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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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