Answer Happy

Consider a random sample of size n X1, ..., Xn from a distrbution for which the joint density of the X1, ..., Xy has the regular exponential form f(21, ..., Im; 0) = b(21, ..., In) exp{c(O'T(11, ..., In)}/a(0). In the case of the regular exponential family, the maximum likelihood (ML) estimate Ô of 0 satisfies the equation, E{T(X1, ..., X.)} T21, ..., In), (1) that is, = [E{T(X1,..., Xn)}le-ê T(21, ..., In). (i) In the case where c(0) = (a scalar), derive the result (1), using the result that the expectation of the score statsitic is zero for all values of the parameter. (ii) In the case where X1, ..., Xn with u and o’ both unknown, obtain the ML estimate of 0 = (1, 02) using (1) i.i.d. N (1,0%)

(iii) The result (1) does not necessarily hold if T(X1, ...,xn) were to be replaced by some other sufficient statistic W for 0. Show this by finding a sufficient statistic W for which (1) does not hold.