DeepAI AI Chat
Log In Sign Up

Cox reduction and confidence sets of models: a theoretical elucidation

by   R. M. Lewis, et al.
Imperial College London

For sparse high-dimensional regression problems, Cox and Battey [1, 9] emphasised the need for confidence sets of models: an enumeration of those small sets of variables that fit the data equivalently well in a suitable statistical sense. This is to be contrasted with the single model returned by penalised regression procedures, effective for prediction but potentially misleading for subject-matter understanding. The proposed construction of such sets relied on preliminary reduction of the full set of variables, and while various possibilities could be considered for this, [9] proposed a succession of regression fits based on incomplete block designs. The purpose of the present paper is to provide insight on both aspects of that work. For an unspecified reduction strategy, we begin by characterising models that are likely to be retained in the model confidence set, emphasising geometric aspects. We then evaluate possible reduction schemes based on penalised regression or marginal screening, before theoretically elucidating the reduction of [9]. We identify features of the covariate matrix that may reduce its efficacy, and indicate improvements to the original proposal. An advantage of the approach is its ability to reveal its own stability or fragility for the data at hand.


Optimal Subsampling Design for Polynomial Regression in one Covariate

Improvements in technology lead to increasing availability of large data...

Implementable confidence sets in high dimensional regression

We consider the setting of linear regression in high dimension. We focus...

Nonparametric Confidence Regions for Level Sets: Statistical Properties and Geometry

This paper studies and critically discusses the construction of nonparam...

Markov Neighborhood Regression for High-Dimensional Inference

This paper proposes an innovative method for constructing confidence int...

Confidence Sets for a level set in linear regression

Regression modeling is the workhorse of statistics and there is a vast l...

Approximate MMAP by Marginal Search

We present a heuristic strategy for marginal MAP (MMAP) queries in graph...

Stable Conformal Prediction Sets

When one observes a sequence of variables (x_1, y_1), ..., (x_n, y_n), c...