Stochastic Comparisons of Symmetric Sampling Designs

Larry Goldstein (), Yosef Rinott () and Marco Scarsini
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Larry Goldstein: University of Southern California
Yosef Rinott: Hebrew University of Jerusalem

Methodology and Computing in Applied Probability, 2012, vol. 14, issue 3, 407-420

Abstract: Abstract 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.

Keywords: Convex order; Stochastic majorization; Stratified sampling; Exchangeable partitions of integers; Primary 60E15; Secondary 62D05, 65C05 (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/s11009-011-9213-3

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