5102-Lecture-06

5102-Lecture-06 - let ˆ θ y = arg max θ lik θ | y Model...

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Lecture 6 Stat 5102-004 31 January 2011
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Bayes Estimators Find a Bayes estimator by minimizing the posterior expected loss. ˆ θ ( y ) = arg min a E[ L ( a , Θ) | y ] Under squared error loss, the posterior mean is the Bayes estimator, L ( a ) = ( a - θ ) 2 E[Θ | y ] = arg min a E[ L ( a , Θ) | y ] ˆ θ ( y ) = E[Θ | y ] and the posterior variance is the minimum posterior expected loss. Var[Θ | y ] = E ±( ˆ θ ( y ) - Θ ) 2 ² ² y ³ STAT 5102 (Theory of Statistics) Lecture 6 1 / 6
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Large Sample Behavior (Binomial example) Θ | y Beta( α + y + n - y ) μ 0 = α α + β The Bayes estimator under squared error loss is ˆ θ = α + β α + β + n μ 0 | {z } a n + n α + β + n | {z } b n Y n . As n → ∞ , a n 0 and b n 1 1 n Y P --→ θ n ( 1 n Y - θ ) D --→ N ( 0 (1 - θ ) ) STAT 5102 (Theory of Statistics) Lecture 6 2 / 6
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Average Behavior (Binomial example) The Bayes estimator under squared error loss is ˆ θ = α + β α + β + n μ 0 | {z } a n + n α + β + n | {z } b n Y n . For each n , E ˆ θ = a n + b n θ and Var ˆ θ = b 2 n n θ (1 - θ ) MSE( θ ) = Bias 2 ˆ θ + Var ˆ θ = [ a n + ( b n - 1) θ ] 2 + b 2 n n θ (1 - θ ) STAT 5102 (Theory of Statistics) Lecture 6 3 / 6
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Maximum Likelihood Estimators Maximum likelihood estimators are defined by maximizing the likelihood. For each y ∈ Y
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Unformatted text preview: , let ˆ θ ( y ) = arg max θ lik( θ | y ) . Model y = ( y 1 ,..., y n ) as a random sample from a uniform on [0 ,θ ] . ● θ lik ( θ |y 29 ● lik( θ | y ) = θ-n I [max y , ∞ ) ( θ ) ˆ θ : Y → ϑ y 7→ max y STAT 5102 (Theory of Statistics) Lecture 6 4 / 6 Maximum Likelihood Estimators Recall that likelihoods are amenable to analysis in the log scale. Since the logarithm is an increasing function, arg max θ ‘ ( θ | y ) = arg max θ lik( θ | y ) . The same arguments maximize both the likelihood and log likelihood. The log likelihood is analytically convenient—especially for I random samples and I exponential families. STAT 5102 (Theory of Statistics) Lecture 6 5 / 6 Normal Example θ = E Y θ lik ( θ |y 29 y θ loglik y STAT 5102 (Theory of Statistics) Lecture 6 6 / 6...
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5102-Lecture-06 - let ˆ θ y = arg max θ lik θ | y Model...

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