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Unformatted text preview: STAT 409 Examples for 09/02/2009 Fall 2009 Def An estimator for is said to be consistent if P , i.e., for all > 0, P - as n . The ( Weak ) Law of Large Numbers: Let X 1 , X 2 , be i.i.d. with mean and standard deviation . Let n n n X ... X X 1 + + = , n = 1, 2, . Then X P n . That is, for all > 0, ( 29 X P lim = - n n . 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 ; 1. Assume k is known. a) Recall that the method of moments estimator ~ of , is k- = Y Y ~ . Show that ~ is a consistent estimator of . b) Recall that the maximum likelihood estimator of , is k n n n i i ln 1 Y ln - = = ....
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