On the Choice of Weights in a Function Space for Quasi-Monte Carlo Methods for a Class of Generalised Response Models in Statistics

Abstract

Evaluation of the likelihood of generalised response models in statistics leads to integrals over unbounded regions in high dimensions. In order to apply a quasi-Monte Carlo (QMC) method to approximate such integrals, one has to transform the original integral into an equivalent integral over the unit cube. From the point of view of QMC, this leads to a known (but non-standard) space of functions for the transformed problem. The "weights" in this function space describe the relative importance of variables or groups of variables. The quadrature error produced via a QMC method is bounded by the product of the worst-case error and the norm of the transformed integrand. This paper is mainly concerned with finding a suitable error bound for the integrand arising from a particular generalised linear model for time series regression, and then determining the choice of weights that minimises this error bound. We obtained "POD weights" ("product and order dependent weights") which are of a simple enough form to permit the construction of randomly shifted lattice rules with the optimal rate of convergence in the given function space setting. © Springer-Verlag Berlin Heidelberg 2013.