May 5

12:30 pm

## A Sharp Multiplier Inequality with Applications to Heavy-Tailed Regression Problems

### Roy Han

General Exam

University of Washington - Statistics

Advisor:Professor Jon A. Wellner

We develop a sharp multiplier inequality used to study the size of the multiplier empirical process $(sum_{i=1}^n xi_i f(X_i))_{f in mathcal{F}}$, where $xi_i$'s and $mathcal{F}$ are multipliers and an indexing function class respectively. We show that in general the size of the suprema of the multiplier empirical process is determined jointly by the growth order of the corresponding empirical process, and the worst size of the maxima of the multipliers.

The new inequality applies to shed light on the long-standing open question concerning the rate behavior of classical ERM procedures in the regression setting with heavy-tailed errors. In particular, the convergence rate of the least squares estimator in a general non-parametric regression model, with usual entropy conditions on the function classes, is shown to be no worse than the maximum of the optimal Gaussian regression rate and the noise level of the errors. We also demonstrate applications in the sparse linear model where both design and errors can be heavy-tailed.