MLE - Detection and Estimation (Spring, 2009) MLE Maximum...

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Detection and Estimation (Spring , 2009) MLE NCTU EE P.1 Maximum Likelihood Estimation ~ MLE 9 A “turn-the-crank” procedure 9 The most popular approach to obtaining practical estimators 9 It is approximately the MVU estimator (actually approximately efficient) for large enough data records Example: DC Levels in WGN x[n] = A + w[n] n = 0, 1, …, N-1 A: unknown, A > 0 w[n] WGN with unknown variance A
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.2 RBLS Approach Sufficient statistic: Assume T(x) is complete. We need to find a function g such that But How to choose g is not obvious! Alternatively, since x[0] is an unbiased estimator, the MVU estimator would take the form too difficult to get!!
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.3 We propose an estimator that is approximately optimal. As N Asymptotically unbiased. Asymptotically efficient Consider The estimator is biased. However, as N The estimator is said to be a consistent estimator . where To find the mean and variance of A ˆ as N , we linearize the estimator about u 0 = E(U) = A + A 2 .
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.4 Asymptotically unbiased. var(x 2 [n]) = 4A 3 +2A 2 , CRLB!! A ˆ is asymptotically efficient!! Moreover, by the central limit theorem, is Gaussian as N . A ˆ is a linear function of A ˆ is Gaussian.
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.5 ~ Finding the MLE The MLE for a scalar parameter is defined to be the value of θ that maximizes p( x ; θ ) for x fixed. Example: DC Level in WGN x[n] = A + w[n] n = 0, 1, …, N-1 A: unknown, A > 0 w[n] WGN with unknown variance A Since A > 0, we choose
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.6 Sometimes, the MLE yields an efficient estimator for finite data records. Example: DC Level in WGN x[n] = A + w[n], n = 0, 1, …, N-1 w[n]: WGN with known variance σ 2 Remark: If an efficient estimator exists, the maximum likelihood procedure will produce it!!
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.7 ~ Properties of the MLE a asymptotically Example: DC Level in WGN x[n] = A + w[n] n = 0, 1, …, N-1 A: unknown, A > 0 w[n] WGN with unknown variance A How large must N be for asymptotic results to apply? Use Monte Carlo computer simulation. Remark: In practice, an analytical expression for the pdf of the MLE is difficult to derive and a computer simulation is usually required.
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Detection & Estimation (Spring, 2009) MLE NCTU EE P.8 Remark: Usually, N does not have to be very large. Example: Phase estimation Assume we wish to estimate the phase φ of a sinusoid embedded in WGN.
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MLE NCTU EE P.9 To find the MLE, we maximize or minimize For f 0 not near 0 or 1/2, Remark: The MLE is a function of the sufficient statistics. Explanation:
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MLE - Detection and Estimation (Spring, 2009) MLE Maximum...

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