# Ratio estimators in sampling

It is intended to estimate the number of dead trees of certain species a forest reserve. The reserve is divided into 200 areas of 1.5 hectares. O number of dead trees is measured using aerial photography (X) in 200 areas having a total count of approximately 15,600 dead trees species. In 10 of the 200 areas, the number of dead trees beyond the assessment by aerial photography is also evaluated by land count (Y). the Results are shown in the table below.

$$\begin{bmatrix}Area:1&2&3&4&5&6&7&8&9&10\\X_i:12&39&24&24&18&30&12&6&36&42\\Y_i:18&42&24&36&24&36&14&10&48&54\end{bmatrix}$$

a)Find using simple random sampling with replacement an estimate for the number of dead trees and also an estimate of their variance.

b) Find the bias for the mean number of dead trees

c) Recalculate estimates without using the auxiliary variable X and compare Results achieved.

d) Recalculate the previous items now considering simple random sampling without replacement

What I did

a)Let $\hat{\tau_y}$ an estimative for total $Y$ and $r=\frac{\overline{y}}{\overline{x}}$ then $$\hat{\tau_y}=r\tau_x=\frac{\overline{y}}{\overline{x}}\tau_x=\frac{30.6}{23.4}15600=20400$$

$$s^2_R=\frac{1}{n-1}\sum (Y_i-rX_i)^2=12.076$$

$$\hat{Var(\tau_y)}=\tau_x^2*\frac{s_R^2}{n\mu_x^2}=15600^2*\frac{12.076}{10*23.4^2}=536711$$

b)Let $B$ the bias $$B=\frac{1}{\mu_x^2}(R\sigma_x^2-p(X,Y)\sigma_x\sigma_y)$$ where $R=\frac{\tau_y}{\tau_x}$ the true ratio and $p(X,Y)$ is the correlation coefficient.Here I am confused if I should just use the sample values $s^2$ and $\overline{x}$ or not

c)$$\hat{\tau_y}=N\overline{y}=200*30.6=6120$$

d)How can I derive the formulas for the case without replacement?

• What's the reference for what you've done so far? – Steve Samuels Nov 6 '15 at 22:17
• @SteveSamuels I'm using this book Book – user72621 Nov 7 '15 at 12:23