# 08_28_09ans - STAT 409 5 Let X 1 X 2 X n be a random sample...

This preview shows pages 1–4. Sign up to view the full content.

STAT 409 Examples for 08/28/2009 Fall 2009 5. Let X 1 , X 2 , … , X n be a random sample of size n from a uniform distribution on the interval ( 0 , θ ) . f ( x ; θ ) = < < otherwise 0 0 1 θ θ x E ( X ) = 2 θ Var ( X ) = 12 2 θ F ( x ; θ ) = < < < θ θ θ 1 0 0 0 x x x x a) Obtain the method of moments estimator of θ , θ ~ . ( 29 2 θ X E = . 2 θ ~ X = . X 2 θ ~ = . b) Is θ ~ unbiased for θ ? That is, does E( θ ~ ) equal θ ? ( 29 ( 29 2 θ X E X E = = . ( 29 ( 29 θ X 2 E θ ~ E = = . c θ ~ is unbiased for θ . c) Compute Var( θ ~ ). X 2 θ ~ = . ( 29 ( 29 ( 29 n 2 σ 4 X Var 4 X 2 Var θ ~ Var = = = . For Uniform ( 0 , θ ), 12 θ 2 2 σ = . ( 29 n = 3 θ θ ~ Var 2 .

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

View Full Document
d) Obtain the maximum likelihood estimator of θ , θ ˆ . Likelihood function: ( 29 n n i θ 1 θ 1 θ L 1 = = = , θ > max X i , ( 29 0 θ L = , θ < max X i . Therefore, θ ˆ = max X i . e) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? F max X i ( x ) = P ( max X i x ) = P ( X 1 x , X 2 x , … , X n x ) = P ( X 1 x ) P ( X 2 x ) P ( X n x ) = n x θ , 0 < x < θ . f max X i ( x ) = F ' max X i ( x ) = n n x n θ 1 - , 0 < x < θ . ( 29 θ 1 θ 0 θ 1 θ θ θ θ ˆ E 1 θ 0 θ 0 1 + = + = = = + - n n n x n dx x n dx x n x n n n n n n . θ ˆ is NOT unbiased for θ .
f) What must c equal if c θ ˆ is to be an unbiased estimator for θ ? ( 29

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.

### Page1 / 8

08_28_09ans - STAT 409 5 Let X 1 X 2 X n be a random sample...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online