8. Prediction Approach: Robustness, Bayesian Methods, Empirical Bayes

The classical design-based approach and its modification by the super-population modelling only narrates the properties of estimators for parameters with respect to selection and theoretical estimation, but nothing about how close is an estimate to the parameter it seeks to estimate or how far it is away from it as assessed from a sample at hand. A third alternative, for example, to estimate a population total observes that a sample at hand gives exactly the value of the sample total but can say nothing about how it differs from the population total unless a relationship among the values of the units in a population is postulated at the outset itself. So, one needs to suppose the vector \(\underline{Y} = (y_1, \ldots , y_i, \ldots , y_N)\) as a random vector implying \(Y= \sum \limits _1^N y_i\) is a random variable. Therefore, Y cannot be estimated but may be predicted by estimating the expectation of Y calculated on modelling the distribution of \(\underline{Y}\). So, a prediction approach is adopted. A fourth approach is Bayesian, postulating a prior distribution for \(\underline{Y}\) and then working out a posterior distribution and a posterior expectation of Y, given the sample data at hand. As certain parameters are involved in the prior and are unknowable in practice, they may be suitably estimated from the sample yielding an empirical Bayes approach. Prediction and Bayesian methods do not need probability sampling. The procedures have their strength only through dependence on the tenability of models. So, a problem of robustness ensues if a procedure is to retain its effectiveness against departures from a specific model postulated. In ensuring robustness, however, a role of a random selection of a sample may often be helpful as we shall see.