Comparison of variance estimators for the ratio estimator based on small sample

Abstract

This paper sheds light on an open problem put forward by Cochran[1]. The comparison between two commonly used variance estimators\(v_1 (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} )\) and\(v_2 (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} )\) of the ratio estimator\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{R} \) for population ratioR from small sample selected by simple random sampling is made following the idea of the estimated loss approach (See [2]). Considering the superpopulation model under which the ratio estimator\(\hat \bar Y_R \) for population mean\(\overline Y \) is the best linear unbiased one, the necessary and sufficient conditions for\(\upsilon _1 (\hat R)\mathop \succ \limits^u \upsilon _2 (\hat R)\) and\(\upsilon _2 (\hat R)\mathop \succ \limits^u \upsilon _1 (\hat R)\) are obtained with ignored the sampling fractionf. For a substantialf, several rigorous sufficient conditions for\(\upsilon _2 (\hat R)\mathop \succ \limits^u \upsilon _1 (\hat R)\) are derived.