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## (Im)Perfect Detectors…

Detecting reliably an event is surely something to worry about in applied science. One of the main models used is the perfect-imperfect detector, obviously underlying a Poisson process…  Two detectors are counting events generated by a source (eg a photon device). The first one detects efficiently (perfect) the events whether the other one lacks efficiency.

Then X~Poi(λ) and Y~Poi(λp), where p is the inefficiency ratio and estimation is (almost) trivial. Assume that m,r are the counts of X,Y respectively. Furthermore, let k be the total observations and n the observations of X.

The mle of λ is m/n as usual. What about p?

The likelihood is proportional to $exp \left[ -\lambda p\left( k-n \right) \right]{{\left( \lambda p \right)}^{r}}$, so taking logarithms and differentiating with respect to p gives us that the mle is $\hat{p}=\frac{mr}{n(k-n)}$.

The real question is : what if $\frac{mr}{n(k-n)}>1$?