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Unformatted text preview: STAT210A HW07 Due: Tuesday, October 20, 2009 7.1. In the inverse binomial sampling procedure, N is a random variable representing the number of trials required to observe x successes in a total of N + x Bernoulli trials (with parameter ). (a) Show that the best (minimum variance) unbiased estimator of is given by * ( N ) = ( x 1) / ( N + x 1). (b) Show that the information contained in N about is I ( ) = ( x 2 x ) / [ 2 (1 )]. (c) Show that var ( * ) > 1 /I ( ). 7.2. Consider a scale family 1 f ( x/ ) , > 0 where f is some fixed density function. (a) Show that the amount of information that a single observation X contains about is given by 1 2 Z yf ( y ) f ( y ) + 1 2 f ( y ) dy. (b) Show that the information X contains about = log is independent of . 7.3. Given a family { p ( x ; )  } and an estimator ( ) with g ( ) = E [ ( X )], the information bound is B ( ) = [ g ( )] 2 /I ( ). Now suppose that)....
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This note was uploaded on 10/17/2009 for the course STAT 210a taught by Professor Staff during the Fall '08 term at University of California, Berkeley.
 Fall '08
 Staff
 Statistics, Bernoulli, Binomial

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