Question

Let X1, X2, ..., Xn be iid random variables with a "Rayleigh” density having the following pdf: 22 -12 10 f(x) = e x > 0 > 0 0 пе a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = V c) (7 points) What is the MLE of 02 + 0 – 10 ? d) (7 points) For a fact, 21–1 X has a Gamma(n, ) distribution. Using this information, find a consistent estimator for 0.

Answer 1

Note: : where g(T(2)O) Factorization Theorem : Joint distribution of(xy...XD): f(20) $(2,0) = g(13)0) h(x) is a function of X.,.. X, and o. n (3) is a function of X,... Xn Ther, T= T(x) = T(x,,...xn) is sufficient for o (x, ... Xn) is the sample of realization (*1, ... kn). n.

as Joint pdf of xy ... Yn : $(3,0) = f(x:) Fi Xy -; In ane izd] : 2 TX 1 Exi? colo on exie) 2 h(2) (Tt), By Factorization theorem, T(*)= I xil is sufficient for o by 1st order central moment : E(xi). To est order rawo moment Lexi 2 4 x ㅈ x ² 2 Txi "2=1 on Equating : SKONNE & Omme % Likelihood of a L() =f(4,0) log (-(0)) long (* 11 2:) - n logo - Iz? a log (((o)) 7,0 7 -2 2 + 1 loge (o)) is massimized at o-txi OMLE do x2 TIL 2 .2 O VIA & 2 n=1 Xi' 2 O MLE + OMLE -10 ECEX"). no, Van &**-no" By invariance property, MLE of 8² 20-10 = dy Note: Let To EXP (9) (3x): - Var(t) = Vas (1 x) = 1 h Van 20 7 T 7 T Pro ý Tis consistent for o. 2 no² →0 asnyoo
Below is mean variance of gamma

x?o BY TX xn Gamma (x,B) : pdf of x fx (2) = e 245 24-1 MGF of x: M26) = E(eta). See I z- where, Tx=(6-1)! Se=($-+) 24-1 ( 7 3 t) a dx a>0, -tro da Ba Ta Ta para (+a Ta =1 (-: kdf of Gamma (a, al (15 stja = (1-st) " where list to b) Mean = E(X)=Mx (4) [-((1-6+)*ac!] E(x2) ditz Mx (+)... = (-«(-0-1) (1-3)*4.2.ß?] = x(a+DB? .: Variance : Var(x) = E(X2) - E x) = (a +D 6-24P2 = ?. aß too tro d2 t=0