TY - JOUR
T1 - Stochastic Comparisons of Symmetric Sampling Designs
AU - Goldstein, Larry
AU - Rinott, Yosef
AU - Scarsini, Marco
PY - 2012/9
Y1 - 2012/9
N2 - 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.
AB - 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.
KW - Convex order
KW - Exchangeable partitions of integers
KW - Stochastic majorization
KW - Stratified sampling
UR - http://www.scopus.com/inward/record.url?scp=84864488016&partnerID=8YFLogxK
U2 - 10.1007/s11009-011-9213-3
DO - 10.1007/s11009-011-9213-3
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AN - SCOPUS:84864488016
SN - 1387-5841
VL - 14
SP - 407
EP - 420
JO - Methodology and Computing in Applied Probability
JF - Methodology and Computing in Applied Probability
IS - 3
ER -