Stochastic Comparisons of Symmetric Sampling Designs
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We compare estimators of the integral of a monotone function f that can be observed only at a sample of points in its domain, possibly with error. Most of the standard literature considers sampling designs ordered by refinements and compares them in terms of mean square error or, as in Goldstein et al. (2011), the stronger convex order. In this paper we compare sampling designs in the convex order without using partition refinements. Instead we order two sampling designs based on partitions of the sample space, where a fixed number of points is allocated at random to each partition element. We show that if the two random vectors whose components correspond to the number allocated to each partition element are ordered by stochastic majorization, then the corresponding estimators are likewise convexly ordered. If the function f is not monotone, then we show that the convex order comparison does not hold in general, but a weaker variance comparison does.
KeywordsConvex order Stochastic majorization Stratified sampling Exchangeable partitions of integers
AMS 2000 Subject ClassificationPrimary 60E15; Secondary 62D05 65C05
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- Goldstein L, Rinott Y, Scarsini M (2011) Stochastic comparisons of stratified sampling techniques for some Monte Carlo estimators. Bernoulli (forthcoming)Google Scholar