Suppose there are two random variables $X$ and $Y$. Define $Z = \frac{X}{Y}$.
This link (https://www.stat.cmu.edu/%7Ehseltman/files/ratio.pdf) shows us how to take the approximate expectation and variance of $Z$:
$$ E(\frac{X}{Y}) = \frac{\mu_{x}}{\mu_{y}} - \frac{\sigma_{x,y}}{\mu_{x}^2} + \frac{\sigma_{y}^2 \cdot \mu_{x}}{\mu_{y}^3}$$
$$Var(\frac{X}{Y}) = \left(\frac{\mu_{x}^2}{\mu_{y}^2}\right) \left(\frac{\sigma_{x}^2}{\mu_{x}^2} - \frac{2 \cdot \sigma_{x,y}}{\mu_{x} \cdot \mu_{y}} + \frac{\sigma_{y}^2}{\mu_{y}^2}\right)$$
My Question: Suppose we have some observed data from $X = x_1, x_2, ... x_n$ and $Y = y_1, y_2....y_n$ and now want to estimate these same quantities. I would have naively assumed that each quantity in the above formulae could have been replaced with their estimated value based on the observed data (but I am not sure if this is correct):
$$\hat{E}\left(\frac{X}{Y}\right) = \frac{\hat{\mu}_{x}}{\hat{\mu}_{y}} - \frac{\hat{\sigma}_{x,y}}{\hat{\mu}_{x}^2} + \frac{\hat{\sigma}_{y}^2 \cdot \hat{\mu}_{x}}{\hat{\mu}_{y}^3}$$
$$\hat{Var}\left(\frac{X}{Y}\right) = \left(\frac{\hat{\mu}_{x}^2}{\hat{\mu}_{y}^2}\right) \left(\frac{\hat{\sigma}_{x}^2}{\hat{\mu}_{x}^2} - \frac{2 \cdot \hat{\sigma}_{x,y}}{\hat{\mu}_{x} \cdot \hat{\mu}_{y}} + \frac{\hat{\sigma}_{y}^2}{\hat{\mu}_{y}^2}\right)$$
In the presence of dealing with limited data and the estimation of sample mean and sample variance - do we know if the above formulae are any better than simply taking:
$$\hat{E}\left(\frac{X}{Y}\right) = \frac{\hat{\mu}_{x}}{\hat{\mu}_{y}}$$
$$\hat{Var}\left(\frac{X}{Y}\right) = \frac{\hat{\sigma}_{x}^2}{\hat{\sigma}_{y}^2} $$
Logically, I would have thought the above formulae (i.e. without the "hats") are "better" (i.e. closer to the population value, smaller variances, etc.) since they are taking into consideration covariances - but based on some previous questions I posted (e.g. Average Result of Dividing Two Two Dice Rolls?), I am not sure about this.
Thanks!
- Note: I understand that in many cases, the "true" values of $E(Z)$ and $Var(Z)$ might not be known - therefore further complicating this question.
