[Paper Review] The Brouwer Lecture 2005: Statistical estimation with model selection

Lucien Birgé|arXiv (Cornell University)|May 8, 2006

Statistical Methods and Inference47 references3 citations

TL;DR

This paper presents a theoretical framework for statistical estimation using model selection, focusing on finite-dimensional models like histograms and regression models. It establishes risk bounds for model selection procedures and demonstrates that optimal performance—close to the best individual model—is achievable when models are chosen based on approximation properties and complexity, particularly via entropy-based penalties.

ABSTRACT

The purpose of this paper is to explain the interest and importance of (approximate) models and model selection in Statistics. Starting from the very elementary example of histograms we present a general notion of finite dimensional model for statistical estimation and we explain what type of risk bounds can be expected from the use of one such model. We then give the performance of suitable model selection procedures from a family of such models. We illustrate our point of view by two main examples: the choice of a partition for designing a histogram from an n-sample and the problem of variable selection in the context of Gaussian regression.

Motivation & Objective

• To establish a theoretical foundation for statistical estimation using model selection in finite-dimensional models.
• To analyze the trade-off between approximation accuracy and model complexity in statistical estimation.
• To demonstrate how model selection can achieve near-optimal performance across diverse models, including histograms and regression.
• To connect statistical estimation with approximation theory, particularly through entropy and metric dimension arguments.
• To provide a unified framework for model selection that balances bias and variance using complexity penalties.

Proposed method

• Uses finite-dimensional models (e.g., piecewise constant functions on partitions) as statistical approximations to unknown densities.
• Applies the concept of risk decomposition: total risk = approximation error + estimation error.
• Introduces a penalty term based on model complexity (dimension and metric entropy) to control overfitting.
• Employs the principle of minimizing a penalized risk criterion to select the best model from a family of models.
• Leverages results from approximation theory (e.g., DeVore and Yu, 1990) to identify models with good approximation properties in Besov spaces.
• Uses entropy-based complexity measures to bound the performance of model selection, ensuring near-optimality.

Experimental results

Research questions

• RQ1 How can model selection procedures achieve near-optimal risk performance across a family of finite-dimensional models?
• RQ2 What is the relationship between model complexity (dimension, entropy) and estimation risk in statistical models?
• RQ3 How do approximation properties of models (e.g., regular vs. irregular partitions) affect the performance of model selection?
• RQ4 In what sense can model selection achieve performance close to the best individual model in a family?
• RQ5 How can results from approximation theory be systematically applied to improve statistical estimation via model selection?

Key findings

• Model selection based on a penalized risk criterion achieves risk bounds that are nearly optimal, up to constant factors, compared to the best individual model in the family.
• The performance of model selection is governed by a trade-off between approximation error and estimation error, with the latter controlled by a complexity penalty proportional to the model's metric entropy.
• For families of models with good approximation properties (e.g., irregular partitions or wavelet-based models), the penalty can be set proportional to the model dimension, ensuring near-optimality.
• The use of Besov space approximation properties allows for the construction of richer model families that significantly improve estimation accuracy over standard regular partitions.
• Theoretical results from approximation theory—particularly those on entropy and approximation rates—directly inform the design of optimal model selection procedures.
• The framework unifies various statistical methods (histograms, wavelet thresholding, regression) under a common model selection principle based on complexity and approximation.