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09_23_11_2 - 1 If the random variable Y denotes an...

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1. If the random variable Y denotes an individual’s income, Pareto’s law claims that P ( Y y ) = θ y k , where k is the entire population’s minimum income. It follows that f Y ( y ) = 1 θ θ 1 θ + y k , y k ; θ > 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 θ ~ of θ , is k - = Y Y θ ~ . Show that θ ~ is asymptotically normally distributed ( as n ) . Find the parameters. ( Assume θ > 2. ) μ = ( ) 1 θ θ θ 1 θ Y E θ θ 1 θ θ - = = = - + k dy y k dy y k y k k . ( ) 2 θ θ θ 1 θ Y E 2 1 θ θ 1 θ θ 2 2 - = = = + - + k dy y k dy y k y k k . σ 2 = Var ( Y ) = 2 2 1 θ θ 2 θ θ - - - k k = ( )( ) 2 2 1 θ 2 θ θ - - k . Consider g ( x ) = k x x - . Then g ( Y ) = θ ~ Y Y = - k , g ( μ ) = k k k - - - 1 θ θ 1 θ θ = θ . g ' ( x ) = ( ) 2 k x k - - . g ' ( μ ) = 2 1 θ θ - - - k k k = ( ) k 2 1 θ - - .
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