Estimation of the CDF of a finite population in the presence of a calibration sample

We compare the performance of a number of estimators of the cumulative dist ribution function (CDF) for the following scenario: imperfect measurements are taken on an initial sample from a finite population and perfect measure ments are obtained on a small calibration subset of the initial sample. The estimators we considered include two naive estimators using perfect and im perfect measurements: the ratio, difference and regression estimators for a two-phase sample; a minimum MSE estimator; Stefanski and Bay's SIMEX estim ator (1996); and two proposed estimators. The proposed estimators take the form of a weighted average of perfect and imperfect measurements. They are constructed by minimizing variance among the class of weighted averages sub ject to an unbiasedness constraint. They differ in the manner of estimating the weight parameters. The first one uses direct sample estimates. The sec ond one tunes the unknown parameters to an underlying normal distribution. We compare the root mean square error (RMSE) of the proposed estimator agai nst other potential competitors through computer simulations. Our simulatio ns show that our second estimator has the smallest RMSE among the nine comp ared and that the reduction in RMSE is substantial when the calibration sam ple is small and the error is medium or large.