Minimum MSE estimation derivation

[EmmaSaunders1]

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



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


from


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


Thank you

[EmmaSaunders1]

Shouldnt the term just be zero - I cant understand it's presence - are there any conditions in which it is not zero??

[EmmaSaunders1]

For anyone who is interested - the last term


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


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


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


Is a minimum and reduces to


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


Then


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


which is the average mean square error