Theoretical Statistics and Mathematics Unit, ISI 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.