By Badong Chen
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0 @2 > : @γ2 HðejγYÞγ50 , 0 if ρ , 0:6 if ρ . 0:6 ð3:59Þ which implies that if jρj , 0:6, the MMSE estimator (γ 5 0) will be a local minimum of the error entropy in the direction of ϕðYÞ 5 Y, whereas if jρj . 0:6, it becomes a local maximum. 2, if ρ 5 0:9, the error entropy HðejγYÞ achieves its global minima at γ % 6 0:74. 3 depicts the error PDF for γ 5 0 (MMSE estimator) and γ 5 0:74 (linear MEE estimator), where μ 5 1; ρ 5 0:9. We can see that the MEE solution is in this case not unique but it is much more concentrated (with higher peak) than the MMSE solution, which potentially gives an estimator with much smaller variance.
According to , we have Hðe 2 γϕðYÞjgMEE Þ 2 HðejgMEE Þ 5 γE½ψðejgMEE ÞϕðYÞ 1 oðγϕðYÞÞ ð3:52Þ where oð:Þ denotes the higher order terms. 53) yields E½ψðejgMEE ÞϕðYÞ 5 0; ’ ϕAG ð3:54Þ Remark: If the error is zero-mean Gaussian distributed with variance σ2 , the score function will be ψðejgMEE Þ 5 2 e=σ2 . In this case, the score orthogonality condition reduces to E½eϕðYÞ 5 0. This is the well-known orthogonality condition for MMSE estimation. In MMSE estimation, the orthogonality condition is a necessary and sufficient condition for optimality, and can be used to find the MMSE estimator.
This is an unconventional risk function because the role of the weight function is to privilege one solution versus all others in the space of the errors. There is an important relationship between the MEE criterion and the traditional MSE criterion. The following theorem shows that the MSE is equivalent to the error entropy plus the KL-divergence between the error PDF and any zero-mean Gaussian density. 1 Let Gσ ð:Þ denote a Gaussian pﬃﬃﬃﬃﬃﬃ ð1= 2πσÞexpð2 x2 =2σ2 Þ, where σ . 0. 4 The loss functions of MEE corresponding to three different error PDFs.