{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

10_07 - STAT 410 Examples for Fall 2011 Def An estimator θ...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

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

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

Unformatted text preview: STAT 410 Examples for 10/07/2011 Fall 2011 Def An estimator θ ˆ is said to be unbiased for θ if E( θ ˆ ) = θ for all θ . 3. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; θ ) = - ≤ ≤ ⋅ otherwise 1 θ 1 θ θ 1 x x 0 < θ < ∞ . Recall: The method of moments estimator of θ is θ ~ = 1 X 1 X X 1- =- , the maximum likelihood estimator of θ is θ ˆ = ∑ = ⋅- n i i n 1 X ln 1 . d) Is θ ˆ unbiased for θ ? That is, does E( θ ˆ ) equal θ ? ( ) ( ) ∫ ∫ = =- ∞ ∞- ⋅ ⋅ ⋅ 1 θ θ 1 X 1 θ 1 ln θ ln X ln E ; dx x x dx x f x . Integration by parts: ∫ ∫- = b a b a du v a b v u dv u Choice of u : L ogarithmic A lgebraic T rigonometric E xponential u = ln x , dv = dx x dx x 1 θ 1 θ θ 1 θ 1 θ 1-- ⋅ ⋅ = , du = dx x 1 , v = θ 1 x . ( ) ∫ ∫ - = = ⋅ ⋅ ⋅ ⋅- 1 θ 1 θ 1 1 θ θ 1 1 1 1 ln θ 1 ln X ln E dx x x x x dx x x = θ 1 θ 1 1 1 θ 1 1 1 θ 1 1 θ 1- = - =- = - ⋅ ⋅ ∫ ∫- x dx x dx x x . Therefore, ( ) ( ) ( ) ∑ ∑ = =-- =- = ⋅ ⋅ n i n i i n n 1 1 θ 1 X ln E 1 θ ˆ E = θ , that is, θ ˆ is an unbiased estimator for θ . OR F X ( x ) = x 1 / θ , 0 < x < 1....
View Full Document

{[ snackBarMessage ]}

Page1 / 7

10_07 - STAT 410 Examples for Fall 2011 Def An estimator θ...

This preview shows document pages 1 - 3. Sign up to view the full document.

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