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# hw2 - ECN 725 ASSIGNMENT 2 Due March 4(Thursday Dr AHN 1(10...

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1 ECN 725 ASSIGNMENT 2 Dr. AHN Due March 4 (Thursday) 1. (10 pts.) Suppose that 1 2 0 1 1 2 ( | , ) E y x x x x where 1 x and 2 x are stochastically independent and 1 2 ( ) ( ) 0 E x E x . Show that 1 2 ( |1, , ) Proj y x x depend on neither of 1 x nor 2 x . 2. (25 pts.; 5 pts. on each.) There are certain conditions under which maximum likelihood (ML) estimators are efficient and asymptotically normal (if a large sample is used). The detailed conditions are often called “regularity conditions”. If some of the conditions are violated, maximum likelihood estimator may not be asymptotically normal. One example is the case of uniformly distributed population with the probability density, 1 ( ) 1(0 ) o o f x x , where 1( ) is an index function which equals one if the argument in the parenthesis is correct and equals zero if not. Let 1 { ,..., } T x x be a random sample. 1) Find the ML estimator. 2) Show that the ML estimator is biased. [Hint: You need to find out the pdf of the ML estimator. You need to study the distributions of “order statistics”. Google them!] 3) Propose an unbiased estimator.

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