Question

2. Let X1, X2, ...,Xbe i.i.d. Poisson with parameter .. (a) Find the maximum likelihood estimator of . Is the estimator minimum variance unbi- ased? (b) Derive the asymptotic (large-sample) distribution of the maximum likelihood estimator. (c) Suppose we are interested in the probability of a zero: Q = P(Xi = 0) = exp(-). Find the maximum likelihood estimator of O and its asymptotic distribution.

Answer 1

a) The pmf of X is f(x) ,x = 0,1,2,. Let X, X, X be a random sample of sizen is taken from above poisson distributi on 2 The lkelihood function is L= П/, 3) i-1 =/ (, A)(, 2).(, 2) -A e -A П) i-1 Σκ. -In ) In L 1n e1n A- -NA i-1 =-nAlne x In A- In (T(x)) _ i-1 d ln L -02-0-0 1 x = 0-n d A Therefore, MLe of Ais A Var A Var (x Var Var )= = 2 CRLB Therefore, x is sati sfying the CRLB condition Hence xis MVUBEof A 1

b) The likelihood function is L T,A) i-1 =f (x, A)(A)(%.A) ,-а джа e eMA +In AIn (( 1)) x; In A-In (T(x) = -nA ln e i-1 =-nA +S, n A- In (7()) where s Σx , dIn L 1 1 = - +S d A so equating this to zero then A s. /n x Using the Central Limit theorem (-n2) as E (X) Var (X)= A N(0,A) AN(A, AIn) c) By invariance property of MLE is MLE of then for any functionf (0),the MLE off(0) isf [8 if so MLE of A is A = x, the functi on of A is e, the MLE of e"^ is e'