May 2024 Estimating a regression function in exponential families by model selection
Juntong Chen
Author Affiliations +
Bernoulli 30(2): 1669-1693 (May 2024). DOI: 10.3150/23-BEJ1649


Let (W1,Y1),,(Wn,Yn) be n pairs of independent random variables. We assume that, for each i{1,,n}, the conditional distribution of Yi given Wi belongs to a one-parameter exponential family with parameter γ(Wi)R. The statistical goal is to estimate these conditional distributions. We consider a model selection procedure which works based on a general assumption that each of the model is VC-subgraph. We establish a non-asymptotic risk bound for the resulting estimator with respect to a Hellinger-type distance. By leveraging this result, we extend several findings previously explored in Gaussian regression to the regression in exponential families. Specifically, we address the curse of dimensionality by imposing structural assumptions, such as general additive and multiple index structures, on γ. We also study model selection for ReLU neural networks, and provide a concrete example of how ReLU neural networks can achieve a significantly faster convergence rate than traditional models. When γ is close to a composition of several Hölder functions, we show that under a suitable parametrization of the exponential family, our estimator achieves the same rate of convergence as in the Gaussian case. Combining with a lower bound, the rate is minimax optimal up to a logarithmic term. Finally, we apply the model selection procedure to address adaptation and variable selection problems in exponential families.

Funding Statement

The author was supported by European Union’s Horizon 2020 research and innovation program under grant agreement No 811017.


The author is grateful to her supervisor Prof. Yannick Baraud for helpful discussions and constructive suggestions. The author also thanks the referees and the editors for their suggestions and comments, which have contributed to the improvement of this paper.


Download Citation

Juntong Chen. "Estimating a regression function in exponential families by model selection." Bernoulli 30 (2) 1669 - 1693, May 2024.


Received: 1 April 2022; Published: May 2024
First available in Project Euclid: 31 January 2024

MathSciNet: MR4699568
Digital Object Identifier: 10.3150/23-BEJ1649

Keywords: Generalized additive structure , Model selection , multiple index structure , regression in exponential family , ReLU neural networks , Variable selection


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Vol.30 • No. 2 • May 2024
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