 A function arising in the estimation of unobserved probabilityH. Radjavi and C.C.A. Sastri Statistics & Probability Letters, 2006, vol. 76, issue 7, 682-688 Abstract: We study a function that arises naturally in the estimation of unobserved probability in random sampling. To define it, let [mu]=([mu]1,[mu]2,...) be an infinite discrete probability measure, with [mu](i)=[mu]i. Let [mu]i>0 and [mu]i[greater-or-equal, slanted][mu]i+1[for all]i. Given s[set membership, variant][0,1], let [sigma](s)=inf{[mu](A):[mu](A)[greater-or-equal, slanted]s}. Several questions arise: what are the measures [mu] for which [sigma] is continuous? What are the measures for which [sigma](s)=s for all s? When [sigma] is not continuous, can one say more about its discontinuities? Work by Rényi yields complete answers to the first two questions, and surprisingly the two classes of measures turn out to be the same. The last question is more complex, and we answer it by considering different cases. In that process, we discover a class of measures which have the interesting property that, for them, events are uniquely determined by their probabilities. We call such measures sharp. Analogous results hold for the function [tau](s)=sup{[mu](A):[mu](A)[less-than-or-equals, slant]s}. Keywords: Unobserved; probability; Discrete; measures; Estimation; Sharp; measures (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:76:y:2006:i:7:p:682-688 Ordering information: This journal article can be ordered from Access Statistics for this article Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul More articles in Statistics & Probability Letters from Elsevier |
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