## Electronic Journal of Statistics

### Posterior concentration rates for mixtures of normals in random design regression

#### Abstract

Previous works on location and location-scale mixtures of normals have shown different upper bounds on the posterior rates of contraction, either in a density estimation context or in nonlinear regression. In both cases, the observations were assumed not too spread by considering either the true density has light tails or the regression function has compact support. It has been conjectured that in a situation where the data are diffuse, location-scale mixtures may benefit from allowing a spatially varying order of approximation. Here we test the argument on the mean regression with normal errors and random design model. Although we cannot invalidate the conjecture due to the lack of lower bound, we find slower upper bounds for location-scale mixtures, even under heavy tails assumptions on the design distribution. However, the proofs suggest to introduce hybrid location-scale mixtures for which faster upper bounds are derived. Finally, we show that all tails assumptions on the design distribution can be released at the price of making the prior distribution covariate dependent.

#### Article information

Source
Electron. J. Statist. Volume 11, Number 2 (2017), 4065-4102.

Dates
First available in Project Euclid: 24 October 2017

https://projecteuclid.org/euclid.ejs/1508810899

Digital Object Identifier
doi:10.1214/17-EJS1344

Zentralblatt MATH identifier
06805087

Subjects
Primary: 62G20: Asymptotic properties
Secondary: 62G08: Nonparametric regression

#### Citation

Naulet, Zacharie; Rousseau, Judith. Posterior concentration rates for mixtures of normals in random design regression. Electron. J. Statist. 11 (2017), no. 2, 4065--4102. doi:10.1214/17-EJS1344. https://projecteuclid.org/euclid.ejs/1508810899

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