# Seminar at SMU Delhi

October 4, 2012 (Thursday) , 3:30 PM at Webinar [Note unusual date]
Speaker: Subhashis Ghosal, North Carolina State University
Title: Improving converge rates for estimating location and size of maximum of a nonparameteric regression function using a two-stage sampling procedure
Abstract of Talk
We consider the problem of estimating the location and the size of the maximum of a smooth multivariate regression function. For an $\alpha$-smooth regression function $f$, the standard rate for estimating the location $\mu$ and the size $f(\mu)$ are $n^{-(\alpha-1)/(2\alpha+d)}$ and $n^{-\alpha/(2\alpha+d)}$, where $d$ stands for the dimension. We propose using a two-stage procedure to improve these rates. We first use a part of the sampling budget and a preliminary estimator of $\mu$ to identify a small region near the preliminary estimator where the remaining sampling budget is to be used.to get more observations. We fit an appropriate polynomial regression model to estimate $\mu$ and $f(\mu)$ in the second stage. We establish that the first stage rates can be improved to $n^{-(\alpha-1)/(2\alpha)}$ and $n^{-1/2}$ respectively for $\alpha>1+\sqrt{1+d/2}$. These rates are optimal in the class of all possible sequential estimators, although the two-stage procedure is much simpler to implement than a fully sequential Robbins-Monro type procedure. Interestingly, the two-stage procedure resolves the curse of dimensionality problem to some extent, as the dimension $d$ does not control the second stage convergence rates, provided that the function class is sufficiently smooth. We further develop a multi-stage generalization that attains the optimal rate for any smoothness level $\alpha>2$ starting with a preliminary estimator with any power-law rate at the first stage. We also show by simulation that the two-stage procedure has much better accuracy than a standard procedure. The technique is potentially useful in an oil exploration problem. This talk is based on joint work with Eduard Belitser and Harry van Zanten.