Detection and Estimation of Jumps in Regression Curves
and Surfaces

Professor Anirban DasGupta 

Abstract

In the third lecture of this series, first the two dimensional case
will be summarized. The two dimensional case is motivated by the
problem of image restoration. One can view it as the problem of 
estimating
an image intensity function, which changes in certain ways on jump
location curves. The work appears to fall into one of two categories,
namely, methods for detecting or progressive tracking of the JLC,
and second, writing formal models for the intensity function
and propose estimation methods which have some sort of an asymptotic
property, such as uniform consistency in some class, or asymptotic
minimaxity in some class. The main work on the tracking aspect is
due to Peter Hall, Peihua Qiu, and their respective coauthors.

We then begin to address the testing and confidence interval problems.
These appear to be the theoretically hardest aspects in the entire area.
Typically, the null distribution of the likelihood ratio statistic
for a change is that of the supremum of a suitable random field.
The methods to approximate the tail areas, that is the P-values,
involve extremely sophisticated use of the theory of random walks,
and use of equally sophisticated methods of geometry, but not plain
geometry. The main work on these aspects is due to David Siegmund
and his numerous coauthors. The area of distributions of suprema
of random fields is currently undergoing rapid progress, and
is deserving of an independent seminar series.

The connection of the likelihood ratio tests to the suprema of random
walks was understood even in the change point problems where there is
no regression setup, i.e., there are no covariates. The regression case
is easier to understand with a preliminary exposure to the nonregression
change point problem. Therefore, most of the likelihood ratio theory
for the regression setup will be deferred to the fourth lecture, while
the classical likelihood ratio theory for the canonical nonregression
case will be reviewed in this lecture. An informative survey by
P.K. Bhattacharya is in the IMS volume on change point problems.