Athanassios N. Avramidis

To estimate the density f of a conditional expectation μ(Z) = E[X|Z], Steckley and Henderson (2003) sample independent copies Z ₁ ,...,Z _m ; then, conditional on Z _i , they sample n independent samples of X, and their sample mean ̅X _i is an approximate sample of μ(Z _i ). For a kernel density estimate ̂f of f based on such samples and a bandwidth (smoothing parameter) h, they consider the mean integrated squared error (MISE), ∫(̂f(x)-f(x)) ² dx, and find rates of convergence of m, n and h that optimize the rate of convergence of MISE to zero. Inspired by the cross-validation approach in classical density estimation, we develop an estimate of MISE (up to a constant) and select the h that minimizes this estimate. While a convergence analysis is lacking, numerical results suggest that our method is promising.

In specifying a multivariate discrete distribution via the the NORmal To Anything (NORTA) method, a problem of interest is: given two discrete unbounded marginals and a target value r, find the correlation of the bivariate Gaussian copula that induces rank correlation r between these marginals. By solving the analogous problem with the marginals replaced by finite-support (truncated) counterparts, an approximate solution can be obtained. Our main contribution is an upper bound on the absolute error, where error is defined as the difference between r and the resulting rank correlation between the original unbounded marginals. Furthermore, we propose a simple method for truncating the support while controlling the error via the bound, which is a sum of scaled squared tail probabilities. Examples where both marginals are discrete Pareto demonstrate considerable work savings against an alternative simple-minded truncation.

A multi-skill staffing problem in a call center where the agent skill sets are exogenous and the call routing policy has well-specified features of overflow between different agent types is addressed. Constraints are imposed on the service level for each call class, defined here as the steady-state fraction of calls served within a given time threshold, excluded. An approximation of these service levels is developed that allows an arbitrary overflow mechanism and customer abandonment. A two-stage heuristic that finds good solutions to mathematical programs with such constraints is developed. The first stage uses search methods supported by the approximation. Because service level approximation errors may be substantial, the solution is adjusted in a second stage in which performance is estimated by simulation. Realistic problems of varying size and routing policy are solved. The proposed approach is shown to be competitive with (and often better than) previously available methods. [Supplementary materials are available for this article. Go to the publisher's online edition of IIE Transactions for the following free supplemental resource: Appendix]

A popular approach for modeling dependence in a finite-dimensional random vector X with given univariate marginals is via a normal copula that fits the rank or linear correlations for the bivariate marginals of X. In this approach, known as the NORTA method, the normal distribution function is applied to each coordinate of a vector Z of correlated standard normals to produce a vector U of correlated uniform random variables over (0,1); then X is obtained by applying the inverse of the target marginal distribution function for each coordinate of U. The fitting requires finding the appropriate correlation ρ between any two given coordinates of Z that would yield the target rank or linear correlation r between the corresponding coordinates of X. This root-finding problem is easy to solve when the marginals are continuous but not when they are discrete. In this paper, we provide a detailed analysis of this root-finding problem for the case of discrete marginals. We prove key properties of r and of its derivative as a function of ρ. It turns out that the derivative is easier to evaluate than the function itself. Based on that, we propose and compare alternative methods for finding or approximating the appropriate ρ. The case of discrete distributions with unbounded support is covered as well. In our numerical experiments, a derivative-supported method is faster and more accurate than a state-of-the-art, nonderivative-based method. We also characterize the asymptotic convergence rate of the function r (as a function of ρ) to the continuous-marginals limiting function, when the discrete marginals converge to continuous distributions.