# MVU - Detection and Estimation(Spring 2009 MVU Minimum...

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Detection and Estimation (Spring , 2009) MVU NCTU EE P.1 Minimum Variance Unbiased Estimation ± MVU Estimators ~ Unbiased Estimators Unbiased estimator : on the average, the estimator will yield the true value of the unknown parameter. θ all for ) ˆ ( = E Let ) ( ˆ x g = , where x = [x[0] x[1] … x[N-1]] T Unbiased = = dx p g E ) ; ( ) ( ) ˆ ( x x Remark: Generally, we seek unbiased estimators (necessary but not sufficient for good estimator) Example: x[n] = A + w[n] n = 0, 1, …, N-1 unbiased!

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Detection & Estimation (Spring, 2009) MVU NCTU EE P.2 biased! Bias of estimator = ) ( ) ˆ ( θ b E =
Detection & Estimation (Spring, 2009) MVU NCTU EE P.3 ~ Minimum Variance Criterion Need optimality criterion to assess performance Mean Square Error (MSE) Good estimator has small MSE. Unfortunately, adoption of this natural criterion leads to unrealizable estimators. MSE is composed of errors due to the variance of the estimator as well as the bias!! Example: for some constant a. The optimal value of a depends on the unknown parameter A. The estimator is not realizable.

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Detection & Estimation (Spring, 2009) MVU NCTU EE P.4 Any criterion that depends on the bias will lead to an unrealizable estimator!! Generally, we constrain the bias to be zero and find the estimator which minimizes the variance. Minimum Variance Unbiased (MVU) estimator. ~ Existence of MVU Estimator θ ˆ must have the smallest variance for all values of . In general, the MVU estimator does not always exist. Example: Assume we have two independent observations x[0] and x[1]. The two estimators are unbiased.
Detection & Estimation (Spring, 2009) MVU NCTU EE P.5 Remark: It’s possible that there may not exist even a single unbiased estimator. ~ Finding the MVU Estimator Approaches: 1. Determine the Cramer-Rao lower bound ( CRLB ) and check to see if some estimator satisfies it. (Chapters 3 and 4) Remarks: 1. If an estimator exists whose variance equals the CRLB for each value of θ , then it must be the MVU estimator. 2. It may happen that no estimator exists whose variance equals the bound.

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Detection & Estimation (Spring, 2009) MVU NCTU EE P.6 2. Apply the Rao-Blackwell-Lehmann-Scheffe ( RBLS ) theorem. (Chapter 5) First find a sufficient statistic then find a function of the sufficient statistic which is an unbiased estimator of θ . 3. Further restrict the class of estimators to be not only unbiased but also linear . Then, find the minimum variance estimator within this restricted class. (Chapter 6) ~ Extension to a Vector Parameter Assume T p ] [ 2 1 θ " = θ is a vector of unknown parameters. An estimator T p ] ˆ ˆ ˆ [ ˆ 2 1 " = θ is unbiased if By define We define an unbiased estimator to have the property . MVU estimator: ,...,p , i i 2 1 ), ˆ var( = is minimum.
Detection & Estimation (Spring, 2009) MVU NCTU EE P.7 ± Cramer-Rao Lower Bound (CRLB) Find a lower bound on the variance of an unbiased estimator.

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MVU - Detection and Estimation(Spring 2009 MVU Minimum...

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