lec6 - Lecture 6 Let us compute Fisher information for some...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Let us compute Fisher information for some particular distributions. Example 1. The family of Bernoulli distributions B ( p ) has p.f. f ( x | p ) = p x (1 - p ) 1 - x and taking the logarithm log f ( x | p ) = x log p + (1 - x ) log(1 - p ) . The second derivative with respect to parameter p is ∂p log f ( x | p ) = x p - 1 - x 1 - p , 2 ∂p 2 log f ( x | p ) = - x p 2 - 1 - x (1 - p ) 2 then we showed that Fisher information can be computed as: I ( p ) = - 2 ∂p 2 log f ( X | p ) = X p 2 + 1 - X (1 - p ) 2 = p p 2 + 1 - p (1 - p ) 2 = 1 p (1 - p ) . The MLE of p is ˆ p = ¯ X and the asymptotic normality result from last lecture becomes n p - p 0 ) N (0 , p 0 (1 - p 0 )) which, of course, also follows directly from the CLT. Example. The family of exponential distributions E ( α ) has p.d.f. f ( x | α ) = ± αe - αx , x 0 0 , x < 0 and, therefore, log f ( x | α ) = log α - αx 2 ∂α 2 log f ( x | α ) = - 1 α 2 . 24
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.

Page1 / 4

lec6 - Lecture 6 Let us compute Fisher information for some...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online