SOME SAMPLING SYSTEMS PROVIDING UNBIASED

SUMMARY. In this paper, modifications of many of the selection procedures commonly adopted in practice, namely, equal probability sampling, varying probability sampling, stratified sampling and multi-stage sampling have been proposed, which, while retaining the form of the usual ratio estimators, make t__3m unbiased. For many of the situations commonly met with in practice, this modification of a given sampling scheme consists essentially in first selecting one unit with probability proportional to its value of the characteristic occurring in the denominator of the ratio and then the remaining units in the sample according to the original scheme of sampling. The expressions for unbiased variance estimators of the unbiased ratio estimators have been given for some of the more important sampling schemes. Further the selection and estimation procedures which provide unbiased ratio estimators in the case of a certain general class of population parameters together with the expressions for its sampling variance and variance estimators have also been considered. As the relationship between two characteristics is usually of much interest, estimation of ratios of certain population parameters has become quite important in a large number of surveys. The method of ratio estimation is also being used to estimate population totals, since a ratio estimator is more efficient than the con ventional unbiased estimator under certain circumstances not uncommon in actual practice. The usual procedure of using the ratio method in estimating any population ratio or total has been to take the ratio of unbiased estimators of the numerator and the denominator and in the latter case multiply it by the population total of the supplementary var?ate taken in the denominator. A disadvantage of this method is that the estimator so obtained is biased for many of the selection procedures commonly adopted in surveys. Further a completely satisfactory (at least to the present authors) treatment of the errors and biases of a ratio estimator is not yet available. For small samples, at least, the bias is not likely to be small. In recent years attempts have been made to give selection and estimation procedures which provide unbiased ratio estimators. Lahiri (1951) has given a method of selecting a sample with probability proportional to its total size (pps) (sum of the sizes of the units in the sample) which is essentially similar to his method of selecting a unit with pps, namely, of selecting a unit with equal probability and including that unit in the sample if a number chosen at random from one to an upper bound of the units is less than or equal to the size of the selected unit. By 'size' here is meant the value of the supplementary var?ate under consideration. Obviously this method avoids the need for completely enumerating all possible samples and finding their total sizes and the cumulated sizes. Once a sample is chosen with pps it is easy to obtain an unbiased ratio estimator. The disadvantage of the selection procedure given by Lahiri is that it involves rejection of some draws.