### Abstract:

We review some results in problems of estimating a finite population total (mean) through a sample sur¬vey. Section 2 considers inference under a fixed population model and Section 3 addresses the same prob¬lem when the finite population is looked upon as a sample from a superpopulation and technique of theo¬ry of prediction are used. Since the probability density function of data obtained from a sample survey equals the selection probability of the sample, thus making the likelihood function 'flat', use of the likeli¬hood, when a prior is assumed for the finite population parameters, restricts one to model-based inference, in case a non-informative sampling design (s.d.) is used for the survey. The data obtained through a set (sample) are minimal sufficient (though not complete sufficient) for inference and hence the use of Rao-Blackwellization provide improved estimators. Noting the non-existence of a uniformly minimum vari¬ance unbiased estimator for population total in general, review is made of the results on admissibility of estimators for a fixed s.d. in the relevant classes. If, however, the survey population is looked upon as a sample from a superpopulation £, optimum strategies are available in certain classes. Under the prediction-theoretic approach, a purposive sampling design becomes an optimal one under a wide class of superpop¬ulation models. This is in direct conflict with the classical probability sampling-based theory. However, these model-dependent optimal strategies fail (invoke large bias or large mean square error (mse)) if the assumed models turn out to be wrong. Use of probability sampling salvages the situation. A class of strate¬gies, which depend both on superpopulation model and sampling design, have been suggested. Finally, the problem of asymptotic unbiased estimation of design variance of these strategies under multiple regres¬sion superpopulation models have been reviewed.