Estimation of symmetric functions of a finite population (Q792714)

Consider a finite population U consisting of N units, with \(x_ i\) denoting the variate value of the \(i^{th}\) unit. Let \(\pi\) denote a permutation of the units (1,2,...,N) and \(\Pi\) denote the set of N! such permutations. Let \(\beta\) denote a subset of U and S denote the collection of all subsets of S. A probability function \(\rho\) on S is called a sampling design with \(\rho\) (\(\beta)\) denoting the probability measure of \(\beta\). A real valued function \(\phi =\phi(\underset \tilde{} x)\) is called a parameter, where \(\underset \tilde{} x=(x_ 1,...,x_ N)\) and an estimate of \(\theta\) is \(e=e(\beta,\underset \tilde{} x)\), a real valued function which depends on \(\underset \tilde{} x\) through the x-values of the units included in \(\beta\). For estimating \(\theta\) the loss function is given by \(\lambda (e,\theta)\), a convex function of e for all \(\theta\). An average risk of e with respect to a uniform prior on the labels of U is given by \[ R(e,\rho,\underset \tilde{} x)=N!^{-1}\sum_{\pi \in \Pi}\sum_{\beta \in S}\rho(\beta)\lambda(e(\beta,\pi,\underset \tilde{} x),\quad \theta(\pi \underset \tilde{} x)). \] A design \(\rho\) is said to be symmetric if \(\rho(\beta)=\rho(\pi \beta)\) for all \(\beta\in S\) and \(\pi\in \Pi\). It is shown that for any symmetric design \(\rho\), the sample mean minimizes the average risk R(e,\(\rho\),\(\underset \tilde{} x)\) among all unbiased estimators of the population mean \(\bar x\). An estimator \(e(\beta\),\(\underset \tilde{} x)\) associated with a design \(\rho\) is said to be linearly invariant if \(e(\beta\),\(\underset \tilde{} x)=\sum_{i\in \beta}b(\beta,i)x_ i\) for all \(\beta\in S\) for which \(\rho(\beta)>0\), where \(\sum_{i\in \beta}b(\beta,i)=1\). A best linearly invariant estimator in the class of linearly invariant estimators of a fixed average size is given which minimizes the average risk.