Let X and Y be independentrandomvariableswith X Poissons dis- tributedwithmean and Y exponentiallydistributedwithrate , where

Let X and Y be independentrandomvariableswith X Poissons dis-

tributedwithmean λ and Y exponentiallydistributedwithrate λ, where λ > 0, and let (X1, Y1), ..., (Xn, Yn) be asamplefromthisdistribution.

a) Determineamomentestimator ˜λn for λ based onthestatistic t(x, y) = x − y and thesample (X1, Y1), ..., (Xn, Yn) and finditsasymptoticdistribution.

b) FindtheasymptoticdistributionoftheMLE ˆλn of λ.

c) Comparetheasymptoticdistributionsoftheestimators ˜λn and˜λn.