# 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.
Keywords :
Document type :
Journal articles
Domain :

https://hal.archives-ouvertes.fr/hal-00756074
Contributor : Yannick Baraud <>
Submitted on : Thursday, November 22, 2012 - 2:34:21 PM
Last modification on : Monday, October 12, 2020 - 10:27:29 AM

### Identifiers

• HAL Id : hal-00756074, version 1

### Citation

Yannick Baraud, Christophe Giraud, Sylvie Huet. GAUSSIAN MODEL SELECTION WITH AN UNKNOWN VARIANCE. Annals of Statistics, Institute of Mathematical Statistics, 2009, 37 (2), pp.630-672. ⟨hal-00756074⟩

Record views