Abstract
Based on the natural logarithm of known population mean of an auxiliaryvariable, x, the study introduces logarithmic ratio and product-type estimatorsof the population mean of the study variable, y, in simple random samplingwithout replacement (SRSWOR) scheme. Part of the eciency conditions forthe proposed logarithmic estimators to be more ecient than the existing ex-ponential ratio and product-type estimators, as well as the customary ratio andproduct-type estimators, is that the natural logarithm of the known populationmean of the auxiliary variable, x, must be greater than 2. Generally, there is ahigh tendency for the proposed logarithmic estimators to be more ecient thanexisting customary and exponential ratio and product-type estimators whenthe natural logarithm of the auxiliary variable population mean is greater than2. The theoretical results are illustrated and conrmed using some numericaldatasets.
Highlights
The use of auxiliary information to improve estimates of population parameters of the study variable is well-known in sample surveys
There is a high tendency for the proposed logarithmic estimators to be more efficient than existing customary and exponential ratio and product-type estimators when the natural logarithm of the auxiliary variable population mean is greater than 2
Theoretical results obtained in the present study indicate that efficiency conditions of logarithmic estimators over the customary ratio and product-type estimators are affected when Ln(X ) > 1 or X > exp(1) ≈ 2.718, while for the exponential ratio and product-type estimators, the efficiency conditions of the logarithmic estimators are affected when Ln(X ) > 2 or X > exp(2) ≈ 7.389
Summary
The use of auxiliary information to improve estimates of population parameters of the study variable is well-known in sample surveys. The exponential ratio-type estimator, tR is biased for Ywith bias and mean square error obtained up to first order of approximations as: 1− f Bias(tR) = n. The exponential product-type estimator, tP is biased for Ywith bias and mean square error obtained up to first order of approximations as: 1− f Bias(tP) = n (11). The present study looks beyond the customary and exponential ratio and product-type estimators proposed by Cochran (1940), Murthy (1964) and Bahl and Tuteja (1991), all of which uses the known value of the auxiliary variable population mean. The exponential-type estimators are known to perform better than the corresponding customary ratio and product-type estimators, in terms of having smaller mean square errors, under certain efficiency conditions.
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