9.11 MEAN-SQUARE-ERROR ESTIMATION

Consider two random variables X and Y that are correlated with joint pdf fX,Y(x, y). Assume that X is measurable whereas we do not have direct access to Y. A simple example is

(9.144) Numbered Display Equation

where V is not measurable and is independent of Y. Such a model is useful for many problems where an estimate of Y is needed but additive noise V corrupts the samples. From Chapter 4, we know that when Y and V are independent (which is often the case in many problems), the pdf of X is obtained from the following convolution:

(9.145) Numbered Display Equation

In order to estimate Y from X, the joint pdf fX,Y(x, y) is required. In the notation of Section 9.10, the unknown “parameter” is , and there is only one sample X such that N = 1. For the simple model above, the joint pdf is

(9.146) Numbered Display Equation

Using the techniques in Chapter 4:

(9.147) Numbered Display Equation

where in the last expression we have used the fact that Y and V are independent. Thus

(9.148)

and

(9.149)

Returning to the general problem of estimating Y from

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