Deviation measures in generalized linear regression

Linear regression is traditionally based on the minimization of variance, or equivalently, standard deviation, but other approaches are possible in which standard deviation is replaced by something more general. A one-to-one correspondence is now known between risk measures, such as have been introduced for various applications in finance, and a large class of deviation measures characterized by simple axioms. Included in that class are asymmetric measures coming from conditional valueatrisk and other currently attractive notions. This paper looks at deviation in that wide sense, formulating the associated problem of regression and investigating the existence and uniqueness of the coefficients that constitute a solution. Such coefficients are characterized in ways that provide a key to their computation.