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Unformatted text preview: Week 5 Lecture 9 A lower bound by Tsybakov Parameter space Θ = { θ ,θ 1 ,...,θ M } (1) d ( θ i ,θ j ) ≥ 2 s , for all 0 ≤ i 6 = j ≤ M. Usually s is the rate of convergence you have obtained by a specific procedure, and d is a distance related to the loss function. Reduction to bounds in probability For any ˆ θ which may not be in Θ, define ˆ θ * = arg min θ j ∈ Θ d ˆ θ,θ j Then inf ˆ θ sup Θ E θ d 2 ˆ θ,θ ≥ s 2 inf ˆ θ sup Θ P θ d ˆ θ,θ ≥ s ≥ s 2 inf ˆ θ sup ≤ j ≤ M P θ j ˆ θ * 6 = θ j = s 2 inf ˆ θ ∈ Θ sup ≤ j ≤ M P θ j ˆ θ 6 = θ j Usually we construct the parameter in a way such that the minimax probability of error p e,M Δ = inf ˆ θ ∈ Θ sup ≤ j ≤ M P θ j ˆ θ 6 = θ j ≥ c for some fixed constant c > 0, then a lower bound cs 2 is obtained. Lower bound for minimax probability of error p e,M ≥ sup τ> τM 1 + τM α τ (2) where α τ = 1 M ∑ M j =1 P θ j ( A j ) with A j = n d P θ d P θ j > τ o ....
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This note was uploaded on 11/06/2009 for the course STAT 680 at Yale.
 '09
 HarrisonH.Zhou

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