Question

2. Asymptotic Maximum Likelihood. 25 Let X1, ..., Xn be independently Poisson distributed with parameter 1, i.e. fx, (x) = exto is X= 0, 1, 2, ... =0,1,2,... (a) Derive the maximum likelihood estimator în of 1 based on n measurements. 5 (b) Show that în is consistent. 5 (c) Is în (asymptotically) efficient? 5 (d) Derive the asymptotic distribution of vn(în – 1). 10

Answer 1

X, , X ,... Xn ind pla) de( wenye xoves x (A) Likelihood function is given by L(A- A=1 logh = am) + Mi legy a na legge: 1) Blog -an 3 - 0 :. MLE of t is Ť

cho E(*) - ħ ŽE(Xi) V(x) = v( h 32 x) ñ vix:) and -Yu so as nog : ŷ = X is a consisdent estimador (C) An the estimadon is said class of unbiased to be efficient if in estimadons it has minimum variance. X is an unbiased estimador don > * is has the smaller variance. & : Ť is an efficient.

(d) The Cendral limit theorem says that provided Var(x) enists and is less than infinity, the sample mean is approximadely normally dischibuded with the connect mean an a variance that goes down as Yn :) XNn(= (x) VM (x)) - N(0,1) in (F-E(x)) V Vai(x) Here E(x) = 2 Val(x) = 7 vn (F-1) ôn= te (0,7)