Ratio method of estimation in sample surveys
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Abstract
The ratio method of estimation in sample surveys involves the use of estimators for population parameters which are linear functions of the ratio of dependent random variables. Only large sample approximations for the bias and sampling variance are available for measuring the accuracy of ratio estimators. This study is concerned with three aspects of the properties of ratio estimators as they are used in sample surveys;First, using a theorem of Cramer, regarding the asymptotic distribution of functions of sample moments for random samples from joint continuous distributions f(x,y) having finite first and second moments, it was found that the ratio of the sample means was asymptotically normally distributed with mean equal to the ratio of the true means and variance given by the usual approximate formula. It is noted that the argument of Fisher for obtaining interval statements about the true quantity can be used, if the t distribution can be assumed to hold for samples from finite populations. This assumption is probably fairly realistic;Second, exact expressions for the bias and variance of ratio estimators have been obtained under various assumptions regarding the joint distribution of the variables sampled. In particular these assumptions restricted the types of regression and conditional variance relationships exhibited by f(x,y). For the variance laws considered and the mean square regression of y on x linear, it was found that exact expressions for the bias and variance of ratio estimators depend on the existence of the first and second moments of the distribution of the reciprocal of the sample mean of the denominator variable. The usual approximate formula for the variance was then compared with the exact expressions when f(x) followed the Pearson Type III and the truncated binomial distributions. For these distributions and whenever the regression and variance relationships considered prevail, the sample size required to achieve reasonable accuracy with the approximate variance formula depends on the magnitude of the coefficient of variation of the denominator variable of the ratio estimator. The larger this coefficient the slower is the convergence of the approximation to the exact variance expression;Third, a systematic comparison of the ratio estimator with other possible methods of estimation using the information available on a supplementary variable was conducted. The comparison was restricted to situations in which specific conditions on the form of the regression and on the residual variance law were satisfied by the joint distribution of the variables involved. The ratio method of estimation, as a general method of estimation, was found to compare favorably.