Estimation of symmetric functions of a finite population
type: Item
instance of: scholarly article
title: Estimation of symmetric functions of a finite population
zbMATH Open document ID: 0537.62012
DOI: 10.1007/BF01915189
author: N. E. Zubov
published in: Metrika
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/175937
review text: 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.
Mathematics Subject Classification ID: 62D05
Mathematics Subject Classification ID: 62F10
zbMATH DE Number: 3854186
zbMATH Keywords: finite population
zbMATH Keywords: permutations
zbMATH Keywords: sampling design
zbMATH Keywords: loss function
zbMATH Keywords: convex function
zbMATH Keywords: average risk
zbMATH Keywords: uniform prior
zbMATH Keywords: symmetric design
zbMATH Keywords: sample mean
zbMATH Keywords: unbiased estimators
zbMATH Keywords: best linearly invariant estimator
reviewed by: Khursheed Alam
MaRDI profile type: Publication
cites work: Q3241532
cites work: The best strategy for estimating the mean of a finite population
cites work: Finite Population Sampling--On Labels in Estimation
cites work: A minimax approach to randomization and estimation in survey sampling
cites work: On the completeness of the class of fixed size sampling strategies
OpenAlex ID: W2081737895
description: scientific article; zbMATH DE number 3854186