stat210a_fall07_hw4

# stat210a_fall07_hw4 - UC Berkeley Department of Statistics...

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UC Berkeley Department of Statistics STAT 210A: Introduction to Mathematical Statistics Problem Set 4 Fall 2007 Issued: Thursday, September 20 Due: Thursday, September 27 Problem 4.1 A random variable X has the Pareto distribution P ( c,k ) if its CDF takes the form 1 - ( k/x ) c for x > k > 0 and c > 0. (a) Show that the distributions P ( c, 1) are a 1-dimensional exponentialfamily with suﬃcient statistic T = log X . (b) Show that T has an exponential distribution with parameters log( k ) and 1 /c . Problem 4.2 Consider a scale family 1 θ f ( x/θ ) , θ > 0 where f is some ﬁxed density function. (a) Show that the amount of information that a single observation X contains about θ is given by 1 θ 2 Z • yf 0 ( y ) f ( y ) + 1 2 f ( y ) dy. (b) Show that the information X contains about ξ = log θ is independent of θ . (c) For the Cauchy distribution C (0 ), show that I ( θ ) = 1 / (2 θ 2 ). Problem 4.3

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stat210a_fall07_hw4 - UC Berkeley Department of Statistics...

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