Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

9.12 MEAN-ABSOLUTE-ERROR ESTIMATION

Consider correlated random variables X and Y with joint pdf fX,Y(x, y) of which X is measurable. The mean-absolute-error (MAE) risk function is

(9.182)

which is minimized by again recognizing that we need only examine the inner integral of the following expression:

(9.183)

The absolute value is handled by splitting the integral into the sum of two integrals so that the estimator is expressed as follows:

(9.184)

The minimum is achieved by differentiating the integrals with respect to h(x) and applying Leibniz's integral rule (see Appendix E):

(9.185)

(9.186)

where, of course, fY|X(y|x) does not depend on h(x). Setting the sum of the right-hand sides equal to zero, we recognize that the function minimizing the MAE is the conditional median:

(9.187)

which means

(9.188)

As was the case for the MSE estimator, it is not always easy to find fY|X