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Admissibility of unbiased estimators in finite population sampling for samples of size at most two. (English) Zbl 0533.62016
Consider a sampling design p of size at most two from a finite population U, and \(X=(x_ 1,x_ 2,...,x_ N)\) is a vector of real numbers associated with the units 1,2,...,N of U. In this paper, the author proposes a sufficient condition for an unbiased estimator to be admissible. For the sampling design p, let \(e^*\) be an unbiased estimator of a real-valued parameter \(\theta\) (X) of the form \(e^*=e^*(s,X)=e^*\!_ i(x_ i)\) if the sample \(s=\{i\}\), \(=e^*\!_{ij}(x_ i,x_ j)\) if \(s=\{i,j\}.\)
If there exist non-zero constants \(q_ 1,q_ 2,...,q_ N\), satisfying: (1) at any point X for which \(x_ 1/q_ 1=x_ 2/q_ 2=...=x_ N/q_ N\), \(e^*(s,X)=\theta(X)\), for all s with \(p(s)>0\); and (2) for given constants \(c_ 1,c_ 2\) with \(c^ 2\!_ 1+c^ 2\!_ 2\neq 0\), \(e^*\) can be formed as \(e^*(s,Z)=c_ 1e'(s,Z)+c_ 2e''(s,Z)\), where \(Z=(x_ 1/q_ 1,x_ 2/q_ 2,...,x_ N/q_ N)\), such that (i) for every \(i=1,2,...,N\), e’\({}_{ij}(x_ i/q_ i,0)=a_ i(x_ i/q_ i)\) for all j \((>i)\), e’\({}_{ji}(0,x_ i/q_ i)=a_ i(x_ i/q_ i)\) for all j \((<i)\); and (ii) for every real t, there exist positive numbers \(f_ 1,f_ 2,...,f_ N\) such that \(f_ ie''\!_{ij}(t,0)=f_ je''\!_{ij}(0,t)\), for all (i,j) with \(1\leq i<j\leq N\), then \(e^*\) is admissible within the class of unbiased estimators of \(\theta\) (X). The result is also used to check admissibility of several unbiased estimators of population totals.
Reviewer: H.-J.Chang

MSC:
62D05 Sampling theory, sample surveys
62C15 Admissibility in statistical decision theory
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