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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).
 Fall '08
 Stedinger

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