Regression analysis: likelihood, error and entropy

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Abstract

In a regression with independent and identically distributed normal residuals, the log-likelihood function yields an empirical form of the \(\mathcal{L}^2\)-norm, whereas the normal distribution can be obtained as a solution of differential entropy maximization subject to a constraint on the \(\mathcal{L}^2\)-norm of a random variable. The \(\mathcal{L}^1\)-norm and the double exponential (Laplace) distribution are related in a similar way. These are examples of an “inter-regenerative” relationship. In fact, \(\mathcal{L}^2\)-norm and \(\mathcal{L}^1\)-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 \(\mathscr {E}\), 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 \(\mathscr {E}\). 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 \(\mathscr {E}\) and the so-called trimmed \(\mathcal{L}^p\)-norms.

Keywords

Regression Likelihood Entropy Error measure M-estimator L-estimator 

Mathematics Subject Classification

90C90 90C25 90C15 

Notes

Acknowledgements

We are grateful to the referees for the comments and suggestions, which helped to improve the quality of the paper. The first author thanks the University of Leicester for granting him the academic study leave to do this research.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK
  2. 2.Department of Mathematical SciencesStevens Institute of TechnologyHobokenUSA

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