Detection and Estimation of Jumps in Regression Curves
and Surfaces

Professor Anirban DasGupta

Abstract

In the fourth and the final lecture of this series, we touch on four
themes. First, the first order extreme value theory for the null
distribution of parametric likelihood ratio statistics will be
described. This is one possible approach for approximating the
P-values, popularized by the Hungarian school. A second approach is to
treat it as a problem of large deviations for the maxima of suitable
random fields. 
This is the approach popularized by David Siegmund and his coauthors.

Distribution theory being so complicated, approximation of the P-values
by using Monte Carlo approximations to the critical values, obtained via
randomization tests or the bootstrap, becomes a natural idea. In
certain parametric models, the randomization tests and the bootstrap
are known to be strongly consistent. The results extend to the regression
setup, and some time series models.

The talk concludes with some proposals for broad open problems in
the area. One must be sensitive to the huge past literature in
the general area, and seek out problems that are still worthwhile.
One such problem is to find out how well can one detect and
estimate multiple changes, and what are the provable consistency
properties. There is also scope for worthwhile numerical work.
For example, one can take a few interesting models, and try to
find out which of the four methods, namely the first order extreme
value theory, the large deviation method, randomization tests,
and the bootstrap, give the most accurate final answers. Quite
possibly, no general preferences exist, although it has not been
studied.