An estimator of the number of change points based on a

2
Example 1.3 (Linear processes) Consider fZn(i) : 1 i ng as in (1.1), but with bj = j A,
and
nj Yn(j) (i) := Pk=?nj+i +1 ek;n=( j A) : 1 i nj ? nj?1 ; ? where fau : u = 0; 1; : : :g is a real sequence such that
(1) ( 1 Zn(i) := a1i(+ b1 Yn+ ai)(i ? n ) + b Y (j) (i) :: n i< in1 n ; j = 2; : : : ; l + 1 (1.1) Zn nj?1) j j ?1 j n j ?1 j where aj 6= aj?1, bj and nj = j n], 0 = 0 < 1 < : : : < l < l+1 = 1, are unknown parameters, and where Yn(j) (i) i=1;:::;nj ?nj? are unobservable stochastic sequences satisfying a (uniform) weak invariance principle. Namely, we assume that, for every n 1 ; there exist l + 1 independent Wiener processes fWn(j)(t) : 0 t nj ? nj?1g, j = 1; : : : ; l + ary after re-de nition on a suitable probability space, 1
1 i maxnj?1 nj ?
Yn(j)(i) ? Wn(j)(i) = OP (n ) (n ! 1)
(1.2)
with some < 1=2 : That means we have a piecewise linear function with an additive measurement noise. Various statistical models satisfy condition (1.2) above as will be demonstrated by a few examples:
Then (1.2), with = 1=(2 + ) ; follows directly from Example 1.1 in combination with Theorem 3.1 of Csorg}, Horvath and Steinebach (1987). o Motivated by a change-point analysis of time series, the following generalization of Example 1.1 is of particular interest. It also demonstrates that, although the increments of the observed process fZn(i) : 1 i ng may have a special dependency structure before and after ( the change-points nj ; the approximating Wiener processes fWnj)(t) : 0 t nj ? nj?1g, j=1,. . . ,l+1, can be chosen to be independent.
An estimator of the number of change points based on a weak invariance principle
Christoph Kuhna
a
Center for Mathematical Sciences, Munich University of Technology, D-80290 Munich, Germany e-mail: kuehn@mathematik.tu-muenchen.de, fax: +49 89 289 28464
( 2 Var("k;n) = 1 : k n1 ; k n ; j = 2; : : : ; l + 1; 2 j j : nj ?1 < > 0, and E j"k;nj2+ < 1 (for some exponent > 3=2 so that
j
2
> 0) : We further assume that there exists an
1 1
A=
Let
1 X
u=0
au 6= 0 : au "k?u;n ;
ek;n =
1 X
u=0
where f"k;n : k = 0; 1; 2; : : : ; k n1 g and f"k;n : k = 0; : : : ; n; nj?1 < k nj g, j=2,. . . ,l+1 are l + 1 independent sequences, each i.i.d. (white noise), with E"k;n = 0 ;
Example 1.1 (Partial sums) For every n
(j 1 ; let fXi;n) : i = 1; 2; : : :g; j = 1; : : : ; l + 1, be l + 1 independent sequences of i.i.d. (independent, identically distributed) random variables (j ( ( ( with P fXi;n) xg = P fX1j1) xg for all x 2 IR ; n 1 ; EX1;j1) = j , V ar(X1;j1) ) = j2 > 0 ; ; being xed. For some 0 = 0 < 1 < : : : < l+1 = 1 ; consider fZn(i) : 1 i ng as in (1.1) with aj = j , bj = j , and (j ) Yn(j) (i) := Pik=1(Xk;n ? j )=
( with P (X1;j1)
0) = 1. Set aj = 1= j ; bj =
j
= 3=2; j
8 9 < X (j) = ( Nnj) (t) := max :i 0 : Xk;n t; ; 0 t < 1 ; 1 k i
(j ) t= Yn(j) (t) := Nn (t) ?=2 j ; 1 i nj ? nj?1; j = 1; : : : ; l + 1 : 3 j= j
Abstract
1 Introduction
The aim of this paper is to discuss the problem of estimating the number of change points in a general context. It extends work of Yao (1988) and Lee (1995) concerning only sequences of independent normal random variables. In particular, we combine an auxiliary result of Yao (1988) for i.i.d.-normal random variables with an invariance principle for partial sums to improve a later result of Yao and Au (1989) where an i.i.d.-sequence with a more restrictive moment condition was considered. As in many other asymptotic studies of change-point problems it turns out that certain invariance principles are very useful and allow for reducing the statistical analysis to that of an asymptotic Gaussian model. Recently, Horvath and Steinebach (1999) pursued this idea in a general vein by simply taking advantage of a weak invariance principle in the testing of a change in the mean or variance of a rather general stochastic process. In particular, cases of dependent observations are included in this framework. Assume that we observe a stochastic sequence (Zn(i))i=1;:::;n having the following structure:
(1.3)
au = O u?
(u ! 1) :
If, in addition, the "k;n's have smooth densities and fau : u = 0; 1; : : :g satis es some regularity conditions, then the invariance principle (1.2), with = 1=(2 + ) follows from Horvath (1997). For details and exact conditions confer Lemmas 2.1 and 2.2 of Horvath (1997). We need an additional technical condition: For all j = 1; : : : ; l + 1 and " > 0 there exist n; C0; C1 > 0 such that for all n n: ~ ~
j
: 1 i nj ? nj?1; j = 1; : : : ; l + 1 :
( If, in addition, E jX1j1) j2+ < 1, for some > 0, then the Komlos, Major and Tusnady ; (1975) strong approximation implies (1.2) for fYn(j) (i) : 1 i nj ? nj?1 g, with = 1=(2+ ) : (j Example 1.2 (Renewal Processes) Let the random variables Xi;n) be as in Example 1.1, but
We study an estimator of the number of change points in the drift of a stochastic process based on the Schwarz criterion. In a general statistical model where the additive measurement noise satis es a certain weak invariance principle (examples included are partial sums, renewal processes, and linear processes in time series analysis) consistency can be shown under the condition that the number of jumps is not greater than a given upper bound. AMS subject classi cation: 62F12; 60F17. Keywords: Change points; Change in the mean; Schwarz criterion; Invariance principle; Partial sum; Renewal process; Linear process.