Estimation Theory Maximum Likelihood Estimator (MLE) Wenhui Xiong NCL UESTC whxiong@uestc.edu.cn
whxiong@uestc.edu.cn Estimation Theory Maximum Likelihood Estimator (MLE) Wenhui Xiong NCL UESTC
Review MVU Estimator 。CRLB→factorize a1np(x9=I(8)(g(x)-0) 80 RBLS: Use sufficient statistics:T(x)ET] unbiased estimator over sufficient statistics:g[T(x)] BLUE:if the unknown is a linear function of the data 6=(HC-1Ⅲ）-1HPC-1x whxiong@uestc.edu.cn 2
whxiong@uestc.edu.cn Review MVU Estimator 2 RBLS: Use sufficient statistics: T(x) unbiased estimator over sufficient statistics: g[T(x)] E[µ · jT(x)] BLUE: if the unknown is a linear function of the data CRLB factorize µ ^ = (HT C¡1 H) ¡1 HT C¡1 x @lnp(x; µ) @µ = I(µ)(g(x) ¡ µ)
How about CRLB method? Example:Estimating DC in WGN with unknown variance A k出=g即{立∑取-4月 whxiong@uestc.edu.cn 3
whxiong@uestc.edu.cn 3 How about CRLB method? p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
How about CRLB method? Example:Estimating DC in WGN with unknown variance A x=即{T公- olnp(x;A) ∂A -+∑-用+∑i-心 whxiong@uestc.edu.cn
whxiong@uestc.edu.cn How about CRLB method? @ ln p(x; A) @A = ¡ N 2A + 1 A X(x[n] ¡ A) + 1 2A2 X(x[n] ¡ A) 2 p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
How about CRLB method? Example:Estimating DC in WGN with unknown variance A (x;A)= 卿{7公-月 0px-司 6A -4∑o网-A+京∑州-A {(1-)∑网-NA+立∑2侧+} whxiong@uestc.edu.cn 5
whxiong@uestc.edu.cn 5 = 1 A ½µ 1 ¡ 1 A ¶Xx[n] ¡ NA + 1 2A Xx 2 (n) + N 2 ¾ How about CRLB method? @ ln p(x; A) @A = ¡ N 2A + 1 A X(x[n] ¡ A) + 1 2A2 X(x[n] ¡ A) 2 p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
How about CRLB method? Example:Estimating DC in WGN with unknown variance A pxA=了 四{z区钢-月 胫出效+4∑4w-+∑n- 6A {(-)∑m-NA+a∑)+} =} It is hard to factorize into I(A)(g(x)-A) CRLB method does not apply whxiong@uestc.edu.cn 6
whxiong@uestc.edu.cn 6 It is hard to factorize into I(A)(g(x)-A) = 1 A ½µ 1 ¡ 1 A ¶Xx[n] ¡ NA + 1 2A Xx 2 (n) + N 2 ¾ How about CRLB method? @ ln p(x; A) @A = ¡ N 2A + 1 A X(x[n] ¡ A) + 1 2A2 X(x[n] ¡ A) 2 CRLB method does not apply p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
How about CRLB method? Example:Estimating DC in WGN with unknown variance A (x;A)= 四{∑钢-月 出效+∑4w-+∑- 6A A{(-)∑m-NA+a∑)+} It is hard to factorize into I(A)(g(x)-A) CRLB method does not apply 82Inp(x;A) A2 CRLB provides bound ∂A2 (A+7)N whxiong@uestc.edu.cn 7
whxiong@uestc.edu.cn 7 It is hard to factorize into I(A)(g(x)-A) = 1 A ½µ 1 ¡ 1 A ¶Xx[n] ¡ NA + 1 2A Xx 2 (n) + N 2 ¾ How about CRLB method? @ ln p(x; A) @A = ¡ N 2A + 1 A X(x[n] ¡ A) + 1 2A2 X(x[n] ¡ A) 2 CRLB method does not apply @ 2 ln p(x; A) @A2 = A2 (A + 1 2 )N CRLB provides bound p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
How about CRLB method? Example:Estimating DC in WGN with unknown variance A (x;A)= 四方亿-4 效+∑4-+∑- 6A {(-)∑m-NA+a∑)+} It is hard to factorize into I(A)(g(x)-A) CRLB method does not apply A2 CRLB provides bound var (A+)N whxiong@uestc.edu.cn 8
whxiong@uestc.edu.cn 8 It is hard to factorize into I(A)(g(x)-A) = 1 A ½µ 1 ¡ 1 A ¶Xx[n] ¡ NA + 1 2A Xx 2 (n) + N 2 ¾ How about CRLB method? @ ln p(x; A) @A = ¡ N 2A + 1 A X(x[n] ¡ A) + 1 2A2 X(x[n] ¡ A) 2 CRLB method does not apply CRLB provides bound p(x; A) = 1 (2¼A) N=2 exp ½ ¡ 1 2A ³Xx[n] ¡ A ´2 ¾ Example: Estimating DC in WGN with unknown variance A
Then RBLS? x利=即京(-2NA-N4 whxiong@uestc.edu.cn 9
whxiong@uestc.edu.cn Then RBLS? 9
Then RBLS? p6x0=2ap24(r训-2NA+w4 脚{（∑州-} h(x) g(∑x2(m),A) whxiong@uestc.edu.cn
whxiong@uestc.edu.cn Then RBLS? = 1 (2¼A) N=2 exp ½ ¡ 1 2 µ 1 A Xx 2 [n] + NA ¶¾ | {z } g( Px2(n);A) ¢exp(Nx¹) | {z } h(x)