Stratified random sampling; gain in precision due to stratification in the case of proportional allocation

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In the current sample survey, the stratification plays an important role theoretically as well as in practical application. The main object of stratification has been thought to improve the precision in estimating the population mean and most of the studies on the stratified random sampling so far seem to have been made within its limitation. But in practice we consider the problem of estimating several population characteristics including means, variances, covariances, correlation coefficients etc. In our recent paper [4], [6] we explored the problem from a general point of view by estimating a fairly general class of functionals of the population distribution. Let ~ be a family of all continuous or all discontinuous distribution functions over p (>__ 1) dimensional Euclidian space. Let us consider a functional t~(F) over ~ with given symmetric kernel ~(xl , . . . , x~) of degree m such that

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