With simple random sampling, how to approximate variance of R=avg(Y)/avg(X)?

Question

Recenly I am reading "Mathematical statistics and data analysis" written by Rice myself. At page 207, theorem A said: With simple random sampling, approimxate variance of $R=\frac{\overline{Y}}{\overline{X}}$ is $Var(R)\approx\dfrac{r^2\sigma^2_{\overline{X}}+\sigma^2_{\overline{Y}}-2r\sigma^2_{\overline{XY}}}{\mu^2_{x}}$ where $r=\dfrac{\sum^N_{i=1}{y_i}}{\sum^N_{i=1}{x_i}}$. Would anyone give me some hint about the procedure of the proof? Thanks a lot!

Answer 1

The delta method is often used to approximate variances of functions of random variables. The variances for the original random variables are known and go into he expression.

Here R is a ratio of U/V where U and V are sample means. It is just gotten by applying the first order Taylor series terms to the function about a central point and taking the square and then the expectation.

So for a differentiable function f(x) we take Var[f(X)] =E[f(X)-f(a)]$^2$ where f(a) is the mean for the random variable f(X).

Expanding the Taylor series about a gives f(X)≈f(a) +f'(a)(X-a) or

f(X)-f(a)≈f'(a)(X-a)

Then E[f(X)-f(a)]$^2$≈ [f'(a)]$^2$ E[X-a]$^2$ This gives the approximation

Var[f(X)]≈f'(a)]$^2$ Var(X).

In your case we can apply the bivariate extension of the delta method as follows:

here.

Note that they give the approximation for your exact example but include a finite population correction for the case where the sample means were gotten from simple random sampling from a finite population. It is similar but not exactly expressed in the same form as your formula from Rice. But they are probably mathematically equivalent (under some added assumptions about the means of X and Y that Rice must give but you have not mentioned). It is clear that the delta method was used.