Consider thefamilyofdistributionswithdensities with respecttostandardLebesguemeasureon R and considerasample X1, ...,Xn from thisdistribution. a) Determineamomentestimator n for based

Consider thefamilyofdistributionswithdensities

with respecttostandardLebesguemeasureon R and considerasample X1, ...,Xn from thisdistribution.

a) Determineamomentestimator ˜θn for θ based onthestatistic t(x) = x;

b) Findtheasymptoticdistributionofthismomentestimator;

c) Showthatthefamilyisaregularone-dimensionalexponentialfamilyandidentify the canonicalparameter,canonicalstatistic,andcumulantfunction;

d) ShowthattheMLEof θ based onasample X1, ...,Xn is givenas

e) Findtheasymptoticdistributionof ˆθn;

f) Comparetheestimators ˜θn and ˆθn.

Transcribed Image Text:

fo(x) = 0x11(0,1)(2), 0 0 = R+