STAT 400 Lecture AL1 Examples for 6.4, Part 1 Spring 2015 Dalpiaz p.m.f. or p.d.f. f ( x ; ), . – parameter space. 1. Suppose = { 1, 2, 3 } and the p.d.f. f ( x ; ) is = 1: f ( 1 ; 1 ) = 0.6, f ( 2 ; 1 ) = 0.1, f ( 3 ; 1 ) = 0.1, f ( 4 ; 1 ) = 0.2. = 2: f ( 1 ; 2 ) = 0.2, f ( 2 ; 2 ) = 0.3, f ( 3 ; 2 ) = 0.3, f ( 4 ; 2 ) = 0.2. = 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 ( based on only one observation of X ) if … a) X = 1; b) X = 2; c) X = 3; d) X = 4. Likelihood function: L ( ) = L ( ; x 1 , x 2 , … , x n ) = ni 1f ( x i ; ) = f ( x 1 ; ) … f ( x n ; ) It is often easier to consider ln L ( ) = ni 1ln f ( x i ; ). Maximum Likelihood Estimator: θˆ = arg max L ( ) = arg max ln L ( ). Method of Moments: E ( X ) = g ( ). Set X = g ( θ~ ). Solve for θ~.2. Let X 1 , X 2 , … , X n be a random sample of size n from a Poisson distribution with mean , > 0. That is, P ( X = k ) = !λ λ kek , k = 0, 1, 2, 3, … . a) Obtain the method of moments estimator of , λ~. b) Obtain the maximum likelihood estimator of , ˆ. 3. Let X 1 , X 2 , … , X n be a random sample of size n from a Geometric distribution with probability of “success” p, 0 < p < 1. That is, P ( X = k ) = ( 1 – p ) k – 1 p, k = 1, 2, 3, … . a) Obtain the method of moments estimator of p, p~. b) Obtain the maximum likelihood estimator of p, pˆ.4. Let X 1 , X 2 , … , X n be a random sample of size n from the distribution with probability density function f ( x ; ) = otherwise010θ1 θ θ1xx 0 < < . a) Obtain the method of moments estimator of , θ~. Method of Moments: E ( X ) = g ( ). Set X = g ( θ~ ). Solve for θ~. b) Obtain the maximum likelihood estimator of , θˆ. Likelihood function: L ( ) = L ( ; x 1 , x 2 , … , x n ) = ni 1f ( x i ; ) = f ( x 1 ; ) … f ( x n ; ) Maximum Likelihood Estimator: θˆ = arg max L ( ) = arg max ln L ( ).4. (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 . 5. Let X 1 , X 2 , … , X n be a random sample of size n from N ( 1 , 2 ), where = { ( 1 , 2 ) : – < 1 < , 0 < 2 < }. That is, here we let 1 = and 2 = 2 . a) Obtain the maximum likelihood estimator of 1 , 1 θˆ, and of 2 , 2 θˆ. b) Obtain the method of moments estimator of 1 , 1 θ~, and of 2 , 2
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