November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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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
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