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 an independent count with a Poisson distribution of mean $RT$, the rate estimator $N/T$ has variance $R/T$. Its fractional standard deviation is $1/\sqrt{RT}$, so achieving target fractional precision $ϵ$ requires expected count $RT={ϵ}^{-2}$ and exposure $T=1/({ϵ}^{2}R)$.

For independently acquired settings with rates $R_i$, the required total exposure is ${ϵ}^{-2}{\sum }_i{R}_i^{-1}$. Rare settings dominate because they require the same expected count but collect it more slowly. Simultaneous acquisition is a different design, and backgrounds or correlated systematic errors require a different uncertainty calculation.

A rate ratio compares event counts per unit exposure between two conditions. In a Poisson regression with a log link and binary indicator coefficient $\beta$, it equals $e^{\beta }$. Exponentiating an approximate Wald confidence interval for $\beta$ gives an interval for the rate ratio. A rate ratio is distinct from a relative risk or an odds ratio.

Let independent counts satisfy $Y_i\sim \mathrm{Poi}(\theta w_i)$, with positive known exposures $w_i$, $\theta >0$, and $a_n={\sum }_{i=1}^n{w}_i\uparrow a_{\infty }<\infty$. Their likelihood is proportional to ${\theta }^{S_n}{e}^{-a_n\theta }$, where $S_n={\sum }_{i=1}^n{Y}_i$. If $S_n>0$, the positive maximum-likelihood estimator is $S_n/a_n$; when $S_n=0$, no maximum is attained on the open parameter space, while the closed-space maximizer is zero. This extended estimate converges almost surely to $S_{\infty }/a_{\infty }$, where $S_{\infty }\sim \mathrm{Poi}(\theta a_{\infty })$. Finiteness follows from $\mathbb{E}{S}_{\infty }=\theta a_{\infty }$, and the limiting Poisson distribution follows from the finite-sum laws. In particular, the probability of a zero estimate tends to $e^{-\theta a_{\infty }}>0$, proving failure of statistical consistency. The total Fisher information tends to $a_{\infty }/\theta$: infinitely many observations need not provide infinite information.

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,