Assume that the Y -values in the population are related to an auxiliary variable x through the ratio model Yi = βxi + i, for i = 1, 2,...,N, where the random errors i are independent with E( i) = 0 and var( i) = viσ2 R, in which the vi are known and β and σ2 R are unknown constants. A simple random sample of size n is selected. The best linear unbiased predictor of the population total τ is τˆ = Ys + β
s, where Ys is the total of the Y -values in the sample and xs is the total of the x-values not in the sample and
![](https://test.transtutors.com/qimg/7f7d7d46-9254-4563-af92-6107810aa26c.png)
When the variance of Yi is proportional to xi, that is, vi = xi, the best linear unbiased estimator β is the ratio estimator ![](https://test.transtutors.com/qimg/463ebddc-89b6-425d-a4e5-d30d453fc386.png)
![](https://test.transtutors.com/qimg/56d87218-de9d-4fef-93ca-4754c8793cee.png)
Compute the value of
with this estimator with the data in Example 1.
(b) Show that when the variance of Yi is proportional
is the mean-of-the-ratios estimator
![](https://test.transtutors.com/qimg/5b17ab40-8095-4def-bd4c-8750b4a8bca3.png)
Compute the value of τˆ with this estimator with the data in Example 1.