HW03 - 1 1 1 2 2 ( ,..., ) n X T X T U T T--= . Prove that...

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UC Berkeley, Department of Statistics Fall, 2009 STAT 210A: Introduction to Mathematical Statistics HW#3 Due: Tuesday, September 22, 2009 Sufficiency, completeness, Basu’s theorem 3.1. Use completeness for the family ( ,1), N θ R to find the unique solution f to the following integral equation: 2 ( )exp( ) 2 exp( ), for all 2 f x x dx π +∞ -∞ = R . 3.2. Let 1 ,.., n X X be i.i.d. from the uniform distribution on (0, 1), and let 1 max{ ,.., } n M X X = . Show that 1 / and X M M are independent. 3.3. Let 1 ,.., n X X be i.i.d. from the uniform distribution on ( , ) θ θ - , where 0 > is an unknown parameter. (a) Find a minimal sufficient statistic T . (b) Define ( ) (1) , n X V X X = - where X is the sample average. Show that and T V are independent. 3.4. Consider the i.i.d. sampling model 2 ~ ( , ) i X N μ σ , parameterized by 2 ( , ) = . Consider the statistics 1 1 1 ( ) n i i T X X n = = and 2 2 1 1 ( ) ( ) /( 1) n i i T X X T n = = - - . Consider the statistic
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Unformatted text preview: 1 1 1 2 2 ( ,..., ) n X T X T U T T--= . Prove that (a) U is ancillary, and (b) ( for extra points ) U is distributed uniformly over a sphere of radius 1 in ( n- 1) dimensional space. Convexity, Jensens inequality 3.5. The geometric mean of a list of positive constants 1 ,..., n x x is t 1/ 1 2 ( ) , n n x x x x = m and the arithmetic mean is the average 1 ( ) / n x x x n = + + m . Show that t x x . 3.6. Show that if f is defined and bounded on ( , )- , then f cannot be convex. 3.7. Let f and g be positive probability densities on R . Show that ( ) log( ) ( ) 0, ( ) f x f x dx g x > unless f g = almost everywhere. This integral is called the Kullback-Leibler (KL) information....
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HW03 - 1 1 1 2 2 ( ,..., ) n X T X T U T T--= . Prove that...

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