Matstat seminarium torsdagen 1 november 2001 kl 13:15 i MH:227

Christian Rau, Centre for Mathematics and Its Applications,
The Australian National University, Canberra

Likelihood-based confidence bands for fault line estimators

ABSTRACT: 
The problem of estimating a smooth fault line in a bivariate
regression, or density surface, is of considerable importance 
in various applications, such as computerised edge detection or 
the biological and geological sciences. In this talk, we study
properties of a new estimator for a fault line in both settings. 
This estimator is constructed by maximising a likelihood in a 
locally-linear model for the edge. The approach offers a unifying 
thread to the two problems, which are usually considered separately 
from each other. The convergence rate of the estimator comes within 
the known minimax-optimal convergence rate by an arbitrarily small power
of the design intensity. Our main focus is on investigation of the 
local behaviour of this estimator, through which we obtain asymptotic 
confidence bands for the fault line, both pointwise and simultaneous.
The pointwise distance between the fault line and its bias-corrected
estimator has a distribution which equals that of the location of the 
maximum of a Gaussian process with quadratic drift, and thus resembles
a commonly encountered limit of $M$-estimators. A simulation study
on artificial data illustrates finite-sample performance of the method.