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Financial Signal Processing and Machine Learning by Dmitry M. Malioutov, Sanjeev R. Kulkarni, Ali N. Akansu

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Chapter 11Regression Models in Risk Management

Stan Uryasev

University of Florida, USA

This chapter discusses theory and application of generalized linear regression that minimizes a general error measure of regression residual subject to various constraints on regression coefficients and includes least-squares linear regression, median regression, quantile regression, mixed quantile regression, and robust regression as special cases. General error measures are nonnegative positively homogeneous convex functionals that generalize the notion of norm and, in general, are asymmetric with respect to ups and downs of a random variable, which allows one to treat gains and losses differently. Each nondegenerate error measure c11-math-0001 yields the deviation measure c11-math-0002 projected from c11-math-0003 and the statistic c11-math-0004 associated with c11-math-0005. General deviation measures are also nonnegative positively homogeneous convex functionals, which, in contrast to error measures, are insensitive to a constant shift. They generalize the notion ...

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