On some alternative sampling strategies using auxiliary information
Let D be a sampling design of equal probability sampling without replacement from a finite population \(U=(U_ 1,U_ 2,...,U_ N)\), where each unit \(U_ i\) is associated to an unknown real variate \(Y_ i\) and an auxiliary variable \(X_ i\), and let \(\bar X\),Ȳ,\=x and \(\bar y\) be the population mean of X, the population mean of Y, the sample mean of X and the sample mean of Y, respectively. The interest here is in estimating \(\bar Y\). Let \(\bar y_ 1(=\bar y)\), \(\bar y_ 2,\bar y_ 3\) and \(\bar y_ 4\) be the usual mean estimator, ratio estimator, product estimator and regression estimator, respectively, and let \(\bar y_{ij}=\alpha_{ij}\bar y_ i+(1-\alpha_{ij})\bar y_ j\), \(i,j=1,2,3\), \(i\neq j\). For a sampling design D, and \(i,j=1,2,3\), \(i\neq j\), the authors propose combinations \(\bar y^*\!_{ij}=\alpha^*\!_{ij}\bar y_ i+(1-\alpha^*\!_{ij})\bar y_ j\) with optimal weights \(\alpha^*\!_{ij}\) in the sense that \(\bar y^*\!_{ij}\) has minimal mean square error, and prove that, to the first degree approximation, the proposed strategies (D,ȳ\({}^*\!_{ij})\) are as efficient as the classical regression strategy (D,ȳ\({}_ 4)\). Further, these strategies are superior to corresponding ratio or product strategies as long as the difference between the weights used and optimal weights is less than \(| 1\pm \rho C_ y/C_ x| /| \lambda_ i-\lambda_ j|\), where \(\rho\) is the correlation coefficient between Y and X, \(C_ y\) and \(C_ x\) are the coefficients of variation of Y and X, resp., \(\lambda_ 1=0\), \(\lambda_ 2=1\), \(\lambda_ 3=-1.\) In two-phase sampling, let D'\({}_ t\) be the design whose second phase sample is a sub-sample of the first phase sample, and D''\({}_ t\) be the design whose second phase sample is selected, independently of the first, directly from the whole population, and let \(\bar y^ t\!_{ij}\) and \(\bar y^ t\!_ 4\) be \(\bar y_{ij}\) and \(\bar y_ 4\), resp., obtained by replacing \(\bar X\) by the first phase sample mean \=x'. Then, to the first degree approximation, the optimal strategies (D'\({}_ t,\bar y^{t^*}\!\!_{ij})\) for \(i,j=1,2,3\) and \(i\neq j\) are as efficient as the two phase regression strategy (D'\({}_ t,\bar y^ t\!_ 4)\), while the optimal strategies (D''\({}_ t,\bar y^{t^*}\!\!_{ij})\) for \(i,j=1,2,3\) and \(i\neq j\) are superior to (D''\({}_ t,\bar y^ t\!_ 4)\).
equal probability sampling without replacement
scientific article; zbMATH DE number 3846639