## The Annals of Statistics

### Gaussian model selection with an unknown variance

#### Abstract

Let Y be a Gaussian vector whose components are independent with a common unknown variance. We consider the problem of estimating the mean μ of Y by model selection. More precisely, we start with a collection $\mathcal{S}=\{S_{m},m\in\mathcal{M}\}$ of linear subspaces of ℝn and associate to each of these the least-squares estimator of μ on Sm. Then, we use a data driven penalized criterion in order to select one estimator among these. Our first objective is to analyze the performance of estimators associated to classical criteria such as FPE, AIC, BIC and AMDL. Our second objective is to propose better penalties that are versatile enough to take into account both the complexity of the collection $\mathcal{S}$ and the sample size. Then we apply those to solve various statistical problems such as variable selection, change point detections and signal estimation among others. Our results are based on a nonasymptotic risk bound with respect to the Euclidean loss for the selected estimator. Some analogous results are also established for the Kullback loss.

#### Article information

Source
Ann. Statist., Volume 37, Number 2 (2009), 630-672.

Dates
First available in Project Euclid: 10 March 2009

https://projecteuclid.org/euclid.aos/1236693145

Digital Object Identifier
doi:10.1214/07-AOS573

Mathematical Reviews number (MathSciNet)
MR2502646

Zentralblatt MATH identifier
1162.62051

Subjects
Primary: 62G08: Nonparametric regression

#### Citation

Baraud, Yannick; Giraud, Christophe; Huet, Sylvie. Gaussian model selection with an unknown variance. Ann. Statist. 37 (2009), no. 2, 630--672. doi:10.1214/07-AOS573. https://projecteuclid.org/euclid.aos/1236693145

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