Analysis of non-deterministic drum scheduling

A non-deterministic drum scheduler reads a file of drum pages in unpredictable order because it selects, at each step, to read the first not previously read page after the reading heads. In this paper bounds are derived for the average latency time of such a scheduler when reading a file uniformly distributed over the drum circumference under the assumption that read requests are issued at uniformly and independently distributed points of the drum circumference.     

Bounds for the expectation of linear combinations of order statistics with application to pert networks

The bounds derived by Arnold and Groeneveld [l] for the expectation of linear combinations of order statistics are slightly extended, and this allows us to deduce directly as particular cases the upper bounds obtained by Aven [2] for the expectation of the maximum of a number of random variables. These latter bounds are then used to construct simple explicit upper bounds for the mean completion time in Pert networks     

Evaluating Improvements for Spacings of Order Statistics

We evaluate sharp upper bounds for the consecutive spacings of order statistics from an i.i.d. sample, measured in scale units generated by various central absolute moments of the parent distribution. The bounds are based on the projection method combined with the Hölder inequalities. We characterize the probability distributions for which the bounds are attained. We also evaluate the so obtained bounds numerically and compare them with other existing bounds.     

Accuracy of the bootstrap approximation

Summary The sampling distribution of several commonly occurring statistics are known to be closer to the corresponding bootstrap distribution than the normal distribution, under some conditions on the moments and the smoothness of the population distribution. These conditional approximations are suggestive of the unconditional ones considered in this paper, though one cannot be derived from the other by elementary methods. In this paper, probabilistic bounds are provided for the deviation of the sampling distribution from the bootstrap distribution. The rate of convergence to one, of the probability that the bootstrap approximation outperforms the normal approximation, is obtained. These rates can be applied to obtain theL ^p bounds of Bhattacharya and Qumsiyeh (1989) under weaker conditions. The results apply to studentized versions of functions of multivariate means and thus cover a wide class of common statistics. As a consequence we also obtain approximations to percentiles of studentized means and their appropriate modifications. The results indicate the accuracy of the bootstrap confidence intervals both in terms of the actual coverage probability achieved and also the limits of the confidence interval.Research supported in part by NSA Grant MDA 904-90-H-1001     

Reducing model risk via positive and negative dependence assumptions

We give analytical bounds on the Value-at-Risk and on convex risk measures for a portfolio of random variables with fixed marginal distributions under an additional positive dependence structure. We show that assuming positive dependence information in our model leads to reduced dependence uncertainty spreads compared to the case where only marginals information is known. In more detail, we show that in our model the assumption of a positive dependence structure improves the best-possible lower estimate of a risk measure, while leaving unchanged its worst-possible upper risk bounds. In a similar way, we derive for convex risk measures that the assumption of a negative dependence structure leads to improved upper bounds for the risk while it does not help to increase the lower risk bounds in an essential way. As a result we find that additional assumptions on the dependence structure may result in essentially improved risk bounds.     

The efficient frontier for bounded assets

This paper develops a closed form solution of the mean-variance portfolio selection problem for uncorrelated and bounded assets when an additional technical assumption is satisfied. Although the assumption of uncorrelated assets is unduly restrictive, the explicit determination of the efficient asset holdings in the presence of bound constraints gives insight into the nature of the efficient frontier. The mean-variance portfolio selection problem considered here deals with the budget constraint and lower bounds or the budget constraint and upper bounds. For the mean-variance portfolio selection problem dealing with lower bounds the closed form solution is derived for two cases: a universe of only risky assets and a universe of risky assets plus an additional asset which is risk free. For the mean-variance portfolio selection problem dealing with upper bounds, the results presented are for a universe consisting only of risky assets. In each case, the order in which the assets are driven to their bounds depends on the ordering of their expected returns.     

Lower Bounds on the Generalized Central Moments of the Optimal Alignments Score of Random Sequences

We present a general approach to the problem of determining tight asymptotic lower bounds for generalized central moments of the optimal alignment score of two independent sequences of i.i.d. random variables. At first, these are obtained under a main assumption for which sufficient conditions are provided. When the main assumption fails, we nevertheless develop a “uniform approximation” method leading to asymptotic lower bounds. Our general results are then applied to the length of the longest common subsequences of binary strings, in which case asymptotic lower bounds are obtained for the moments and the exponential moments of the optimal score. As a by-product, a local upper bound on the rate function associated with the length of the longest common subsequences of two binary strings is also obtained.     

Reliability and expectation bounds for coherent systems with exchangeable components

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.     

Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives

We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler–Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered.     

Gradient estimates for a class of elliptic systems

Summary Gradient bounds are proved for solutions to a class of second order elliptic systems in divergence form. The main condition on this class is a generalization of the assumption that the system be the Euler-Lagrange system of equations for a functional depending only on the modulus of the gradient of the solution.     

Grüss-type bounds for covariances and the notion of quadrant dependence in expectation

We show that Grüss-type probabilistic inequalities for covariances can be considerably sharpened when the underlying random variables are quadrant dependent in expectation (QDE). The herein established covariance bounds not only sharpen the classical Grüss inequality but also improve upon recently derived Grüss-type bounds under the assumption of quadrant dependency (QD), which is stronger than QDE. We illustrate our general results with examples based on specially devised bivariate distributions that are QDE but not QD. Such results play important roles in decision making under uncertainty, and particularly in areas such as economics, finance, and insurance.     

A Berry-Esséen Theorem for Serial Rank Statistics

Berry-Esséen bounds of the optimal O(n-1/2) order are obtained, under the null hypothesis of randomness, for serial linear rank statistics, of the form a1 (Rt)a2(Rt-k). Such statistics play an essential role in distribution-free methods for time-series analysis, where they provide nonparametric analogues to classical (Gaussian) correlogram-based methods. Berry-Esséen inequalities are established under mild conditions on the score-generating functions, allowing for normal (van der Waerden) scores. They extend to the serial case the earlier result of Does (1982, Ann. Probab., 10, 982-991) on (nonserial) linear rank statistics, and to the context of nonparametric rank-based statistics the parametric results of Taniguchi (1991, Higher Order Asymptotics for Time Series Analysis, Springer, New York) on quadratic forms of Gaussian stationary processes.     

Minimizing loss probability bounds for portfolio selection

In this paper, we derive a portfolio optimization model by minimizing upper and lower bounds of loss probability. These bounds are obtained under a nonparametric assumption of underlying return distribution by modifying the so-called generalization error bounds for the support vector machine, which has been developed in the field of statistical learning. Based on the bounds, two fractional programs are derived for constructing portfolios, where the numerator of the ratio in the objective includes the value-at-risk (VaR) or conditional value-at-risk (CVaR) while the denominator is any norm of portfolio vector. Depending on the parameter values in the model, the derived formulations can result in a nonconvex constrained optimization, and an algorithm for dealing with such a case is proposed. Some computational experiments are conducted on real stock market data, demonstrating that the CVaR-based fractional programming model outperforms the empirical probability minimization.     

Projection Method for Moment Bounds on Order Statistics from Restricted Families: II. Independent Case

The method of projection, proposed in Part I, is applied to derive sharp moment bounds for the expectations of order statistics based onindependentsamples from restricted families of distributions. Three families are considered: life distributions with decreasing failure density, decreasing failure rate, and symmetric unimodal ones. The respective bounds are also numerically compared with those for general populations in both the dependent and independent cases.     

Stability in the Numerical Solution of Linear Parabolic Equations with a Delay Term

This paper is concerned with the stability of numerical processes that arise after semi-discretization of linear parabolic equations wit a delay term. These numerical processes are obtained by applying step-by-step methods to the resulting systems of ordinary delay differential equations. Under the assumption that the semi-discretization matrix is normal we establish upper bounds for the growth of errors in the numerical processes under consideration, and thus arrive at conclusions about their stability. More detailed upper bounds are obtained for -methods under the additional assumption that the eigenvalues of the semi-discretization matrix are real and negative. In particular, we derive contractivity properties in this case. Contractivity properties are also obtained for the -methods applied to the one-dimensional test equation with real coefficients and a delay term. Numerical experiments confirming the derived contractivity properties for parabolic equations with a delay term are presented.     

An Alternative Model for Queueing Systems with Single Arrivals,Batch Services and Customer Coalescence

In this paper we consider a queueing system with single arrivals, batch services and customer coalescence and we use it as a building block for constructing queueing networks that incorporate such characteristics. Chao et al. (1996) considered a similar model and they proved that it possesses a geometric product form stationary distribution, under the assumption that if the number of units present at a service completion epoch is less than the required number of units, then all the units coalesce into an incomplete (defective) batch which leaves the system. We drop this assumption and we study a model without incomplete batches. We prove that the stationary distribution of such a queue has a nearly geometric form. Using quasi-reversibility arguments we construct a network model with such queues which provides relevant bounds and approximations for the behaviour of assembly processes. Several issues about the validity of these bounds and approximations are also discussed.