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.