Least squares estimation in the monotone single index model

By Fadoua Balabdaoui, Cécile Durot, Hanna K. Jankowski. Bernoulli, 2016

50 citations recorded by the source corpus.

We study the monotone single index model where a real response variable $Y $ is linked to a $d$-dimensional covariate $X$ through the relationship $E[Y | X] = \Psi_0(\alpha^T_0 X)$ almost surely. Both the ridge function, $\Psi_0$, and the index parameter, $\alpha_0$, are unknown and the ridge function is assumed to be monotone on its interval of support. Under some regularity conditions, without imposing a particular distribution on the regression error, we show the $n^{-1/3}$ rate of convergence in the $\ell_2$-norm for the least squares estimator of the bundled function $\psi_0({\alpha}^T_0 \cdot),$ and also that of the ridge function and the index separately. Furthermore, we show that the least squares estimator is nearly parametrically rate-adaptive to piecewise constant ridge functions.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/2b65cc8aa14dd44909d0075f55b0aeb74991fbfc

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In Statistics and Probability

Citations in the corpus

Nonparametric Shape-Restricted Regression (2017, background): “[9] studied the LSE in a monotone single index model (see (49)) and showed the n1/3-consistency of the LSEs of m and β∗; also see [106] and [57].”
Score estimation in the monotone single‐index model (2017, methodology): “Following similar arguments, as in the work of Balabdaoui et al. (2016), in the proof of Proposition 2 given in the Supplementary Material, we can easily show that sub-Gaussianity of X implies that”