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Unformatted text preview: STAT 410 Homework #10 Spring 2008 (due Friday, April 4, by 3:00 p.m.) 1. Let &gt; 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 ) = ( ) &amp; &amp; &lt; &lt; otherwise 1 1 1 &amp; &amp; x x . a) Obtain the maximum likelihood estimator of , n &amp; . Likelihood function: L ( ) = ( ) ( ) 1 1 1 1 &amp; &amp; X 1 X 1 &amp; &amp; = =  = n i i n n i i . ln L ( ) = ( ) ( ) = + n i i n 1 X 1 1 ln &amp; &amp; ln . () ( ) ( ) = + = n i d d i n 1 X 1 L ln &amp; ln = 0. ( ) = = n i n i n 1 X 1 ln &amp; . b) Find the CDF of X 1 . F X ( x ) = ( )  x dy y 1 &amp; 1 &amp; = ( ) 1 &amp; x y = ( ) &amp; 1 1 x , 0 &lt; x &lt; 1. F X ( x ) = 0, x &lt; 0, F X ( x ) = 1, x &gt; 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 &gt; 0. f W ( w ) = e w , w &gt; 0. &amp; W 1 has an Exponential distribution with mean &amp; 1 . d) Find E ( W 1 ) and Var ( W 1 ). W 1 has an Exponential distribution with mean &amp; 1 . &amp; E ( W 1 ) = &amp; 1 , Var ( W 1 ) = 2 &amp; 1 . e) Use WLLN ( Theorem 4.2.1 ) and Theorem 4.2.4 to show that n &amp; is a consistent estimator for ( as n ). By WLLN, ( ) &amp; 1 W E W = P . Since g ( x ) = x 1 is continuous at &amp; 1 , W 1 &amp; = n = g ( W ) &amp; 1 g P = . &amp; n &amp; is a consistent estimator for . f) Use CLT ( Theorem 4.4.1 ) and Theorem 4.3.9 to show that n &amp; is asymptotically normally distributed ( as n ). Find the parameters. By CLT,  1 , 1 W 2 &amp; &amp; N n D . Since g ( x ) = x 1 is differentiable and &amp; 1 ' g = 2 0, ( )  &amp; 1 W g g n =  &amp; &amp; n n D ( )  1 , 2 2 2 &amp; &amp; N = ( ) 2 &amp; , N . &amp; , ~ 2 &amp; &amp; &amp; n N n . 2. Let &gt; 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 ) = ( ) &amp; &amp; &lt; &lt; otherwise 1 1 1 &amp; &amp; x x ....
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 Spring '08
 AlexeiStepanov
 Normal Distribution, Probability, Probability theory, probability density function, dy

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