Chapter 2 Minimum Variance Unbiased EstimatorsCh. 2: Minimum Variance Unbiased Est.Example: Estimate DC in White Uniform Noise2.4 Minimum Variance Criterion2.5 Existence of MVU Estimator2.6 Finding the MVU Estimator2.7 Vector ParameterChapter 2Minimum Variance Unbiased EstimatorsMVUCh. 2: Minimum Variance Unbiased Est.Basic Idea of MVU: Out of all unbiased estimates, find the onewith the lowest variance(This avoids the realizability problem of MSE)2.3 Unbiased EstimatorsAn estimator is unbiased if{}θθ=ˆEfor allθExample: Estimate DC in White Uniform Noise[][]1...,,1,0−=+= NnnwAnxUnbiased Estimator:[]{}valueAofregardlessAAEbeforeassamenxNNn==Α∧−=∧∑:110Biased Estimator:⇒<≠≥=⇒≠<=⇒=⇒=≥∨∨∧∨−>∨∑1010,1][][,1:)(110AifAifBiasAAEthenAifAAEAAnxnxthenAifNotenxNANnBiased Est.MVUE = Minimum Variance Unbiased Estimator)ˆ()ˆvar()ˆ(2θθθbmse +=So, MVU could also be called“Minimum MSE Unbiased Est.”(Recall problem with MMSE criteria)Constrain bias to be zero 0 find the estimator that minimizes variance2.4 Minimum Variance CriterionNote:= 0 for MVU2.5 Existence of MVU EstimatorSometimes there is no MVUE… can happen 2 ways:1. There may be no unbiased estimators2. None of the above unbiased estimators has a uniformly minimum varianceEx. of #2Assume there are only 3 unbiased estimators for a problem. Two possible cases:3,2,1),(ˆ== igiixθ∃ an MVU∃ an MVU1ˆθθ}ˆvar{iθ2ˆθ3ˆθ}ˆvar{iθ1ˆθθ2ˆθ3ˆθEven if MVU exists: may not be able to find it!!2.6 Finding the MVU EstimatorNo Known “turn the crank” MethodThree Approaches to Finding the MVUE1. Determine Cramer-Rao Lower Bound (CRLB)… and see if some estimator satisfies it (Ch 3 & 4)(Note: MVU can exist but not achieve the CRLB)2. Apply Rao-Blackwell-Lechman-Scheffe TheoremRare in Practice… We’ll skip Ch. 53. Restrict to Linear Unbiased & find MVLU(Ch. 6)Only gives true MVU if problem is linear2.7 Vector ParameterWhen we wish to estimate multiple parameters we group them into a vector:[]Tpθθθ!21=θ[]Tpθθθˆˆˆˆ21!=θThen an estimator is notated as:{}θθ =ˆEUnbiased requirement becomes:Minimum Variance requirement becomes:For each i…{}estimates unbiased allover minˆvar θθ
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