Ratio-Type Estimators in Double Sampling for Two Stages Designs

Abstract

In the case of simple random sampling without replacement, unbiased ratio-type estimator was first developed by Hartley and Rose (1954). They assumed that the information on the auxiliary character in the population was completely available. But sometimes, it happens that this information is lacking, under this situation. The problem was considered by sukhatma (1962) who resorted to technique of single stage, double sampling and presented several ratio-type estimators out of which one was an unbiased estimate of the population mean. In the present investigation, unbiased ratio-type estimators for two-stage sampling design have been presented when the population mean of the auxiliary variable is known or unknown. For obtaining unbiased ratio-type estimators, a technique called “The extended method of symmetric means” to two-stage designs has been developed which is an extension to the technique developed by Tukey (1956) and generalized for the multi-variate case by Robson (1957). By using this extended method, variances of the several estimators considered, for the same order of approximation, have assumed that the number of primary units and secondary units in the population are large. The condition under which unbiased ratio-type estimators were more efficient than the corresponding biased estimators for the two situations viz. when single phase sampling in two stages is adopted and when two-phase sampling in two-stages is adopted, have been obtained. The estimates of variance in double sampling for two-stage design have been worked out. Using a cost function, the optimum variances for a fixed cost have also been obtained. The results obtained in chapter IV when double sampling s resorted to in two stages, have been illustrated with the help of data on wheat crop. Collected in Patna district of Bihar state during Rabi season 1966-67 under the scheme operated by I.A.R.S. for evolving suitable sampling procedure for obtaining reliable estimates of yield rates of principal cereal crops at community Development Block Level.