Detection and Estimation Theory Lecture 7 Mojtaba Soltanalian- UIC - - PowerPoint PPT Presentation

Detection and Estimation Theory Lecture 7

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- the new roadmap

Sufficient Statistics

• Can we compress the measurements into a lower

dimensional statistic that carries all the useful information?

• “In particular, we want to use this lower dimensional

statistic to estimate θ with the same quality as if we kept all of x. If so, then to study θ we could discard x and retain only the compressed statistic.”

Sufficient Statistics

Sufficient Statistics

• Estimate the DC level, A, of a signal given noisy measurements x[0], x[1], ...

x[N-1] where vs discarding data points All look sufficient!

Sufficient Statistics

Sufficient Statistics

Sufficient Statistics

NOT dependent on A

Sufficient Statistics Neyman-Fisher Factorization Theorem

Neyman-Fisher Factorization Theorem

We will see the details later when we discus the maximum likelihood estimation : “An implication of the theorem is that when using likelihood- based inference, two sets of data yielding the same value for the sufficient statistic T(X) will always yield the same inferences about θ. By the factorization criterion, the likelihood's dependence on θ is only in conjunction with T(X). As this is the same in both cases, the dependence on θ will be the same as well, leading to identical inferences.”

Sufficient Statistics Neyman-Fisher Factorization Theorem

Sufficient Statistics Neyman-Fisher Factorization Theorem

Sufficient Statistics Neyman-Fisher Factorization Theorem

Neyman-Fisher Factorization Theorem

• - Extended Version

Sufficient Statistics and MVUE