Discussion
Let us consider the following methods of estimation, all based on SRSWOR of the same sample size. I Sample mean II Ratio estimator III Regression estimator. Let the corresponding variances be denoted by \(V_{1}^{2}\), \(V_{2}^{2}\) and \(V_{3}^{2}\). Assuming that the sample size is sufficiently large, we have :
Answer: D
The answer is none of the above.
To first order in large samples, with \(\rho\) the correlation and \(C_{x}, C_{y}\) the coefficients of variation,
\[ \begin{aligned} V_{1}^{2} &\propto C_{y}^{2} \\ V_{2}^{2} &\propto C_{y}^{2} + C_{x}^{2} - 2\rho C_{x}C_{y} \\ V_{3}^{2} &\propto C_{y}^{2}(1 - \rho^{2}) \end{aligned} \]
The regression estimator is always at least as good as both others, so \(V_{3}^{2}\) is smallest and option A is wrong. But the order of the other two depends on the data: the ratio estimator beats the sample mean only when \(\rho \gt C_{x}/(2C_{y})\). So option B holds for some populations and option C for others, and neither ordering holds in general.
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