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Unformatted text preview: 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = & & ¡ & & ¢ £ ≤ ≤- ⋅ otherwise 1 1 & & 1 & x x 0 < θ < ∞ . Let Y 1 < Y 2 < … < Y n denote the corresponding order statistics. a) Let V n = n θ Y 1 . Find the limiting distribution of V n . b) Let ( ) Y 1 W n n n- = . Find the limiting distribution of W n . 2. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) & 2 X X ln 1 & & ; x x x f x f ⋅- = = , x > 1, θ > 2. We already know (Homework 8) that the maximum likelihood estimator & ˆ of θ is ¤ = + = n i i x n 1 ln 2 1 & ˆ and the method of moments estimator & ~ of θ is 1 1 2 & ~-- = x x . a) Is & ˆ a consistent estimator for θ ? b) Is & ~ a consistent estimator for θ ? c) Show that & ˆ is asymptotically normally distributed ( as n → ∞ ). Find the parameters....
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