The admissibility of the linear interpolation estimator of the population total (Q1192987)

Consider a finite population of size \(N\). Let \(\lambda=(\lambda_ 1,\lambda_ 2,\dots ,\lambda_ r)\) be a vector of known constants and let \(\theta(\lambda)=\{ y\mid \text{ for each }i=1,2,\dots ,N\), \(y_ i=\lambda_ j\) for some \(j=1,2,\dots ,r\}\) be the parametric space. For any \(y\in \theta\) and a sample \(s=\{ i_ 1,i_ 2,\dots ,i_ n\}\) of labels such that \(1\leq i_ 1\leq i_ 2\leq\dots \leq N,\) let \(y(s)=(y_{i_ 1},y_{i_ 2},\dots ,y_{i_ n})\). We wish to estimate the population total \(Y=\sum^ N_{i=1}y_ i\) with squared error loss. Consider the estimator defined by \[ e^*(s,y)=N y_{i_ 1}\text{ if }n(s)=n=1, \] \[ e^*(s,y)=2^{-1}\{y_{i_ 1}(i_ 1+i_ 2-1)+y_{i_ 2}(2N-i_ 1-i_ 2+1)\},\text{ if }n(s)=n=2, \] \[ e^*(s,y)=2^{-1}\{y_{i_ 1}(i_ 1+i_ 2-1)+\sum^{n- 1}_{j=2}y_{i_ j}(i_{j+1}-i_{j-1})+y_{i_ n}(2N-i_{n-1}- i_ n+1),\text{ if }2<n(s)=n<N, \] \[ \text{and }e^*(s,y)=\sum^ N_{i=1}y_ i,\text{ if }n(s)=n=N. \] The author demonstrates the admissibility of \(e^*(s,y)\) for the parameter space \(\theta=\theta(\lambda)\). As a corollary he then derives the admissibility of \(e^*\) with the parameter space \(\delta=R^ N\). Furthermore, admissible set estimators are constructed from the posteriors which yield the linear interpolators and these are compared with standard frequentist methods.