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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, Jensen’s 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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- Fall '08
- Statistics, Basu, following integral equation, Mathematical Statistics HW, Statistics STAT 210A, positive constants x1