The quantity B used in ratio estimation is sometimes called the ratio-of-means estimator. An alternative that has

The quantity ˆB used in ratio estimation is sometimes called the ratio-of-means estimator.
An alternative that has been proposed is the mean of ratios estimator: Let bi = yi/xi for unit i; then the mean-of-ratios estimator is

a Doyou think the mean-of-ratios estimator is appropriate for the data in Example 4.5? Why, or why not?
b Show that, for the ratio-of-means estimator ˆB, tx ˆB = ty when the entire population is sampled (i.e., S = U).
c Give an example to show that it is possible to have tx ≠ ty when the entire population is sampled.
d Define

Show that for an SRS of size n, the bias of as an estimator of B is

As a consequence, if Sbx ≠ 0 the bias does not decrease as n increases.
e. (Requires linear model theory.) Show that is the weighted least squares estimator of β under the model

When εi’s are independent with mean 0 and variance σ2x2i .