5102-Lecture-07 - I Let ˆ be the mle of I Let g be a...

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Lecture 7 Stat 5102-004 2 February 2011
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Maximum Likelihood Estimators Maximum likelihood estimators are defined by maximizing the likelihood. For each y ∈ Y , let ˆ θ ( y ) = arg max θ lik( θ | y ) . ˆ θ : Y → ϑ y 7→ arg max θ lik( θ | y ) 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. STAT 5102 (Theory of Statistics) Lecture 7 1 / 6
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Poisson Example λ = E Y λ lik ( λ |y 29 y λ log lik ( λ |y 29 y STAT 5102 (Theory of Statistics) Lecture 7 2 / 6
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Poisson Example θ = log E Y θ lik ( θ |y 29 log y θ log lik ( θ |y 29 log y STAT 5102 (Theory of Statistics) Lecture 7 3 / 6
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Invariance of the mle θ = log λ λ log lik ( λ |y 29 y θ log lik ( θ |y 29 log y STAT 5102 (Theory of Statistics) Lecture 7 4 / 6
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Invariance of the mle The maximum likelihood estimator is invariant to reparameterizations.
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Unformatted text preview: I Let ˆ η be the mle of η . I Let g be a bijection, and let θ = g ( η ) . I The mle of θ is ˆ θ = g (ˆ η ) . 1. Densities determine the values of a likelihood function. 2. Relabeling distributions has no effect on their densities. STAT 5102 (Theory of Statistics) Lecture 7 5 / 6 Y 1 , . . . , Y 12 iid ∼ Bern ( p ) ¯ y = 2 3 p lik ( p|y 29 y 0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.25 ϖ ϖ |y y 1-y ϖ = p 1-p 1 2 3 4 5 μ μ n y μ = n p 2 4 6 8 10 12 θ θ logit y θ = logit p-4-2 2 4 STAT 5102 (Theory of Statistics) Lecture 7 6 / 6...
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