Regression analysis: likelihood, error and entropy

In a regression with independent and identically distributed normal residuals, the log-likelihood function yields an empirical form of the L ² -norm, whereas the normal distribution can be obtained as a solution of differential entropy maximization subject to a constraint on the L ² -norm of a random variable. The L ¹ -norm and the double exponential (Laplace) distribution are related in a similar way. These are examples of an “inter-regenerative” relationship. In fact, L ² -norm and L ¹ -norm are just particular cases of general error measures introduced by Rockafellar et al. (Finance Stoch 10(1):51–74, 2006) on a space of random variables. General error measures are not necessarily symmetric with respect to ups and downs of a random variable, which is a desired property in finance applications where gains and losses should be treated differently. This work identifies a set of all error measures, denoted by E, and a set of all probability density functions (PDFs) that form “inter-regenerative” relationships (through log-likelihood and entropy maximization). It also shows that M-estimators, which arise in robust regression but, in general, are not error measures, form “inter-regenerative” relationships with all PDFs. In fact, the set of M-estimators, which are error measures, coincides with E. On the other hand, M-estimators are a particular case of L-estimators that also arise in robust regression. A set of L-estimators which are error measures is identified—it contains E and the so-called trimmed L ^p -norms.