Question

6. Consider a sample X1, ... ,X, U(0,0) and answer following questions. (a) Suggest a reasonable estimator of 8 and explain. (b) Define Y = max(X1,..., X.). Note that Y is also a random variable since it is a function of random variables. Derive a probability of density function of Y. fy(). (Hint: P(Y < y) = P(X1 Sy, X, Sy...,X, Sy); X1,..., X, are independent; Relationship between cdf and pdf for a continuous variable) (c) Derive a closed-form of E[Y] and Var(Y). (Caution: Make sure of the possible range of Y.) (d) Is max(X1,...,xn) an unbiased estimator of e? If not, suggest an unbiased estimator of 8 and prove it's unbiasedness. (e) Calculate the standard error of the estimator you suggested in (d).
hi i am struggling with this problem, thanks

Answer 1

ABW) X1, X2, ---, Xn Lady uro, o) 8 (4): 3: 4 likelihood function of sample values, La frui) = (0)" We arange sample observations in ascendung Older of magnitude as, XOC X(2) <X(3) <--- < X(n) then L= ; OS X(1)</02)2X63)<-- <X() s@ From the above result, L is maximum, when ois minimum but os, (n) Omin = X(n) = Max. of (X1, X2, -- , Xn) Thus M.L.E. of o= 0= X(n) = max. of (X, Xap mo Xn) - Answer

(b) Y= max(* 182, - - ,Xn) codifi of Y = Fyly) = Pe Cysy) = Pe[max. of (X1 X2, -- , Xn) sy] = Pe[ x5 y, x = y, ----,Xnsy] = P(X, sy). P(XSY)---- P(Xnsy) C-X1, X2, -- , Xn are si dependent] = fxily). Fxzly). ----FXn (y) Fyly) = (*)" [*' F(*)= ] po.dofe of y = f/4)= dy Fyly) ulogue -on.n (4) 10 tvly)= molyo; osyse VU 11 on = Answer

20(49)-(130) Saha ep met onze - Recet; 4. ) =613 comment of the) - (R)3 I- TAMI Itu 4 - Itu u hp-taff ch! =(hja 05h50 methu = (hit.. (2) @

Var (4) = E(yº) – (E(y)]2 - 69.) 02 (67) 0] Var (Y) = 02 [(#2) - (n. 1)] = Answer (d) y = max (X1, X2, -- , Xn) smice Ely) = (140) Thus Ely) to Hence y is not an unbiased estimator of o. also. Sawice Ely)= (147) - Answer (#!). Ely)=0 E[(*) * )=0 Thus (n+1).y is an unbiased estimator of 8 Answer Answer

is an unbiased estimator thus ô = (mt) y Var (O) = var [(ht!) y] = (hu? you (4) var () = (*#1 )? [CM+2) - 67.)? 3.02 standard Erroc of the estimator, S, E. = /varlo) = 0 (0:1) 672) - A) na - Answer