The product of two unbiased estimates, $x$ and $y$, from the same data is a biased estimator. However, I don't want to split the data into two sections. I would like to take the average estimate for $x$ and $y$ using different sections of the data. This to me seems intuitively similar to cross-validation and bootstrapping. I'd like to find some theoretical analysis on this process but I've not really been able to find any information. It seems like a pretty general idea that someone must have done before.