###
CHOOSING A LINEAR MODEL WITH A RANDOM NUMBER

OF CHANGE-POINTS AND OUTLIERS

###
HENRI CAUSSINUS AND FAOUZI LYAZRHI

*Laboratoire de Statistique et Probabilités, UMR-CNRS 5583, Université Paul Sabatier,*

118, Route de Narbonne, 31062 Toulouse Cedex, France
(Received July 27, 1995; revised July 30, 1996)

**Abstract.**
The problem of determining a
normal linear model with possible perturbations,
viz. change-points and outliers, is formulated as a
problem of testing multiple hypotheses, and a Bayes
invariant optimal multi-decision procedure is
provided for detecting at most *k* (*k* > 1) such
perturbations. The asymptotic form of the procedure
is a penalized log-likelihood procedure which does
not depend on the loss function nor on the prior
distribution of the shifts under fairly mild
assumptions. The term which penalizes too large a
number of changes (or outliers) arises mainly from
realistic assumptions about their occurrence. It is
different from the term which appears in Akaike's or
Schwarz' criteria, although it is of the same order
as the latter. Some concrete numerical examples are
analyzed.

*Key words and phrases*:
Akaike's
criterion, Bayes decision procedure, change-point,
invariance, maximal invariant, outliers, regression
analysis, Schwarz' criterion.

**Source**
( TeX ,
DVI ,
PS )