Showing papers by "Fred J. Hickernell published in 2002"

Uniform designs limit aliasing

Abstract: SUMMARY When fitting a linear regression model to data, aliasing can adversely affect the estimates of the model coefficients and the decision of whether or not a term is significant. Optimal experimental designs give efficient estimators assuming that the true form of the model is known, while robust experimental designs guard against inaccurate estimates caused by model misspecification. Although it is rare for a single design to be both maximally efficient and robust, it is shown here that uniform designs limit the effects of aliasing to yield reasonable efficiency and robustness together. Aberration and resolution measure how well fractional factorial designs guard against the effects of aliasing. Here it is shown that the definitions of aberration and resolution may be generalised to other types of design using the discrepancy.

Obtaining O( N - 2+∈ ) Convergence for Lattice Quadrature Rules

Abstract: Good lattice quadrature rules are known to have O(N - 2+∈) convergence for periodic integrands with sufficient smoothness. Here it is shown that applying the baker's transformation to lattice rules gives O(N - 2+∈) convergence for nonperiodic integrands with sufficient smoothness. This approach is philosophically different than making a periodizing transformation of the integrand as it results in a different error analysis.

The error bounds and tractability of quasi-Monte Carlo algorithms in infinite dimension

Abstract: Dimensionally unbounded problems are frequently encountered in practice, such as in simulations of stochastic processes, in particle and light transport problems and in the problems of mathematical finance. This paper considers quasi-Monte Carlo integration algorithms for weighted classes of functions of infinitely many variables, in which the dependence of functions on successive variables is increasingly limited. The dependence is modeled by a sequence of weights. The integrands belong to rather general reproducing kernel Hilbert spaces that can be decomposed as the direct sum of a series of their subspaces, each subspace containing functions of only a finite number of variables. The theory of reproducing kernels is used to derive a quadrature error bound, which is the product of two terms: the generalized discrepancy and the generalized variation.Tractability means that the minimal number of function evaluations needed to reduce the initial integration error by a factor s is bounded by Ce-p for some exponent p and some positive constant C. The e-exponent of tractability is defined as the smallest power of e-1 in these bounds. It is shown by using Monte Carlo quadrature that the e-exponent is no greater than 2 for these weighted classes of integrands. Under a somewhat stronger assumption on the weights and for a popular choice of the reproducing kernel it is shown constructively using the Halton sequence that the e-exponent of tractability is 1, which implies that infinite dimensional integration is no harder than one-dimensional integration.

${E(s^2)}$-Optimality and Minimum Discrepancy in 2-level Supersaturated Designs

Abstract: Supersaturated experimental designs are often assessed by the E(s 2 )c ri- terion, and some methods have been found for constructing E(s 2 )-optimal designs. Another criterion for assessing experimental designs is discrepancy, of which there are several different kinds. The discrepancy measures how much the empirical distribution of the design points deviates from the uniform distribution. Here it is shown that for 2-level supersaturated designs the E(s 2 ) criterion and a certain discrepancy share the same optimal designs.

An Historical Overview of Lattice Point Sets

Abstract: Good lattice point sets are an important kind of low discrepancy points for multidimensional quadrature, simulation, experimental design, etc. The theoretical development of lattice point sets began over 40 years ago, but some important gaps in the theory remain. This article reviews the development of lattice point sets and highlights some open problems.

Monte Carlo and quasi-Monte Carlo methods 2000 : proceedings of a conference held at the Hong Kong Baptist University, Hong Kong SAR, China, November 27-December 1, 2000

Abstract: Large Deviations in Rare Events Simulation: Examples, Counterexamples and Alternatives.- Some Applications of Quasi-Monte Carlo Methods in Statistics.- Some New Perspectives on the Method of Control Variates.- Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms.- Quasirandom Walk Methods.- Recent Advances in the Theory of Nonlinear Pseudorandom Number Generators.- QMC Integration - Beating Intractability by Weighting the Coordinate Directions.- Quasi-Monte Carlo - Discrepancy between Theory and Practice.- Efficient Monte Carlo Simulation Methods in Statistical Physics.- An Historical Overview of Lattice Point Sets.- Multicanonical Monte Carlo Simulations.- Pricing American Derivatives using Simulation: A Biased Low Approach.- Fast Evaluation of the Asian Basket Option by Singular Value Decomposition.- Relationships Between Uniformity, Aberration and Correlation in Regular Fractions 3s-1.- Uniformity in Fractional Factorials.- Another Random Scrambling of Digital (t, s)-Sequences.- Fast Generation of Randomized Low-Discrepancy Point Sets.- Obtaining O(N-2+?) Convergence for Lattice Quadrature Rules.- Efficient Bidirectional Path Tracing by Randomized Quasi-Monte Carlo Integration.- Residual Versus Error in Transport Problems.- Construction of Equidistributed Generators Based on Linear Recurrences Modulo 2.- Quasi-Regression and the Relative Importance of the ANOVA Components of a Function.- Variants of Transformed Density Rejection and Correlation Induction.- Using Discrepancy to Evaluate Fractional Factorial Designs.- A Parallel Quasi-Monte Carlo Method for Computing Extremal Eigenvalues.- A Nonempirical Test on the Weight of Pseudorandom Number Generators.- A Kronecker Product Construction for Digital Nets.- Parallel Quasi-Monte Carlo Methods on a Heterogeneous Cluster.- American Option Pricing: A Classification-Monte Carlo (CMC) Approach.- A Software Implementation of Niederreiter-Xing Sequences.- Average Case Complexity of Weighted Integration and Approximation over IRd with Isotropic Weight.- Using MCMC for Logistic Regression Model Selection Involving Large Number of Candidate Models.- Quasi-Monte Carlo Methods in Designs of Spatial Sampling Points.- Improving the Efficiency of the Two-stage Shrinkage Estimators Using Bootstrap Methods.- Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains.- D-optimal Designs Based on Elementary Intervals for b-adic Haar Wavelet Regression Models.- On the Monte-Carlo Simulation of Several Regression Estimators in Nonlinear Time Series.

Designs for smoothing spline ANOVA models

Abstract: Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique This paper considers the design problem for specific types of SS-ANOVA models As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior Three examples for additive and interaction spline models are provided for illustration A comparison of the asymptotic designs, the 2d factorial designs, and the glp designs is given by numerical computation