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# 10_25 - STAT 400 1 Let X 1 X 2 X n be a random sample of...

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STAT 400 Examples for 10/25/2011 Fall 2011 1. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function ( ) ( ) ( ) θ 2 X X ln 1 θ θ ; x x x f x f - = = , x > 1, θ > 1. a) Find the maximum likelihood estimator θ ˆ of θ . L( θ ) = ( ) = - n i i i 1 θ 2 X X ln 1 θ . ln L( θ ) = ( ) = = - + - n i i n i i n 1 1 X ln θ X ln ln 1 θ 2 ln . ( ) = - - = n i i d d n 1 X ln 1 θ 2 θ θ L = 0. = + = n i i n 1 X ln 2 1 θ ˆ . b) Suppose θ > 2. Find the method of moments estimator θ ~ of θ . E(X) = ( ) ( ) ( ) ( ) 2 2 1 θ 2 X 2 θ 1 θ ln 1 θ - - = - = - dx x x x dx x f x . ( ) ( ) 2 2 1 2 θ ~ 1 θ ~ X X 1 - - = = = n i i n . 1 X 1 X 2 θ ~ - - = .

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2. 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 ~ . ( ) 2 θ X E = . 2 θ ~ X = . X 2 θ ~ = . b) Is θ ~ unbiased for θ ? That is, does E( θ ~ ) equal θ ? ( ) ( ) 2 θ X E X E = = . ( ) ( ) θ X 2 E θ ~ E = = . checkbld θ ~ is unbiased for θ . c) Find Var( θ ~ ). X 2 θ ~ = . ( ) ( ) ( ) n 2 σ 4 X Var 4 X 2 Var θ ~ Var = = = . For Uniform ( 0 , θ ) , 12 θ 2 2 σ = . ( ) n = 3 θ θ ~ Var 2 .
d) Obtain the maximum likelihood estimator of θ , θ ˆ . Likelihood function: ( ) n n i θ 1 θ 1 θ L 1 = = = , θ > max X i , ( ) 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 ?

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10_25 - STAT 400 1 Let X 1 X 2 X n be a random sample of...

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