On predicting the finite population distribution function
Consider a finite population of size \(N\) with \(y_ 1, y_ 2, \dots, y_ N\) as values of a study variable \(y\). The finite population distribution function is defined by \[ F_ n(t)= N^{-1} \sum_{j=1}^ N \Delta(t- y_ j) \quad \text{where} \quad \Delta(x)=1 \text{ if } x\geq 0 \text{ and 0 otherwise}. \] The authors discuss the problem of optimal prediction of \(F_ N(t)\) under general linear regression models with normally distributed errors when the error variance is unknown and derive its best unbiased predictor as well as the asymptotic distribution. Under a simple location model the asymptotic relative efficiency of this predictor, relative to that of the ordinary sample empirical distribution function, is derived. Further, for some important special cases, large sample approximations for the prediction variances are obtained for the best unbiased predictors.
ordinary sample empirical distribution function
scientific article; zbMATH DE number 571644
scientific article