home3 - h 2 p 1 p 2 ≤ 1 2 k p 1-p 2 k 1 ≤ h p 1 p 2[2-h...

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Stat 210B Homework Assignment 3 (due February 27) 1. Consider the function class F = { f = d k =1 θ k ψ k } , where ψ k L 2 ( Q ) and where we impose the constraint ( R f 2 dQ ) 1 / 2 R . Show H ( ², Q, F ) d log ± 4 R + ² ² ² . 2. Suppose that a class F ⊂ L 1 ( P ) satis±es the ULLN. Then also conv( F ) satis±es the ULLN, where conv( F ) is the convex hull of F (the set of arbitrary ±nite convex combinations of functions from F ). 3. Given densities p 1 and p 2 with respect to some σ -±nite measure μ , de±ne that the Hellinger distance as follows: h ( p 1 , p 2 ) = ± 1 2 Z ( p 1 - p 2 ) 2 ² 1 / 2 and de±ne the variation distance as follows: k p 1 - p 2 k 1 = Z | p 1 - p 2 | dμ. Establish the following two inequalities:
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Unformatted text preview: h 2 ( p 1 , p 2 ) ≤ 1 2 k p 1-p 2 k 1 ≤ h ( p 1 , p 2 )[2-h 2 ( p 1 , p 2 )] 1 / 2 . 4. Consider two probability measures P 1 and P 2 on a measurable space ( X , A ), dominated by a σ-±nite measure μ and let p 1 and p 2 be the densities of P 1 and P 2 with respect to μ , respectively. Show that the Hellinger distance h ( p 1 , p 2 ) does not depend on the dominating measure μ . 5. Given densities p and p , show h 2 ( p, p ) ≤ 16 h 2 (¯ p, p ) , where ¯ p = ( p + p ) / 2....
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This note was uploaded on 01/25/2011 for the course STAT 201b taught by Professor Michaeljordan during the Fall '05 term at Berkeley.

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