10_20 - STAT 400 p.m.f or p.d.f 1 Fall 2011 Examples for f...

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STAT 400 Examples for 10/20/2011 Fall 2011 p.m.f. or p.d.f. f ( x ; θ ) , θ Ω . Ω – parameter space. 1. Suppose Ω = { 1, 2, 3 } and the p.d.f. f ( x ; θ ) is θ = 1: f ( 1 ; 1 ) = 0.6, f ( 2 ; 1 ) = 0.1, f ( 3 ; 1 ) = 0.1, f ( 4 ; 1 ) = 0.2. θ = 2: f ( 1 ; 2 ) = 0.2, f ( 2 ; 2 ) = 0.3, f ( 3 ; 2 ) = 0.3, f ( 4 ; 2 ) = 0.2. θ = 3: f ( 1 ; 3 ) = 0.3, f ( 2 ; 3 ) = 0.4, f ( 3 ; 3 ) = 0.2, f ( 4 ; 3 ) = 0.1. What is the maximum likelihood estimate of θ ( based on only one observation of X ) if … a) X = 1; b) X = 2; c) X = 3; d) X = 4. Likelihood function: L ( θ ) = L ( θ ; x 1 , x 2 , … , x n ) = = n i 1 f ( x i ; θ ) = f ( x 1 ; θ ) f ( x n ; θ ) It is often easier to consider ln L ( θ ) = = n i 1 ln f ( x i ; θ ) .
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2. 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 0 1 0 θ 1 θ θ 1 x x 0 < θ < . a) Obtain the method of moments estimator of θ , θ ~ . b) Obtain the maximum likelihood estimator of θ , θ ˆ .
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2. (continued) c) Suppose n = 3, and x 1 = 0.2, x 2 = 0.3, x 3 = 0.5. Compute the values of the method of moments estimate and the maximum likelihood estimate for
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