2014/02/09 by Ryan J. Tibshirani, Tibshirani, Ryan J. · 3 citations
Mathematics · #Advanced Statistical Methods and Models #FOS: Mathematics #Fuzzy Systems and Optimization #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.1402.1920
openalex publication_date 2014/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.