lectr16 - 1 16. Mean Square Estimation Given some...

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Unformatted text preview: 1 16. Mean Square Estimation Given some information that is related to an unknown quantity of interest, the problem is to obtain a good estimate for the unknown in terms of the observed data. Suppose represent a sequence of random variables about whom one set of observations are available, and Y represents an unknown random variable. The problem is to obtain a good estimate for Y in terms of the observations Let represent such an estimate for Y . Note that can be a linear or a nonlinear function of the observation Clearly represents the error in the above estimate, and the square of n X X X , , , 2 1 . , , , 2 1 n X X X ) ( ) , , , ( 2 1 X X X X Y n = = (16-1) ) ( . , , , 2 1 n X X X ) ( ) ( X Y Y Y X - =- = (16-2) 2 | | PILLAI 2 the error. Since is a random variable, represents the mean square error. One strategy to obtain a good estimator would be to minimize the mean square error by varying over all possible forms of and this procedure gives rise to the M inimization of the M ean S quare E rror (MMSE) criterion for estimation. Thus under MMSE criterion,the estimator is chosen such that the mean square error is at its minimum. Next we show that the conditional mean of Y given X is the best estimator in the above sense. Theorem1: Under MMSE criterion, the best estimator for the unknown Y in terms of is given by the conditional mean of Y gives X . Thus Proof : Let represent an estimate of Y in terms of Then the error and the mean square error is given by } | | { 2 E ), ( ) ( n X X X , , , 2 1 }. | { ) ( X Y E X Y = = (16-3) ) ( X Y = ). , , , ( 2 1 n X X X X = , Y Y- = } | ) ( | { } | | { } | | { 2 2 2 2 X Y E Y Y E E - =- = = (16-4) PILLAI } | | { 2 E 3 Since we can rewrite (16-4) as where the inner expectation is with respect to Y , and the outer one is with respect to Thus To obtain the best estimator we need to minimize in (16-6) with respect to In (16-6), since and the variable appears only in the integrand term, minimization of the mean square error in (16-6) with respect to is equivalent to minimization of with respect to }] | { [ ] [ X z E E z E z X = }] | ) ( | { [ } | ) ( | { z 2 z 2 2 X X Y E E X Y E Y X n n n n n n n n n n - =- = . X + -- =- = . ) ( } | ) ( | { }] | ) ( | { [ 2 2 2 dx X f X X Y E X X Y E E X (16-6) (16-5) , 2 . , ) ( X f X , } | ) ( | { 2 - X X Y E 2 } | ) ( | { 2 X X Y E - . PILLAI 4 Since X is fixed at some value, is no longer random, and hence minimization of is equivalent to This gives or But since when is a fixed number Using (16-9) ) ( X } | ) ( | { 2 X X Y E - . } | ) ( | { 2 =- X X Y E (16-7) } | ) ( {| =- X X Y E (16-8) ), ( } | ) ( { X X X E = (16-9) ) ( , X x X = ). ( x PILLAI . } | ) ( { } | { =- X X E X Y E 5 in (16-8) we get the desired estimator to be Thus the conditional mean of Y given represents the best...
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lectr16 - 1 16. Mean Square Estimation Given some...

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