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Minimum Variance Unbiased Estimation - Study Guide | ECE 531, Study notes of Electrical and Electronics Engineering

University of Illinois - Chicago Electrical and Electronics Engineering

Material Type: Notes; Class: Detection and Estimation Theory; Subject: Electrical and Computer Engr; University: University of Illinois - Chicago; Term: Spring 2011;

Typology: Study notes

2011/2012

Uploaded on 05/18/2012

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Estimation: chapter 2+3

Minimum variance unbiased estimation + the CRLB

Natasha Devroye

[email protected]

http://www.ece.uic.edu/~devroye

Spring 2011

• Find a few estimators

Estimation: a first example

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

[N-1] where

x[n] are samples of this!

• Compare their performance {• mean?

• variance?

• pdf?

Discover Study notes of Electrical and Electronics Engineering University of Illinois - Chicago

Partial preview of the text

Download Minimum Variance Unbiased Estimation - Study Guide | ECE 531 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

Estimation: chapter 2+

Minimum variance unbiased estimation + the CRLB

Natasha Devroye

[email protected]

http://www.ece.uic.edu/~devroye

Spring 2011

Find a few estimators

Estimation: a first example

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

[N-1] where

x[n] are samples of this!

Compare their performance

mean?
variance?
pdf?

Estimation: a first example

Estimators of the DC level, A

Estimation: definitions

Minimum variance unbiased estimation

Give a counter-example! (b1pg.20)

The Cramer-Rao Lower Bound

the CRLB give a lower bound on the variance of ANY UNBIASED estimator
does NOT guarantee bound can be obtained
IF find an estimator whose variance = CRLB then it’s MVUE
otherwise can use Ch.5 tools (Rao-Blackwell-Lehmann-Scheffe Theorem and

Neyman-Fisher Factorization Theorem) to construct a better estimator from

any unbiased one - possibly the MVUE if conditions are met

The Cramer-Rao Lower Bound (CRLB)

Use?
Intuition?

The Cramer-Rao Lower Bound (CRLB)

CRLB T or F

What is I(θ)?

Why “information”?
- non-negative
- additive for independent observations

Vector form of the CRLB

Vector CRLB for transformations

Example of vector CRLB with transformation

CRLB for General Gaussian Case

When observations are Gaussian and one knows the dependence of the

mean and covariance matrix on the unknown parameters, we know the

closed form of the CRLB (or Fisher information matrix):

CRLB for Gaussians examples

Example: range estimation

Discrete time signal model

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Minimum Variance Unbiased Estimation - Study Guide | ECE 531, Study notes of Electrical and Electronics Engineering

Related documents

Partial preview of the text

Download Minimum Variance Unbiased Estimation - Study Guide | ECE 531 and more Study notes Electrical and Electronics Engineering in PDF only on Docsity!

Estimation: chapter 2+

Minimum variance unbiased estimation + the CRLB

Natasha Devroye

[email protected]

http://www.ece.uic.edu/~devroye

Spring 2011

Estimation: a first example

[N-1] where

Estimation: a first example

Estimation: definitions

Minimum variance unbiased estimation

Give a counter-example! (b1pg.20)

The Cramer-Rao Lower Bound

Neyman-Fisher Factorization Theorem) to construct a better estimator from

any unbiased one - possibly the MVUE if conditions are met

The Cramer-Rao Lower Bound (CRLB)

The Cramer-Rao Lower Bound (CRLB)

CRLB T or F

What is I(θ)?

Vector form of the CRLB

Vector form of the CRLB

Vector CRLB for transformations

Example of vector CRLB with transformation

CRLB for General Gaussian Case

mean and covariance matrix on the unknown parameters, we know the

closed form of the CRLB (or Fisher information matrix):

CRLB for Gaussians examples

Example: range estimation

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