Poisson exposure model

For fixed exposures $x_i\ge 0$, a Poisson exposure model takes independent observationsIt has mean and variance $\theta x_i$ and is therefore heteroscedastic when the exposures differ.

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.

For testing $H_0:\theta =1$ against a larger alternative, approximate size-$0.05$ upper-tail tests reject whenor, respectively,