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Detection and Estimation Theory Lecture 8 Mojtaba Soltanalian- UIC - - PowerPoint PPT Presentation
Detection and Estimation Theory Lecture 8 Mojtaba Soltanalian- UIC - - PowerPoint PPT Presentation
Detection and Estimation Theory Lecture 8 Mojtaba Soltanalian- UIC msol@uic.edu http://msol.people.uic.edu Based on ECE 531 Slides- 2011 (Prof. Natasha Devroye) Finding MVUE- what we discussed Finding MVUE- what we discussed Finding MVUE-
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Finding MVUE- what we discussed
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Finding MVUE- the new roadmap
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Sufficient Statistics
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Sufficient Statistics Neyman-Fisher Factorization Theorem
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Sufficient Statistics and MVUE
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Sufficient Statistics
- - Completeness Example
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Sufficient Statistics
- - MVUE Construction via Completeness
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Rao-Blackwell-Lehmann-Scheffe (RBLS) Theorem
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Rao-Blackwell-Lehmann-Scheffe (RBLS) Theorem
Remarks:
- Given any estimator f that is not a function of a sufficient
statistic, there exists a better estimator if variance is concerned.
- “The conditional expectation averages out (or removes) non-
informative components in the original estimator. We can view this as a filter that eliminates unnecessary components of the data.”
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Rao-Blackwell-Lehmann-Scheffe (RBLS) Theorem
Proof: (for decreasing the variance)
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Rao-Blackwell-Lehmann-Scheffe (RBLS) Theorem
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RBLS Theorem and the MVUE
The Rao-Blackwell Theorem paves the way for decreasing the variance of an unbiased estimator. The question that remains: when can we know that we have obtained the MVUE? Answer: When T is a complete sufficient statistic. In fact, Lehmann-Scheffe Theorem states that If T is complete, there is at most one unbiased estimator that is a function of T. Unique MVUE (UMVUE)
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RBLS Theorem and the MVUE
Let’s go back a little bit!
RBLS
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Vector Versions
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Vector Versions
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Further Examples
(see example 5.8)
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