Consider thesituationinExercise3.6,where (X,Y ) are random variablestakingvaluesin R+ N0 with density with respectto m,

Consider thesituationinExercise3.6,where (X,Y ) are random variablestakingvaluesin R+ × N0 with density

with respectto ν × m, where ν is thestandardLebesguemeasureon R+ and m is countingmeasureon N0. Considernow n independent andidenticallydistributed observations (X1, Y1), ..., (Xn, Yn), where (Xi, Yi)has densityformen(5.24)and (λ, β) ∈ R2 + are unknown.

a) DeterminetheMLEfor (λ, β) and identifywhenitiswell-defined;

b) FindtheasymptoticdistributionoftheMLE (ˆλn, ˆ βn);

c) Determinethemomentestimator (˜λn, ˜ βn) for (λ, β) based onthefunction s(X,Y ) =(X,XY );

d) Findtheasymptoticdistributionofthemomentestimator ˜ βn and compareitto the asymptoticdistributionof ˆ βn.

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(A)(x, y) = e/A (3x) -Br e y! >0, y = 0,1,... (5.24)