# Ratio of unbiased mean estimates

Suppose $$X$$ and $$Y$$ are two quantities and we are interested in estimating the ratio $$X/Y$$ but we need to estimate $$X$$ and $$Y$$ using samples (e.g. survey samples). For example, $$X$$ may be the the amount of money spent by the population on leisure in year $$T$$ and $$Y$$ may be the amount of money spent in year $$T+1$$. It is tempting to estimate $$X$$ and $$Y$$ separately and divide the estimates to get an estimate of $$X/Y$$. If $$\hat{\mu}_x$$ and $$\hat{\mu}_y$$ are empirical means of $$X$$ and $$Y$$ as computed from the samples, the ratio $$r = \hat{\mu}_x/\hat{\mu}_y$$ is not an unbiased estimate of $$X/Y$$. This can be easily seen as follows: $$E\left[\frac{X}{Y}\right] = E\left[X\right] E\left[\frac{1}{Y}\right] \geq \frac{E\left[X\right]}{E\left[Y\right]},$$where the equality follows the independence of $$X$$ and $$Y$$, and the inequality follows from Jensen's inequality, noting that $$1/Y$$ is a convex function of $$Y$$.

If samples of $$X$$ and $$Y$$ are available, an unbiased estimate of $$X/Y$$ can be obtained by generating samples of $$X/Y$$ via statistical bootstrap.

Incidentally, if $$X$$ and $$Y$$ are independent zero mean Gaussian random variables, then their ratio has a Cauchy distribution. However, if $$X$$ and $$Y$$ are not zero mean, then their ratio had a distribution that isn't well known and has a pretty complex distribution.