20props - EECS 501 ESTIMATOR PROPERTIES Fall 2001 Problem...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
EECS 501 ESTIMATOR PROPERTIES Fall 2001 Problem: Let { x 1 ...x N } be iidrv with x i N ( m,σ 2 ) and m, σ 2 unknown . Want: To compute ˆ m MLE and ˆ σ 2 MLE based on observations { X 1 ...X N } . Solution: f x 1 ...x N ( X 1 ...X N ) = Q N i =1 1 2 πσ 2 e - 1 2 ( X i - m ) 2 2 since x i indpt rvs. Set: 0 = ∂m log f x 1 ...x N = ∂m [ - N 2 log(2 π ) - N 2 log σ 2 - 1 2 N i =1 ( X i - m ) 2 2 ] = 1 σ 2 N i =1 ( X i - m ) = 0 ˆ m MLE = 1 N N i =1 X i = samplemean . Set: 0 = ∂σ 2 log f x 1 ...x N = ∂σ 2 [ - N 2 log(2 π ) - N 2 log σ 2 - 1 2 N i =1 ( X i - m ) 2 2 ] = - N 2 1 σ 2 + 1 2 N i =1 ( X i - m ) 2 / ( σ 2 ) 2 = 0 ˆ σ 2 MLE = 1 N N i =1 ( X i - m ) 2 . Replace m in ˆ σ 2 MLE with ˆ m MLE ˆ σ 2 MLE = samplevariance . Note: ˆ σ MLE = p ˆ σ 2 MLE : MLE commutes with nonlinear functions g ( a ). Why?
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.

Ask a homework question - tutors are online