Generalized Deep Regression for Repeated Measurements
TL;DR
This paper studies the estimation of a marginal regression function from independent units with repeated binary, count, or continuous responses using ReLU deep neural networks, and derives an oracle inequality for unequal cluster sizes and a rate for compositionally smooth functions.
Abstract
In this paper, we study the estimation of a marginal regression function from independent units with repeated binary, count, or continuous responses using ReLU deep neural networks. In the model, we assume that the dependence is generated by an unobserved random mean function within each unit. We then fit a neural network with a convex generalized regression loss. We show an oracle inequality by separating conditional measurement variation from between-unit variation. In addition, we prove that with $n$ units and $m$ measurements per unit, ReLU networks can attain an integrated mean squared error of order $n^{-1}+(nm)^{-2\beta/(2\beta+d)}$, up to logarithmic factors, over $\beta$-H\"older classes. We also derive a weighted oracle inequality for unequal cluster sizes and a rate for compositionally smooth functions. For pointwise ensemble inference, we give a projection central limit theorem and prove infinitesimal jackknife consistency under an explicit asymptotic linearity condition. Simulations and real data examples are provided to support our theoretical findings and practical implications.
