Degrees of Freedom and Model Search

TL;DR

This paper investigates the degrees of freedom in statistical model selection, deriving exact formulas for best subset selection under orthogonal predictors and introducing the concept of search degrees of freedom for adaptive procedures.

Contribution

It provides the first exact expression for degrees of freedom in best subset selection with orthogonal predictors and introduces the novel concept of search degrees of freedom.

Findings

Derived an exact formula for degrees of freedom in best subset selection.

Introduced the concept of search degrees of freedom for adaptive model selection.

Extended Stein's formula to discontinuous functions.

Abstract

Degrees of freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed by a given procedure. But, despite this fundamental role in statistics, its behavior not completely well-understood, even in some fairly basic settings. For example, it may seem intuitively obvious that the best subset selection fit with subset size k has degrees of freedom larger than k, but this has not been formally verified, nor has is been precisely studied. In large part, the current paper is motivated by this particular problem, and we derive an exact expression for the degrees of freedom of best subset selection in a restricted setting (orthogonal predictor variables). Along the way, we develop a concept that we name "search degrees of freedom"; intuitively, for adaptive regression procedures that perform variable selection, this is a part of the (total) degrees of freedom that we attribute entirely to the model selection mechanism. Finally, we establish a modest extension of Stein's formula to cover discontinuous functions, and discuss its potential role in degrees of freedom and search degrees of freedom calculations.