home7 - Stat 210B Homework Assignment 7 (due May 8) 1....

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Unformatted text preview: Stat 210B Homework Assignment 7 (due May 8) 1. Consider the following two statistical functionals: (a) φ ( F ) = E F X and (b) φ ( F ) = F − 1 ( p ) = inf { x : F ( x ) ≥ p } , where the domain of φ ( · ) is { F | F differentiable at φ ( F ) with F ′ ( φ ( F )) > } . Show that one of these functionals is continuous (in the sup norm) and the other is not. 2. Given a distribution function F we define the quantile function as F − 1 ( p ) = inf { x : F ( x ) ≥ p } . Define F t,x = (1- t ) F + tδ x . Express F − 1 t,x in terms of F − 1 . 3. Find the influence function of the map F → integraltext t (1- F − ) − 1 dF (the cumulative hazard function). 4. Consider an artificial data set consisting of the 8 numbers 1 , 2 , 3 . 5 , 4 , 7 , 7 . 3 , 8 . 6 , 12 . 4 , 13 . 8 , 18 . 1. Let ˆ θ be the 25% trimmed mean. (a) Calculate bootstrap estimates of the standard error of θ , using bootstrap samples of size B = 25 , 100 , 200 , 500 , 1000 , 2000. Use these to form a numerical estimate of the ideal2000....
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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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home7 - Stat 210B Homework Assignment 7 (due May 8) 1....

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