Mühlemann, Anja (2021). The Role of Loss Functions in Regression Problems. (Thesis). Universität Bern, Bern

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Abstract
In regression analysis, the goal is to capture the influence of one or more explanatory variables X1, . . . ,Xm on a response variable Y in terms of a regression function g : Rm > R. An estimate ĝ of g is then found or evaluated in terms of its ability to predict a prespecified statistical functional T of the conditional distribution L(Y X1, . . . ,Xm). This is done with the help of a loss function that penalizes estimates that perform poorly in predicting T(L(Y X1, . . . ,Xm)). More precisely, it is done by using loss functions that are consistent for T. Clearly, the outcome of the evaluation or estimation strongly depends on the functional T. However, when we focus on a specific functional T a vast collection of suitable loss functions may be available and the result can still be sensitive to the choice of loss function. There are several viable solution strategies to approach this issue. We can, for instance, impose additional properties on the loss function or the resulting estimate so that only one of the possible loss functions remains reasonable. In this doctoral thesis we adopt another approach. The underlying idea is that we would naturally prefer an estimate ĝ that is optimal with respect to several consistent loss functions for T, as then the choice of loss function seems to impact the outcome less severely. In Chapter 1, we consider the nonparametric isotonic regression problem. We show that this regression problem is special in that for identifiable functionals T, solutions which are simultaneously optimal with respect to an entire class of consistent losses exist and can be characterized. There are, however, several functionals of interest that are not identifiable. The expected shortfall is just one prominent example. However, some of those functionals can be obtained as a function of a vectorvalued elicitable functional. In the second Chapter, we investigate when simultaneous optimality with respect to a class of consistent losses holds for these functionals and introduce the solution to the isotonic regression problem for a specific loss in the case where simultaneous optimality is not fulfilled. In parametric regression, on the other hand, different consistent loss functions often yield different parameter estimates under misspecification. This motivates to consider the set of these parameters as a way to measure misspecification. We introduce this approach in Chapter 3 and show how the set of these model parameters can be calculated on the population and on the sample level for an isotonic regression function g.
Item Type:  Thesis 

Dissertation Type:  Cumulative 
Date of Defense:  26 March 2021 
Subjects:  500 Science > 510 Mathematics 
Institute / Center:  08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science 
Depositing User:  Hammer Igor 
Date Deposited:  09 Apr 2021 13:36 
Last Modified:  26 Mar 2022 01:30 
URI:  https://boristheses.unibe.ch/id/eprint/2594 
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