Question

1. Suppose that y E R is a parameter, and {X1, X2, ..., Xm} is a set of positive i.i.d. random variables with density function fx, given by fx.(ar)yey, You observe that X = {X1, X2, ..., Xm} in fact take the values r = {r1, x2, ..., x'm}, respec- tively. Write for the average of the values {x1, x2,.., Tm) a) What is the likelihood function, L(y; x), as a function of y? What is the log-likelihood function, log L(y; x), (again a function of y)? b) What is the score, i.e. log L(y#)] What is it about this example which makes the score a scalar, not a vector? c) Use First Order Conditions and your answers to Question (a), to express the MLE of y, denoted, terms of . (You may assume that Second Order Conditions are satisfied) d) Applying a general result from the lecture to this case, we get that (0 log L(; Xi) Vm )N (0,I-1) where I Cov for any im Use this result together with the Information Equality to find the limiting distribution as moo of the random variable Vm ) - e) Let Ho be the hypothesis that y asymptotic approximations for the mean and variance of when m = 3]. Assuming that Ho is true, use (d) to find 9000

Answer 1

Selm xI,X2Xmf Nariables uith density fincton given by s a o posctive iid random Ye2r : the sikeli hoed funthon the parameter Y orgiren somble values ,a, - *m s LiV,a) trei LIT,x) (mer Y takng og LIr;) e. Jeg Lr,x) Jogy-rxi m m differentiettny d wrt ,we get Score te. 14 i21 For giren ample values ,x2, - * )Xm; we the fxed vale for barameter Y get Seore scaler a

(C) lo get the miE aY, we eaunte [청니다짐 ㅇ farm, ursa] = (foroen i)) m m W A mLE a (d) Since, Informatiom I = ovaIoL (a) E from ti) m Σ 27 M Σ νατ (X) ) Gv are iid = I F then Vix)-

fom the piren esult, ( dN(0 ,I") Tm N(r)ds(o m2 J eunder Ho 3 we have also that giren m=9000 we knaw that, V is fallono asymptotic nmal uith mean E[m (- 0 vRo00 El-3) = 0 EIN-3) Et-3 ECA) Q Ver (m -)= and Var() 11 m. Var Bmce,

Xi NCamma (r, m) (By addilive froperty of CHamma distriburon YLSKSNGtama mr, m) m ia le (mrm ym-emry m E(4) (mmm (mmm gm-2emry put mry mw dy du fere El) m-2 du m m-I m E()

El 티(수) Naw y2 (mmM eMmy 00 Imwm M 1m-3 -(mw)2 m2 m = lmr/2 m (m-t m2 m-1 El2) m ml Now we have to find that NaT (m ) m Var () (#1 m Var m Var mIE(A)-EI4 -ELA m. m m2

m.v2m Ml mr2 m1l m-1 m 9000 and r3 9o00 m2 32 -9.001 ma Var () O.001 gooo-1 M- and E