Let $X = (x\_1,\ldots,x\_n)$ be an i.i.d sample from distribution $F%$ and let $y = \prod\_{i=1}^n x\_i$


Can we derive a randomized, unbiased. estimator $\hat{y}$ of $y$ that *on average* considers only a subsample of $X$?


Weak conditions may be imposed on $F$, for instance we may assume it has finite mean and variance, we may also assume that $\forall i, x\_i >0$, but otherwise nothing is known about $F$.



Unbiased sample from a product
The solution is the Poisson estimator. Let $a\_i$ = $\log x\_i$. We would like to approximate $\exp (\sum\_{i=1}^n a\_i) = \exp (n \hat{a})$


Draw $\kappa \sim \textrm{Poisson}(\lambda)$, then draw with replacement $a\_{i\_1}, \ldots, a\_{i\_\kappa}$ and compute $y = e^{\lambda}\prod\_{j=1}^{\kappa} n a\_{i\_j}$


$$E(y) = \sum\_{k=0}^{\infty} \frac{1}{k!} E\left(\prod\_{j=1}^{k} n a\_{i\_j}\right) = e^{n\hat{a}}$$


