# UMVUE for the Product of Means from Independent Samples

Let $$x_1, \ldots, x_m$$ be i.i.d. samples drawn from a distribution $$P$$, and $$y_1, \ldots, y_n$$ be i.i.d. samples drawn from a distribution $$Q$$. Assume that the samples $$x_i$$ and $$y_j$$ are independent of each other. Suppose $$\bar{X}$$ is the uniform minimum variance unbiased estimator (UMVUE) for $$\mu_X = E_{X \sim P}[X]$$ and $$\bar{Y}$$ is the UMVUE for $$\mu_Y = E_{Y \sim Q}[Y]$$.

What is the UMVUE for the parameter $$\mu_X \mu_Y$$? Is $$\bar{X} \bar{Y}$$ the UMVUE for $$\mu_X \mu_Y$$, or is there a better estimator?

• There is in general no reason that $\bar{X} \bar{Y}$ have expectation $\mu_X \mu_Y$ ... Commented Aug 4 at 22:37
• $x_i$ and $y_j$ are assumed independent so $\bar{X}$ and $\bar{Y}$ are independent hence $\bar{X}\bar{Y}$ will have expectation $\mu_X \mu_Y$. Commented Aug 5 at 16:39
• Thanks, missed that. Do you know something more, like distributions, or if $\bar{X}, \bar{Y}$ are sufficient, complete? Commented Aug 5 at 17:26
• No, we don't know much about distributions. so I'm happy to know of any counterexample as well. Commented Aug 5 at 23:09