## (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 , so taking logarithms and differentiating with respect to p gives us that the mle is .

The real question is : what if ?

**Look it up… **S. S. Chitgopekar, “A note on the estimation of the Poisson parameter,” *International Journal of Mathematics and Mathematical Sciences*, vol. 8, no. 1, pp. 193-196, 1985. [pdf]