Minimum MSE estimation derivation

  • #1
Hello,

Would anyone be-able to recommend a good, easy to read article which outlines MMSE and its derivation. Specifically I am having trouble finding this term


[tex]
+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2
[/tex]

from

[tex]
E({\left | \left | X-z \right | \right |}^2|Y=y)
=\int (x-z)'(x-z)P(x|y)dx\\
=[z'-\int x'P(x|y)dx][z-\int xP(x|y)dx] + \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2
[/tex]

Thank you
 

Answers and Replies

  • #2
Shouldnt the term just be zero - I cant understand it's presence - are there any conditions in which it is not zero??
 
  • #3
For anyone who is interested - the last term

[tex]
+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2
[/tex]

is necessary to account for the difference between E(x^2) and [E(x)]^2. When Z = E[x|Y=y] the term

[tex]
E({\left | \left | X-z \right | \right |}^2|Y=y)
[/tex]

Is a minimum and reduces to

[tex]
+ \int x'xP(x|y)dx-\left \| \int xP(x|y)dx \right \|^2 =
[/tex]

Then

[tex]
E({\left | \left | X \right | \right |}^2|Y=y)-E(X|Y=y)^2\\
=E({\left | \left | X \right | \right |}^2|Y=y)-{\left | \left | \hat{X} \right | \right |}^2
[/tex]

which is the average mean square error
 

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