Journal of Statistical Planning and Inference 137 (2007) 1243 – 1259
www.elsevier.com/locate/jspi
On the detection of changes in autoregressive time series I.
Asymptotics
Marie Hušková
a
, Zuzana Prášková
a
, Josef Steinebach
b, ∗
a
Charles University in Prague, Department of Statistics, Sokolovská 83, CZ–186 75 Praha 8, Czech Republic
b
Universität zu Köln, Mathematisches Institut,Weyertal 86-90, D-50931 Köln, Germany
Received 9 September 2005; received in revised form 15 January 2006; accepted 6 February 2006
Available online 3 May 2006
Abstract
Various test statistics are discussed which can be used for detecting changes in the parameters of an autoregressive time series. In
this first part of our study, the limiting behavior of the test statistics is derived under the null hypothesis of no change as well as under
alternatives. In a forthcoming second part of our investigation, these asymptotic results will be compared to some corresponding
bootstrap procedures, and a small simulation study will be conducted.
© 2006 Elsevier B.V.All rights reserved.
MSC: Primary 62G20; secondary 60F17; 62M10
Keywords: Change point analysis; Time series; Extreme value asymptotics
1. Introduction
We consider the time series model with a change after an unknown time point m, i.e.,
Y
i
= Y
i,n
= x
T
i
+ x
T
i
I {i>m}+ e
i
, i = p + 1,...,n, (1.1)
where
x
i
= x
i,n
= (Y
i -1,n
,...,Y
i -p,n
)
T
, i = p + 1,...,n, j = 1,...,p,
(p <) m = m
n
( n), = (
1
,...,
p
)
T
and =
n
= (
1n
,...,
pn
)
T
= 0 are unknown parameters, and e
1
,...,e
n
are
independent identically distributed (i.i.d.) random errors having a positive variance Var(e
1
) =
2
and satisfying further
conditions specified below.
The function I {A} in (1.1) denotes the indicator of the set A. For the sake of convenience, we suppress the index
n in the observations Y
i,n
as well as in the parameters m
n
and
n
(and in variables depending on the latter) whenever
possible, but keep in mind that in the limiting results below, as n →∞, both m
n
and
n
may be changing when n is
increasing.
∗
Corresponding author.
E-mail addresses: huskova@karlin.mff.cuni.cz (M. Hušková), praskova@karlin.mff.cuni.cz (Z. Prášková), jost@math.uni-koeln.de
(J. Steinebach).
0378-3758/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jspi.2006.02.010