Let X = (X1, ...,Xn) be independentandPoissondistributedwith E(Xj) = Nj where > 0 is unknownand Nj

Let X = (X1, ...,Xn) be independentandPoissondistributedwith E(Xj) = λNj where λ > 0 is unknownand Nj are knownconstants.Modelsofthistypearise,for example,inriskstudieswhere Nj is thenumberofindividualsatriskingroup j, Xj the numberofevents,e.g.accidentsorcasualties,and λ is theriskrate.

a) Showthattheabovemodeldefinesaregularexponentialfamilywithcanonical parameter θ = log λ;

b) Identifythecanonicalstatisticandfinditsmeanandvariance;

c) Findthemaximumlikelihoodestimateˆλ of λ;

d) FindtheFisherinformationfor λ and thevarianceofˆλ.

An alternativeestimatoristhe averagerate

e) Showthat ˜λ is anunbiasedestimatorof λ and determineitsvariance;

f) Comparethevariancesoftheestimators ˆλ and˜λ to theCramér–Raolowerbound;
g) HowarethefindingsaboverelatedtothoseinExercise4.3?

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