Estimating the Population Mean From a Simple Random Sample When Some Responses are Missing

We develop a design-based prediction approach to estimate the finite population mean in a simple setting where some responses are missing. The approach is based on indicator sampling random variables that operate on labeled units (subjects). Missing data mechanisms are defined that may depend on a subject, or on a selection (such as when the study design assigns groups of selected subjects to different interviewers). Using an approach usually reserved for model-based inference, we develop a predictor that equals the sample total divided by the expected sample size. The methods are direct extensions of best linear unbiased prediction (BLUP) in finite population mixed models. When the probability of missing is estimated from the sample, the empirical estimator simplifies to the mean of the realized non-missing responses. The different missing data mechanisms are revealed by the notation that accounts for the labels and sample selections. The mean squared error (MSE) of the empirical estimator, counter-intuitively, is smaller than the MSE if the probability of missing is known