Estimators for a Poisson exposure model

For $S_r={\sum }_i{x}_i^r$, unweighted least squares through the origin and maximum likelihood giveBoth estimators are unbiased.

The two variances areSince $S_2^2\le S_1{S}_3$ by Cauchy-Schwarz, maximum likelihood has no larger variance, with equality when all positive exposures are equal.