Edge Preserving Smoothing by Local Mixture Modelling

Smooth models became more and more popular over the last couple of
years. Standard smoothing methods however can not cope with
discontinuities in a function or its first derivative. In
particular, this implies that structural changes in data may be
hidden in smooth estimates. Recently, Chu, Glad, Godtliebsen &
Marron (1998) suggest local M estimation as edge preserving
smoother. The basic idea behind local M estimation is that
observations beyond a jump are considered as outliers and
down-weighted or neglected in the estimation. We pursue a
different, but related idea here and treat observations beyond a
jump as tracing from a different population which differs from the
current one by a shift in the mean. This means we impose locally a
mixture model where mixing takes place due to different mean
values. For fitting we apply a local version of the EM algorithm.
The advantage of our approach shows in its general formulation. In
particular, it easily extends to non Gaussian data. The procedure
is applied in two examples, the first concerning the analysis of
structural changes in the duration of unemployment, the second
focusing on disease mapping.