Hw10ans - STAT 410 Homework #10 Spring 2008 (due Friday,...

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Unformatted text preview: STAT 410 Homework #10 Spring 2008 (due Friday, April 4, by 3:00 p.m.) 1. Let > 0 be an unknown parameter and let X 1 , X 2 , , X n be independent random variables, each with the probability density function f ( x ) = ( ) & & < <-- otherwise 1 1 1 & & x x . a) Obtain the maximum likelihood estimator of , n & . Likelihood function: L ( ) = ( ) ( ) 1 1 1 1 & & X 1 X 1 & &- = =- -- = n i i n n i i . ln L ( ) = ( ) ( ) =-- + n i i n 1 X 1 1 ln & & ln . () ( ) ( ) =- + = n i d d i n 1 X 1 L ln & ln = 0. ( ) =-- = n i n i n 1 X 1 ln & . b) Find the CDF of X 1 . F X ( x ) = ( ) -- x dy y 1 & 1 & = ( ) 1 & x y-- = ( ) & 1 1 x-- , 0 < x < 1. F X ( x ) = 0, x < 0, F X ( x ) = 1, x > 1. c) Let W i = ln ( 1 X i ), i = 1, 2, , n . Find the CDF and the PDF of W 1 . F W ( w ) = P ( W w ) = P ( ln ( 1 X ) w ) = P ( X 1 e w ) = 1 e w , w > 0. f W ( w ) = e w , w > 0. & W 1 has an Exponential distribution with mean & 1 . d) Find E ( W 1 ) and Var ( W 1 ). W 1 has an Exponential distribution with mean & 1 . & E ( W 1 ) = & 1 , Var ( W 1 ) = 2 & 1 . e) Use WLLN ( Theorem 4.2.1 ) and Theorem 4.2.4 to show that n & is a consistent estimator for ( as n ). By WLLN, ( ) & 1 W E W = P . Since g ( x ) = x 1 is continuous at & 1 , W 1 & = n = g ( W ) & 1 g P = . & n & is a consistent estimator for . f) Use CLT ( Theorem 4.4.1 ) and Theorem 4.3.9 to show that n & is asymptotically normally distributed ( as n ). Find the parameters. By CLT, - 1 , 1 W 2 & & N n D . Since g ( x ) = x 1 is differentiable and & 1 ' g = 2 0, ( ) - & 1 W g g n = - & & n n D ( ) - 1 , 2 2 2 & & N = ( ) 2 & , N . & , ~ 2 & & & n N n . 2. Let > 0 be an unknown parameter and let X 1 , X 2 , , X n be independent random variables, each with the probability density function f ( x ) = ( ) & & < <-- otherwise 1 1 1 & & x x ....
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This homework help was uploaded on 04/13/2008 for the course STAT 410 taught by Professor Alexeistepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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Hw10ans - STAT 410 Homework #10 Spring 2008 (due Friday,...

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