Discussion
The bais in ratio estimator decreases with :
Answer: A
The bias decreases with increasing the sample size \(n\).
The ratio estimator \(\hat{R}\bar{X} = (\bar{y}/\bar{x})\bar{X}\) is biased because the expectation of a ratio is not the ratio of expectations. To first order its bias is
\[ \begin{aligned} B &\approx \frac{1 - f}{n}\bar{Y} \\ &\quad (C_{x}^{2} - \rho C_{x}C_{y}) \end{aligned} \]
which is of order \(1/n\), so it shrinks as \(n\) grows, and the ratio estimator is consistent. Hartley and Ross showed that the bias divided by the standard error of \(\hat{R}\) is at most the coefficient of variation of \(\bar{x}\), which also tends to 0 as \(n\) increases.
Decreasing \(n\) increases the bias, so 'both' is wrong, and a larger population with \(n\) fixed only moves the factor \(1 - f\) towards 1.
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