Section7 - Likelihood function i n n i x x x f L max 1 1 ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = ∏ = i x L max< = Plot the graph of L θ against θ

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CEE 304 – Section Notes (10/18/06) Example problems comparing MLE’s and MM: 1. Given the probability mass function of a discrete random variable X: x P(X = x) 0 0.9(1-p) 1 0.9p x 0 0.1 where x 0 is a fixed, known number. Suppose we have a sample {x 1 , … , x N } consisting of: n 1 zeros n 2 ones n 3 x 0 ’s Such that the total sample size N = n 1 + n 2 + n 3 . Find p using both method of moments and maximum likelihood estimators. Answer: Method of Moments Estimator of p E[X] = 0[0.9(1-p)] + 1[0.9p] + x 0 (0.1) = 0.9p + 0.1 x 0 Let x = 0.9 + 0.1 x p ˆ 0 Yields: 9 . 0 1 . 0 ˆ 0 x x p = Maximum Likelihood Estimator Solve max () [] [ ] [ ] { } 3 2 1 1 . 0 9 . 0 1 9 . 0 n n n p p ==> 2 1 2 ˆ n n n p + = 1
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2. Suppose my waiting time X for a bus is uniformly distributed on [0, θ ] with pdf: f(x) = 1/ θ for 0 x θ , 0 otherwise If the results x 1 , … , x n of a random sample from this distribution are observed, find maximum likelihood and method of moment estimators for the parameter θ . Method of Moments () X E x = = θ/ 2 => x 2 = θ ) Maximum Likelihood () ()
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Unformatted text preview: Likelihood function: { } i n n i x x x f L max , 1 1 ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = ∏ = ( ) { } i x L max , < = Plot the graph of L( θ ) against θ . Then find the θ , which is corresponding to the maximum value of L( θ ). It shows that . { } i x max ˆ = 3. Let {x 1 , … , x n } be a random sample of independent random variables from a distribution with density function: f X (x) = 0.564 a 1.5 x 0.5 exp(-ax), x ≥ 0 For this distribution E[X] = 1.5/a. What are the maximum likelihood and method of moments estimators of a? Answer: Method of Moments = E[X] = 1.5/a ==> x a / 5 . 1 ˆ = x Maximum Likelihood L(a) = = ∏ ( ) i n i X x f ∏ = 1 ( ) [ ] = − n i i ax x a 1 5 . 5 . 1 exp 564 . lnL(a) = n ln(0.564) + 1.5n ln(a) + 0.5 Σ ln(x i ) – a Σ x i ( ) [ ] ln = a L da d => x x n a i i / 5 . 1 / 5 . 1 ˆ = = ∑ 2...
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This note was uploaded on 02/02/2008 for the course CEE 3040 taught by Professor Stedinger during the Fall '08 term at Cornell University (Engineering School).

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Section7 - Likelihood function i n n i x x x f L max 1 1 ≥ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = = ∏ = i x L max< = Plot the graph of L θ against θ

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