# How to estimate the absolute expected difference?

Suppose we have two random variables $X$ and $Y$ with unknown distributions. I am looking for an unbiased estimator for the absolute expected difference: $$| E \{ X - Y \} | .$$ For instance, suppose we have unbiased independent observations $x_1, \ldots, x_n$ and $y_1, \ldots, y_m$ ($n,m \geq 1$), how can we use these data to construct an unbiased estimator for the above quantity?

Preferably, I would like to find an unbiased estimator for the general case, including perhaps only mild assumptions such as the existence of the first (few) moment(s). However, I am happy with any progress including:

• solutions for specific distributions with unknown parameters (e.g., under the assumption that $X$ and $Y$ are both distributed according to a normal distribution, but the means and variances are unknown),
• efficient biased estimators, for instance that minimize the mean squared error (again, suggestions for the general case and for specific distributions are welcome), and
• any thoughts on good estimators for special cases.

Thanks for any input!

• Are you looking for it to never be biased in all cases, on average, or in a specific case?
– John
Aug 31 '11 at 7:57
• Is there any reason you suspect the absolute value of the obvious estimator for $E[X-Y]$ would be biased? Aug 31 '11 at 9:12
• If you plug in sample means for $X$ and $Y$ you will get a consistent estimator. Take for simplicity $m=n=1$: $E|x_1 - y_1| \geq |Ex_1 - Ey_1| = |\mu_x - \mu_y|$ the bias indeed is present, but I think it is crucial only in very small samples... Aug 31 '11 at 9:32
• Why do you want an unbiased estimator?
– Karl
Sep 1 '11 at 1:29
• There seem to be some discrepancies between the question asked and the comments. Is it an estimator for $\vert E[X-Y]\vert$ that is desired (as in the question) or an estimator for $E\vert X - Y\vert ]$? It is not obvious to me that $\vert E[X-Y]\vert = \vert E[X] - E[Y]\vert = \vert \mu_X - \mu_Y \vert$ is the same as $E\vert X - Y\vert ]$. Oct 12 '11 at 1:18

Bootstrap bias correction was invented to adjusted for bias in the estimation of $f(Z)$ (my $Z$ is your $X-Y$). The idea is very simple: create $B$ bootstrap resamples from your data, and calculate $f_b=f(Z_b)$ for each one of them. Then the bootrstrap estimate of the bias is $\bar{f_b}-f(Z)$, where $\bar{f_b}$ is the average of $f_b$'s. Finally, subtract this bias from $f(Z)$ do get the bias corrected estimate $2f(Z) - \bar{f}_b$.