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Unformatted text preview: 1. If the random variable Y denotes an individuals income, Paretos law claims that P ( Y y ) = y k , where k is the entire populations minimum income. It follows that f Y ( y ) = 1 1 + y k , y k ; > 0. Assume k is known. Let Y 1 , Y 2 , , Y n be a random sample of size n . a) Recall that the method of moments estimator ~ of , is k- = Y Y ~ . Show that ~ is asymptotically normally distributed ( as n ). Find the parameters. ( Assume > 2. ) = ( ) 1 1 Y E 1 - = = = - + k dy y k dy y k y k k . ( ) 2 1 Y E 2 1 1 2 2- = = = +- + k dy y k dy y k y k k . 2 = Var ( Y ) = 2 2 1 2 --- k k = ( )( ) 2 2 1 2 -- k . Consider g ( x ) = k x x- . Then g ( Y ) = ~ Y Y =- k , g ( ) = k k k--- 1 1 = ....
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This note was uploaded on 10/13/2011 for the course STAT 409 taught by Professor Stephanov during the Fall '11 term at University of Illinois at Urbana–Champaign.
- Fall '11