- Email: [email protected]
Limit theorems for a class of tests of gradual changes
Journal of Statistical Planning and Inference 89 (2000) 57–77
www.elsevier.com/locate/jspi
Limit theorems for a class of tests of gradual changes a Department
b FB
Marie Huskovaa; ∗ , Josef Steinebachb
of Statistics, Charles University, Sokolovska 83, CZ-18600 Praha, Czech Republic Mathematik & Informatik, Philipps-Universitat, Hans-Meerwein-Str., D-35032 Marburg, F.R. Germany
Received 22 June 1999; received in revised form 21 December 1999; accepted 11 January 2000
Abstract We study a class of asymptotic tests for detecting various types of gradual changes in a location model. The results of Jaruskova (1998. J. Statist. Plann. Inference 70, 263–276) and Huskova (1998. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability and Statistics. Elsevier, Amsterdam, pp. 577–584.) are extended to other families of change alternatives. Under the null hypothesis of “no change”, limiting extreme value distributions of Gumbel type are derived for some asymptotic log-likelihood ratio test statistics. Consistency under certain alternatives is also c 2000 Elsevier Science B.V. All rights reserved. discussed. MSC: 62 F 05; 62 J 05 Keywords: Change-point test; Linear and nonlinear regression; Gradual change; Limiting extreme value distribution; Weighted embedding
1. Introduction Consider the following location model with a gradual change after an unknown time point m: i−m + ei ; i = 1; : : : ; n; (1.1) Yi = + n + where a+ = max(0; a); ; = n ; m = mn (6n) are unknown parameters, and e1 ; : : : ; en are i.i.d. random variables with Eei =0; var ei =2 ¿ 0, and E|ei |2+ ¡ ∞; i =1; : : : ; n; for some ¿ 0. The parameter ¿ 0 is supposed to be known. We are interested in testing the hypotheses H0 : m = n
vs:
H1 : m ¡ n; 6= 0;
This work was supported by the grant GACR-201=97=1163 and CES:J13=98:113200008. Corresponding author. E-mail addresses: [email protected] .cuni.cz (M. Huskova), [email protected] (J. Steinebach).
∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 0 9 4 - X
(1.2)
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or the one-sided case H0 : m = n
H+ 1 : m ¡ n; ¿ 0:
vs:
(1.3)
Most of the testing procedures suggested so far deal with models of “abrupt” changes i.e. = 0 (cf. e.g. the monograph of Csorg˝o and Horvath, 1997). Jaruskova (1998) discussed the case of = 1 and derived some limiting extreme value distributions of Gumbel type for test statistics which are equivalent to the log-likelihood ratio test statistic if e1 ; : : : ; en are normally distributed. Huskova (1998), also for =1, proposed a new class of testing procedures by making use of heavier weights in the normalization. This results in limiting weighted sup-norm distributions, under H0 , and asymptotic normality, respectively, under certain local alternatives. Moreover, for ∈ (0; 1); Huskova (1999) discussed the problem of estimating the change-point m. A Bayesian type of statistics for polynomial changes in regression models has been studied by Jandhyala and MacNeill (1997). Here we study a class of test procedures based on the partial sums of weighted residuals: n P (1.4) Sk = (xik − xk )Yi ; k = 1; : : : ; n; i=1
where
xik = xik ( ; n) =
i−k n
+
n 1P xik ; xk = xk ( ; n) = n i=1
;
i; k = 1; : : : ; n; (1.5)
k = 1; : : : ; n:
The test statistic we are interested in is as follows: |Sk | 1 Pn ; max T n = ˆn 16k6n−1 ( i=1 (xik − xk )2 )1=2
(1.6)
where ˆn is an estimator of with the property ˆn − = oP ((log log n)−1 )
as n → ∞:
(1.7)
Here and in the sequel, log x = ln max(x; 1). Note that T n as given in (1.6) is based on an asymptotic log-likelihood ratio approach. Namely, it will turn out that the limiting extreme value distribution of T n coincides with that of its corresponding counterpart Tn in case and 2 are known, e.g. = 0 and 2 = 1. That is, it will be enough to discuss the asymptotics of Tn =
max
16k6n−1
|Sk | 2 1=2 i=1 xik )
(
(1.8)
k = 1; : : : ; n
(1.9)
Pn
with Sk =
n P i=1
xik Yi ;
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59
which is equivalent to studying the log-likelihood ratio statistic for (1.1) and (1.2) if m is unknown and the {ei } are i.i.d. N(0; 1) normal random variables. This idea is well known and has also been pursued for the testing of possible changes in the trend of a regression model. Confer the “direct test” of Chu and White (1992), in which design points similar to (1.5) (with ¿1) are assumed (see also Csorg˝o and Horvath, 1997, Chapter 3:2:1). The resulting null distributions, however, are di erent from our main results (compare Theorems 3:2:1–3:2:3 in Csorg˝o and Horvath (1997) and Theorems 3.1–3.3 in Section 3 below). The paper is organized as follows. Section 2 contains some weak embeddings for the weighted partial sum processes n P [nt] − i Zi ; t ∈ [0; 1] (1.10) Sn (t) = g n i=1 and Sn (t) =
n P i=1
g
[nt] − i n
(Zi − Z n );
t ∈ [0; 1];
(1.11)
where g = g(y); y ∈ R; is a weight function, and {Zi }i=1; 2; ::: are i.i.d. with E Z1 = Pn 0; var Z1 = 1, and E|Z1 |2+ ¡ ∞ for some ¿ 0; Z n = (1=n) i=1 Zi . These weak invariance principles hold true with a rather general weight function g, and thus may be of independent interest. , to derive our main In Section 3, the above embeddings are applied, with g(y) = y+ results on the limiting distributions for the statistics Tn and T n of (1.8) and (1.6). There a key observation is that the limit behaviour of Tn under H0 is given by the exceedance probability of the limit process ) ( R1 (2 + 1)1=2 t (y − t) dW (y) : 06t ¡ 1 ; (1.12) (1 − t) +1=2 where {W (y): y¿0} denotes a standard Wiener process. The Gaussian process in (1.12) can be transformed into a stationary process, so that the results on the exceedence probability of a stationary Gaussian process on an increasing time interval can be applied. As a consequence, one gets limiting extreme value distributions of Gumbel type under H0 . It turns out that these asymptotics are di erent according to the three cases ¿ 12 ; = 12 , and 0 ¡ ¡ 12 . The limiting null distributions of {T n } are the same as those for {Tn } and can, in fact, be derived from the latter. Finally, in Section 4, we brie y discuss consistency of the proposed testing procedures. In a concluding remark, we mention some possible extensions of our main results which will be discussed elsewhere. 2. Embeddings of weighted partial sum processes Here we prove some weighted embeddings for the processes {Sn (t): t ∈ [0; 1]} and {S n (t): t ∈ [0; 1]} as de ned in (1.10) and (1.11) of Section 1. The weight function
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g = g(y); y ∈ R; is assumed to satisfy the following assumptions: g is nondecreasing; square integrable and absolutely continuous on [0; 1] with derivative g0 ; g(y) = 0 (y60); g(y) ¿ 0 (y ¿ 0); lim sup R t t↓0
0
tg2 (t) g2 (y) dy
¡ ∞:
(2.1) (2.2)
Sometimes, assumption (2:2) will be replaced by tg(t) ¡ ∞: lim sup R t g(y) dy t↓0 0
(2.3)
Remark 2.1. (a) It is an easy consequence of Jensen’s inequality that, under (2.1), assumption (2:3) implies (2:2). Note that, for any 0 ¡ t61; Z t 2 Z 1 1 t 2 g (y) dy ¿ g(y) dy ; (2.4) t 0 t 0 wherein strict inequality follows from the fact that g cannot be constant in view of assumption (2:1). (b) Note that assumptions (2:2) and (2:3) are satis ed if g is regularly varying at 0 with index ¿ 0. Our main results in Section 3 will be based on the following weighted embedding of partial sums: Theorem 2.1. Let {Zi }1=1; 2::: be i.i.d. r.v.’s with EZ1 =0; var Z1 =1; and E|Z1 |2+ ¡ ∞ for some ¿ 0. Let the weight function g = g(y) satisfy conditions (2:1) and (2:2). Then there exists a sequence of Wiener processes {Wn (t): t ∈ [0; 1]}; n=1; 2; : : : ; such that as n → ∞; P Rt 0 n−1 g (y)W (t − y) dy g(([nt] − i)=n)Z n i i=1 − 0 Rt n sup t Pn−1 = OP (1) (2.5) 2 1=2 2 1=2 ( 0 g (y) dy) ( i=1 g (([nt] − i)=n) 1=n6t61 for any 06 6 12 − 1=(2 + ). Moreover; Z t 1 n−1 P [nt] − i 0 √ Zi − g g (y)Wn (t − y) dy sup n n i=1 06t61 0 =oP (n1=(2+)−1=2 ):
(2.6)
Remark 2.2. Using integration by parts (cf. e.g. Arnold (1973, (4.5.12) on p. 92) or Karatzas and Shreve (1988, (3.8) on p. 155)), we have, for t ∈ [0; 1]; Z t Z t g(t − y) dWn (y) = [g(t − y)Wn (y)]t0 + g0 (t − y)Wn (y) dy 0
0
Z =
t 0
0
g (t − y)Wn (y) dy =
Z 0
t
g0 (y)Wn (t − y) dy:
(2.7)
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
61
Remark 2.3. Note that the process {V (t): t ∈ [0; 1]} with Z V (t) =
t
0
g(t − y) dW (y);
(2.8)
where {W (y): y ∈ [0; 1]} is a Weiner process, de nes a zero mean Gaussian process with Z s∧t g(s − y)g(t − y) dy; (2.9) cov(V (s); V (t)) = 0
where s ∧ t = min(s; t); 06s; t61. Moreover, as n → ∞; |V (t)| = OP ((log log n)1=2 ): 1=2 1=n6t61 (var V (t)) sup
(2.10)
; y ∈ R; satis es conditions (2.1) – (2.3) for any Remark 2.4. The function g(y) = y+ ¿ 0.
Proof of Theorem 2.1. It is an immediate consequence of the Komlos et al. (1975,1976) strong approximations that on a suitable probability space there exists a Wiener process {W (y): y¿0} such that, as n → ∞, 1=(2+) k P 1 Zi − W (k) = OP (1); 16k6n k i=1
(2.11)
k P max Zi − W (k) = oP (n1=(2+) ): 16k6n
(2.12)
max
i=1
Also, by Theorem 1:2:1 of Csorg˝o and Revesz (1981), as T → ∞, for any Wiener process {W (y): y¿0}, sup
sup |W (t + y) − W (t)| = O((log T )1=2 ) a:s:
06t6T 06y61
(2.13)
If not mentioned otherwise the estimations below hold for n → ∞. A brief calculation shows that k−i n−1 k−1 P P P i−1 k −i i Zi = −g g Zj g n n n i=1 i=1 j=1 =
i−1 i −g W (k − i) g n n i=1 k−1 P i−1 i 1=(2+) −g (k − i) g + OP n n i=1 k−1 P
(2.14)
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uniformly in k = 1; : : : ; n (by (2.11)). Now, for k = [nt], i−1 i −g W (k − i) g n n i=1
k−1 P
=
k−1 P
Z
i=1
Z =
i=n
(i−1)=n
(k−1)=n
0
g0 (y)(W (nt − ny) + W ([nt] − i) − W (nt − ny)) dy
k −1 (log k)1=2 ; g0 (y)W (n(t − y)) dy + OP g n
(2.15)
uniformly in k =1; : : : ; n (by (2.13)). On combining (2.14) and (2.15), we get uniformly in t ∈ [1=n; 1];
n−1 P
g
i=1
[nt] − i n
Z =
Zi
([nt]−1)=n
0
[nt] − 1 1=(2+) (nt) : g (y)W (n(t − y)) dy + OP g n 0
(2.16)
Next we check that, uniformly in t ∈ [1=n; 1], Z
([nt]−1)=n 0
g0 (y)W (n(t − y)) dy =
Z
t
0
g0 (y)W (n(t − y)) dy + OP (g(t))
(2.17)
and P 2 1 [nt]−1 g n i=1
Z t 1 i = : g2 (y) dy + O g2 (t) n n 0
(2.18)
For the proof of (2.17), note that Z 06
t
Z
0
([nt]−1)=n
g (y)|W (n(t − y))| dy6
t
([nt]−1)=n
g0 (y) sup |W (x)| dy = OP (g(t)): x∈[0;2]
On the other hand, P 2 1 [nt]−1 g n i=1 =
=
Z t i − g2 (y) dy n 0
[nt]−1 P i=1 [nt]−1 P i=1
Z
i=n
(i−1)=n
Z
i=n
(i−1)=n
g2
i 1 − g2 (y) dy + O g2 (t) n n
Z
i=n
y
! 0
2g (x)g(x) d x
1 dy + O g2 (t) n
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
=
Z
[nt]−1 P i=1
1 6 n
Z
i=n
(i−1)=n t
0
2g0 (x)g(x)
Z
x
(i−1)=n
1 2g (x)g(x) d x + O g (t) n 0
dy
2
63
1 d x + O g2 (t) n
1 = O g (t) n 2
:
√ A combination of (2.14) – (2.18) yields, uniformly in t ∈ [1=n; 1]; with Wn (y) = (1= n) W (ny) Rt 0 Pn−1 g (y)Wn (t − y) dy i=1 g(([nt] − i)=n)Zi − 0 Rt Pn−1 2 ( 0 g2 (y) dy)1=2 ( i=1 g (([nt] − i)=n)1=2 =
n−1=2 P[nt]−1 2 ((1=n) i=1 g (i=n))1=2 Z +
t 0
= OP
+ OP
n−1 P i=1
g(( [nt] − i)=n)Zi − n1=2
Z 0
t
g0 (y)Wn (t − y)dy
) ( Rt P[nt]−1 ( 0 g2 (y) dy)1=2 − ((1=n) i=1 g2 (i=n))1=2 g (y)Wn (t − y) dy Rt Pn−1 ( 0 g2 (y) dy)1=2 ((1=n) i=1 g2 (i=n))1=2 0
t 1=2 g(t)((nt)1=(2+) + g(t)=n) Rt (nt)1=2 ( 0 g2 (y) dy)1=2
!
tg2 (t) t 1=2 g(t)(log log(nt))1=2 Rt Rt ( 0 g2 (y) dy)1=2 (nt) 0 g2 (y) dy
! :
(2.19)
For the second rate above note that, by the law of iterated logarithm, uniformly in t ∈ [1=n; 1]; sup |W (n(t − y))| = OP ((nt log log(nt))1=2 ) as n → ∞:
06y6t
(2.20)
In view of conditions (2.1) and (2.2), the rate in (2.19) amounts to (log log(nt))1=2 = OP ((nt)(1=(2+))−(1=2) ) OP (nt)(1=(2+))−(1=2) + (nt) which immediately implies (2.5) for any 06 6 12 − 1=(2 + ). Making use of (2.12) instead of (2.11) in the above proof, assertion (2.6) is an immediate consequence of (2.14) – (2.18). Rt Proof of Remark 2.3. Since var V (t) = 0 g2 (y) dy = O(tg2 (t)) for t ↓ 0; and V (t) = Rt 0 g (y)W (t − y) dy, the law of iterated logarithm for the Wiener process gives (2.10) 0 by a similar argument as in (2.20). Next we prove an analogue of Theorem 2.1 for the centered weighted sums: Theorem 2.2. Let the assumptions of Theorem 2:1 be satis ed; but with condition (2:2) on the weight function g replaced by (2:3). Then there exists a sequence
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
{Bn (t): t ∈ [0; 1]}; n = 1; 2; : : : ; of Brownian bridges such that; as n → ∞; Pn−1 i=1 g(([nt] − i)=n)(Zi − Z n ) sup t Pn−1 n Pn−1 1 2 2 1=2 ( 1=n6t61 i=1 g (([nt] − i)=n) − n ( i=1 g(([nt] − i)=n) ) Rt 0 g (y)Bn (t − y) dy 0 − Rt Rt = OP (1) ( g2 (y) dy − ( g(y) dy)2 )1=2 0
(2.21)
0
for any 06 6 12 − 1=(2 + ). Moreover; Z t 1 n−1 P [nt] − i 0 (Zi − Z n ) − g g (y)Bn (t − y) dy sup √ n n 06t61 i=1 0 =OP (n(1=(2+))−(1=2) ):
(2.22)
Remark 2.5. Note that 2 n−1 n−1 P P 2 k −i P 1 n−1 k −i k −i (Zi − Z n ) = − g g g var n n n i=1 n i=1 i=1 and that the process {V (t): t ∈ [0; 1]} with Z t g0 (y)B(t − y) dy; V (t) =
(2.23)
0
where {B(y): y ∈ [0; 1]} is a Brownian bridge, de nes a zero mean Gaussian process with Z s Z t Z s∧t g(s − y)g(t − y) dy − g(s − y) dy g(t − y) dy; cov(V (s); V (t)) = 0
0
0
(2.24)
s; t ∈ [0; 1]. Moreover, Z t g0 (t − s)B(y) dy V (t) = 0
= [g(t − y)B(y)]t0 + Z =
0
t
Z
t
0
g(t − s) dB(y)
g(t − y) dB(y):
(2.25)
Proof of Theorem 2.2. The proof follows along the lines of the proof of Theorem 2.1. Details will be omitted. Note that, by condition (2.3) and Remark 2.1, Z t Z t g2 (y) dy tg2 (t); g(y) dy tg(t); 0
0
where at bt means that at =bt is bounded away from 0 and ∞ as t ↓ 0. So, Z t 2 Z t Z t g2 (y) dy − g(y) dy = g2 (y) dy(1 + o(1)); t ↓ 0: 0
0
0
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
65
Moreover, n−1 P i=1
g
2
[nt] − i n
1 − n
n−1 2 n−1 P P 2 [nt] − i [nt] − i (1 + o(1)) g = g n n i=1 i=1
as n → ∞. This suces for the corresponding modi cation of the proof of Theorem 2.1. The following corollaries of Theorems 2.1 and 2.2 will be used later on. Corollary 2.1. Under the assumptions of Theorem 2:1; as n → ∞; Pn−1 | i=1 g((k − i)=n)Zi | max P log n6k6n ( n−1 g2 ((k − i)=n))1=2 i=1
−
|
sup
log n=n6t61
0 g (y)W (t − y) dy| n 0 Rt = oP ((log log n)−1=2 ): 2 1=2 ( 0 g (y) dy)
Rt
(2.26)
Proof. It follows immediately from (2.5) in Theorem 2.1 that the left-hand side of (2.26) is of order OP ((log n)− ) for some ¿ 0. This proves (2.26). Corollary 2.2. Under the assumptions of Theorem 2:2; as n → ∞; Pn−1 | i=1 g((k − i)=n)(Zi − Z n )| max P P log n6k6n ( n−1 g2 ((k − i)=n) − (1=n)( n−1 g((k − i)=n))2 )1=2 i=1
i=1
g0 (y)Bn (t − y) dy| − sup Rt Rt = oP ((log log n)1=2 ): 2 2 1=2 log n=n6t61 ( 0 g (y) dy − ( 0 g(y) dy) ) |
Rt 0
(2.27)
Proof. It follows immediately from (2.21) in Theorem 2.2. Making use of Theorems 2.1, 2.2 and Corollaries 2.1, 2.2, we are now in a position to derive our main results.
3. Limiting extreme value distributions Our main results are based on the following theorem. Theorem 3.1. Let Z1 ; Z2 ; : : : be i.i.d. r.v.’s with EZ1 = 0; EZ12 = 1; and E|Z1 |2+ ¡ ∞ for some ¿ 0. Then ! Pk−1 (k − i) Z i x i=1 (3.1) lim P an max Pk−1 2 1=2 6x + bn = exp(−e ) n→∞ 26k6n ( i ) i=1
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
and lim P an max
n→∞
|
!
Pk−1
26k6n
i=1 (k − i) Zi | Pk−1 2 1=2 6x ( i=1 i )
+ bn
= exp(−2ex )
(3.2)
for all x; where an = (2log log n)1=2 ; and in case (i) ¿ 12 : 1 bn = 2log log n + log 4
2 + 1 2 − 1
1=2 ! ;
(ii) = 12 : 1 bn = 2 log log n + log log log log n − log(4); 2 (iii) 0 ¡ ¡ 12 : 1 − 2 C 1=(2 +1) H2 +1 log log log n + log bn = 2 log log n + 2(2 + 1) 1=2 22 =(2 +1)
! ;
with H2 +1 as in Remark 12:2:10 of Leadbetter et al. (1983); and Z ∞ y ((y + 1) − y − y −1 ) dy: C = −(2 + 1) 0
Under H0 ; extreme value limiting distributions for {Tn } and {T n } of (1.8) and (1.6), with weights xik as in (1.5), are now immediate from Theorem 3.1. Note that xik can be replaced by x˜ik = (i − k) + (i; k = 1; : : : ; n); since the denominator 1=n cancels out by the normalization. Theorem 3.2. Let Y1 ; : : : ; Yn satisfy the location model of (1:1) for n = 1; 2; : : : . Then; under H0 : m = n; Pn i=1 (i − k)+ (Yi − ) Pn 6x + bn = exp(−e−x ) (3.3) lim P an max 2 1=2 n→∞ k=1;:::; n−1 ( i=1 (i − k)+ ) and
lim P an
n→∞
max
k=1;:::; n−1
|
Pn i=1 (i − k)+ (Yi − )| Pn 6x + bn = exp(−2e−x ) 1=2 ( i=1 (i − k)2 +)
(3.4)
for all x; where an and bn = bn ( ) are as in Theorem 3:1 according to the cases ¿ 12 ; = 12 ; 0 ¡ ¡ 12 . If is replaced by Yn and by a suitable estimator ˆn satisfying (1.7), then the asymptotics of Theorem 3.1 retain, i.e.
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
67
Theorem 3.3. Under the assumptions of Theorem 3:2; Pn i=1 (i − k)+ (Yi − Y n ) Pn P 6x + b lim P an max n n 2 2 1=2 n→∞ k=1;:::; n−1 ˆn ( i=1 (i − k)+ − (1=n)( i=1 (i − k)+ ) ) = exp(−e−x ) and
(3.5)
Pn | i=1 (i − k) + (Yi − Y n )| Pn Pn 6x + bn lim P an max 2 2 1=2 n→∞ k=1;:::; n−1 ˆn ( i=1 (i − k)+ − (1=n)( i=1 (i − k)+ ) )
=exp(−2e−x );
(3.6)
for all x; where an and bn = bn ( ) are as in Theorem 3:1 according to ¿ 1 1 2; 0 ¡ ¡ 2.
1 2;
=
Remark 3.1. In case = 1, the results of Theorems 3.2 and 3.3 have been obtained by Jaruskova (1998). For the proof of Theorem 3.1 we make use of the following extreme value asymptotics: Proposition 3.1. Consider a zero mean; unit variance; stationary Gaussian process {X (t): t¿0} with autocorrelation r( ) = EX (t)X (t + ); t; ¿0. Assume that lim r( ) log = 0;
(3.7)
→∞
r( ) = 1 − C + o( ) Then
as ↓ 0:
lim P aT sup X (t)6x + bT
T →∞
06t6T
(3.8) = exp(−e−x )
(3.9)
for all x; where aT = (2 log T )1=2 ;
(3.10)
2− log log T + log(C 1= H (2)−1=2 2(2− )=(2 ) ); (3.11) 2 √ H as in Remark 12:2:10 of Leadbetter et al. (1983); H1 = 1; H2 = 1= . bT = 2 log T +
Proof. Confer Leadbetter et al. (1983, Theorem 12:3:5). We need yet another auxiliary result from extreme value theory: Proposition 3.2. Consider {X (t): t¿0} as in Proposition 3:1; and let 1 − r( ) be regularly varying of index 2; as ↓ 0. De ne v = vT to be the largest solution of the equation 1 =1 (3.12) (2 log T ) 1 − r v
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and set u = uT = Then;
!1=2 √ Tv 2 2 log p : 2 log T
(3.13)
√ lim P uT sup X (t)6x + uT2 = exp(−e−x = )
T →∞
06t6T
(3.14)
for all x. Proof. This is a special case of Berman (1992, Theorem 10.6.1). Note that, in Berman √ (1992, (3.3.24)), ! = 1= : Proof of Theorem 3.1. We rst note that, as n → ∞, Pk−1 Pk−i Pk−1 | i=1 (i − (i − 1) ) j=1 Zj | | i=1 (k − i) Zi | = max max Pk−1 Pk−1 26k6log n 26k6log n ( i=1 i2 )1=2 ( i=1 i2 )1=2 k (k log log k)1=2 = OP max 26k6log n k +1=2 = OP ((log log log n)1=2 ) = oP ((log log n)1=2 ) which means that it suces to investigate the max over k ∈ [log n; n] only. In view of Corollary 2.1, the corresponding asymptotics can be derived from those of Rt Rt | 0 y −1 Wn (t − y) dy| | 0 y −1 Wn (t − y) dy| = sup (2 + 1)1=2 : sup Rt 2 +1 )1=2 2 1=2 (t ( 0 y dy) log n=n6t61 log n=n6t61 This is equivalent to investigating Rs | 0 x −1 W (s − x) d x| (2 + 1)1=2 sup (s2 +1 )1=2 log n6s6n by substituting s = nt; x = ny. We now put s = et , and consider the process {X (t): t¿0} given by X (t) = =
(2 + 1)1=2 (2 + 1)1=2
R et
x −1 W (et 0 et( +1=2)
R et
(et − 0 et( +1=2)
− x) d x
y) dW (y)
:
(3.15)
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
69
Note that {X (t): t¿0} is a zero mean Gaussian process with covariance function R es s (e − y) (et − y) dy cov(X (s); X (t)) = (2 + 1) 0 s( +1=2) t( +1=2) e e Z 1 = (2 + 1)e(s−t)=2 (1 − z) (1 − zes−t ) d z; 06s6t: 0
Clearly, {X (t): t¿0} is stationary with unit variance, and, in order to get the limiting extreme value distribution, it will be enough to check the behaviour of r(t), where r(t) − 1 = cov(X (s); X (s + t)) − 1 Z 1 (1 − z) (1 − ze−t ) d z − 1 = (2 + 1)e−t=2 0
(
= (2 + 1) e−t=2
+ (e
−t=2
Z − 1)
0
Z 0 1
1
(1 − z) ((1 − ze−t ) − (1 − z) ) d z ) 2
(1 − z) d z
:
(3.16)
Obviously, since r(t) = O(e−t=2 ) as t → ∞, condition (3.7) is satis ed for any ¿ 0. Concerning the asymptotics for t → 0, we prove the following lemma. Lemma 3.1. For r(t) as in (3:16); we have; as t → 0; if (i) ¿ 12 : r(t) = 1 −
2 +1 2 2 8(2 −1) t + o(t ); 1+o(1) (ii) = 12 : r(t) = 1 − 4 t 2 log( 1t ); (iii) 0 ¡ ¡ 12 : r(t)R= 1 − C t 2 +1 + o(t 2 +1 ), ∞ with C = −(2 + 1) 0 y ((y + 1) − y − y −1 ) dy.
Proof. (i) ¿ 12 : On setting t2 + o(t 2 ) 2 a Taylor series expansion gives = 1 − e−t = t −
(t ↓ 0);
(1 − ze−t ) − (1 − z) = (1 − z + z) − (1 − z) = (1 − z) −1 z + ( − 1)
2 z 2 −2 ; (1 − z + z) 2
R1 where 0 ¡ ¡ 61. Since, for ¿ 12 ; 0 (1 − z)2 −2 z 2 d z = (2 − 1) (3)= (2 + 2), we obtain, by the dominated convergence theorem, as t ↓ 0, Z 1 (1 − z) ((1 − ze−t ) − (1 − z) ) d z 0
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
Z =
1
0
(1 − z)2 −1 z d z +
( − 1) 2 2
Z
1
0
(1 − z)2 −2 z 2 d z(1 + o(1))
−1 1 + 2 + o(2 ): 2(2 + 1) 2(2 − 1)(2 + 1) R1 So, with (2 + 1) 0 (1 − z)2 d z = 1, t2 t −1 1 2 2 2 t − + o(t ) +t r(t) − 1 = 1 − + o(t) + o(t ) 2 2 2 2(2 − 1) =
t2 t − + + o(t 2 ) 2 8 −1 1 1 1 − + − + o(t 2 ) = −t 2 4 2(2 − 1) 4 8 2 + 1 + o(t 2 ): 8(2 − 1)
= −t 2
Note that, for = 1; (2 + 1)=8(2 − 1) = 3=8, which coincides with the results of Jaruskova (1998). (iii) 0 ¡ ¡ 12 : With as before, and on substituting 1 − z = y; −d z = dy, we have Z 1 (1 − z) ((1 − z + z) − (1 − z) ) d z 0
= 2 +1 = 2 +1 Z +
Z
1=
0
(Z
y ((y + 1 − y) − y ) dy
1=
0 1=
0
y ((y + 1 − y) − y − y −1 (1 − y)) dy )
y
2 −1
(1 − y) dy
= 2 +1 (I1 + I2 ):
Now, for 0 ¡ ¡ 1=2, 0¿(y + 1 − y) − y − y −1 (1 − y)¿ and
Z
1
0
y
2 −1
Z dy ¡ ∞;
1
∞
( − 1) −2 y 2
y2 −2 dy ¡ ∞:
Therefore, by dominated convergence, Z ∞ y ((y + 1) − y − y −1 ) dy: lim I1 = →0
0
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
71
Moreover,
y2 +1 y2 − I2 = 2 2 + 1
1= 0
2 1 1 : = 2(2 + 1)
So, r(t) − 1 = −C t 2 +1 (1 + o(1)) +
t t 1 1 − + O(t 2 ) (t + O(t 2 )) + − + O(t 2 ) 2 2 2
= −C t 2 +1 (1 + o(1)) + O(t 2 ) = −C t 2 +1 + o(t 2 +1 ) since 2 + 1 ¡ 2. (ii) = 12 : Consider I1 and I2 as in case (iii). Here, I2 =
1 4
and
Z I1 = O(1) +
1
= O(1) + I˜1
1=
y
1=2
1=2
(y + 1 − y)
−y
1=2
1 − y−1=2 (1 − y) 2
dy
(t ↓ 0):
First note that, by the same estimations as above, Z 1 (− 1 ) 1= 1 1 1 ˜I 1 ¿ 2 2 : ds = − ln 2 s 8 1 On the other hand, with y0 such that 16y0 61=; (y=y + 1)3=2 ¿1 − for all y¿y0 , Z (1 − ) 1= (1 − y)2 dy I˜1 6 − 8 y y0 1= 2 y2 (1 − ) ln y − 2y + =− 8 2 y0 1 (1 − ) ln (1 + o(1)) =− 8
(t ↓ 0):
Since ¿ 0 can be chosen arbitrarily small, this results in t 1 1 + + − + O(t 2 ) r(t) − 1 = − 2 ln 4 2 2 1 (1 + o(1)) 2 (t ↓ 0): t ln =− 4 t This proves Lemma 3.1.
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
We are now in a position to complete the proof of Theorem 3.1. It is enough to consider MT = max X (t) 06t6T
and
M˜ T = max |X (t)| 06t6T
with X (t) from (3.15), and T = log n. In cases (i) and (iii), the limiting extreme value distributions of {MT } and {M˜ T } are now immediate from Proposition 3.1 and Lemma 3.1. The asymptotic of {M˜ T } follows from that of {MT } by symmetry and asymptotic independence of max and min in case of the Gaussian process {X (t)}. The case = 12 can be treated by Proposition 3.2 together with Lemma 3.1(ii). Recall v = vT ; u = uT from (3.12) and (3.13). A simple calculation shows that, as T → ∞, (1 + o(1)) log T log log T: 4 On choosing 1=2 1 log T log log T ; v˜ = 4 ! √ T v= ˜ 2 T (log log T )1=2 √ u˜ 2 = 2 log = 2 log (2 log T )1=2 4 v2 =
= 2 log T + O(log log log T ) = (1 + o(1))2 log T; we have u − u˜ =
√
u2
p 1 1 v 2 =o − u˜ = O log u˜ v˜ u˜
and by Theorem 10:6:1 of Berman (1992), u˜T (MT − u˜T ) = u˜T (MT − uT ) + OP (u˜T (uT − u˜T )) = (1 + o(1))uT (MT − uT ) + oP (1); since, as T → ∞,
√ P(uT (MT − uT )6x) → exp(−e−x = )
for all x. So, the same asymptotic applies to u˜ T (MT − u˜ T ). Moreover, (2 log T )1=2 (MT − u˜ T ) = (2 log T )1=2 MT − u˜ T (2 log T )1=2 and by an expansion of u˜ T , 1=2 (log log T )1=2 √ u˜ T (2 log T )1=2 = (2 log T )2 + 4 log T log 4 √ 1 = 2 log T + log log log T − log(4 ) + o(1) 2 = b˜˜T + o(1):
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
73
So, with a˜T (2 log T )1=2 and b˜˜T as above, √ P(a˜ M 6x + b˜˜ ) → exp(−e−x = ) T
T
T
for all x, which is equivalent to P(a˜T MT 6x + b˜T ) → exp(−e−x ) √ for all x, if b˜T = b˜˜T + log( ). This completes the proof of Theorem 3.1. Note that MT = max06t6T X (t) can be replaced by maxlog T 6t6T X (t), since X (t) = OP ((log log T )1=2 ) = oP ((log T )1=2 ) (T → ∞):
max
06t6 log T
Proof of Theorem 3.2. On replacing i = n + 1 − j; k = n + 1 − ‘, assertion (3.3) is immediate from Theorem 3.1, since D
{Zj }j=1; :::; n :={(Yn+1−j − )=}j=1; :::; n ={(Yj − )=}j=1; :::; n satis es the required assumptions. Proof of Theorem 3.3. Since, by condition (1.7), 1 1 − = oP ((log log n)−1 ) (n → ∞); ˆn it suces to prove (3.5) with ˆn replaced by . Assume w.l.o.g., = 0 and 2 = 1, and set i = n + 1 − j; k = n + 1 − ‘; Yn+1−j = Zj , i.e. we consider P‘−1 j=1 (‘ − j) (Zj − Z n ) : max P‘−1 P ‘−1 2 1=2 2 ‘=2;:::; n ( j=1 j − (1=n)( j=1 j ) ) We have ‘−1 P j=1 ‘−1 P j=1
j 6
‘ +1 ; +1
j 2 6
‘−1 P j=1
‘2 +1 ; 2 + 1
j ¿
‘−1 P j=1
(‘ − 1) +1 ; +1
j 2 ¿
(‘ − 1)2 +1 ; 2 + 1
so that, by Jensen’s inequality, for ‘6n=log n, as n → ∞, !2 ‘−1 ‘−1 P 2 ‘−1 P 2 1 ‘−1 P P 2 1 : j ¿ j − j ¿ j 1+O n j=1 log n j=1 j=1 j=1 Moreover, if ‘6n=log n, for some c0 ¿ 0, P‘−1 j=1 (‘ − j) 6c0 ‘1=2 : P‘−1 P‘−1 ( j=1 j 2 − (1=n)( j=1 j )2 )1=2 On setting S‘ =
‘−1 P j=1
(‘ − j) Zj ;
S‘ =
‘−1 P j=1
(‘ − j) (Zj − Z n );
(3.17)
(3.18)
74
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
we obtain
1=2 P‘−1 var S‘ S‘ S‘ j=1 (‘ − j) Z n = − : (var S‘ )1=2 (var S‘ )1=2 var S‘ (var S‘ )1=2 √ Since |Z n | = OP (1= n); (3.17) – (3.19) imply S‘ 26‘6n=log n (var S ‘ )1=2 S‘ 1 1 + OP = max 1+O : 26‘6n=log n (var S‘ )1=2 log n (log n)1=2
(3.19)
max
(3.20)
On the other hand, uniformly for n=log n6‘6n; as n → ∞; ‘−1 P 2 log n ‘2 +1 1+O ; j = 2 + 1 n j=1 ‘ +1 j = +1 j=1
‘−1 P
log n 1+O n
so that 16
var (S‘ ) = O(1); var (S‘ )
and max
n=log n6‘6n
|
(3.21)
P‘−1
− j) Z n | = OP (1): (var S‘ )1=2
j=1 (‘
(3.22)
Finally, since max
log(n=log n)6t6log n
D
|X (t)| =
max
06t6log log n
|X (t)| = OP (log log log n);
also |S‘ | = OP (log log log n); n=log n6‘6n (var S‘ )1=2
(3.23)
|S‘ | = OP (log log log n): n=log n6‘6n (var S ‘ )1=2
(3.24)
max
and max
A combination of (3.19) – (3.24) completes the proof of Theorem 3.3. Remark 3.2. In view of Theorem 3.3, asymptotic critical values for the testing of H0 vs. H1 , or H0 vs. H1+ , respectively, can now be chosen as appropriate upper quantiles of the two-sided or one-sided Gumbel distribution. The latter tests are consistent as will be outlined next.
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
75
4. Consistency In this section, we brie y discuss consistency of the asymptotic tests suggested above. It holds that, for the location model of (1.1), under H1 , as n → ∞, Pn | i=1 (i − k) + (Yi − )| P Pn − bn → +∞ (4.1) an max 1=2 k=1;:::; n−1 ( i=1 (i − k)2 ) + with an and bn as in Theorem 3.1, provided +1=2 , p √ n − mn log log n → +∞: |n | n n Moreover, under H1+ , as n → ∞, Pn P i=1 (i − k)+ (Yi − ) Pn − bn → +∞; an max 2 1=2 k=1;:::; n−1 ( (i − k) ) + i=1 provided √
n n
n − mn n
+1=2 ,
p log log n → +∞:
We also have, under H1 , as n → ∞, Pn | i=1 (i − k) + (Yi − Yn )| P Pn Pn − bn → +∞; an max 2 2 1=2 k=1;:::; n−1 ˆn ( (i − k) − (1=n)( (i − k) ) ) + + i=1 i=1 provided ˆn = OP (1) and (4.2) holds, and, under H1+ , as n → ∞, Pn P i=1 (i − k)+ (Yi − Yn ) Pn Pn − bn → +∞; an max 2 2 1=2 k=1;:::; n−1 ˆn ( (i − k) − (1=n)( (i − k) ) ) + + i=1 i=1
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
provided ˆn = OP (1) and (4.4) holds. This proves consistency of the asymptotic tests rejecting H0 for large values of the test statistics, if critical values are chosen from the asymptotic distributions in Theorems 3.2 and 3.3, respectively. We prove (4.5) and (4.6), for example. The other statements can be proved in a Pn similar vein. Note that, with m = mn ; = n ; xim = ((i − m)=n) + ; xm = (1=n) i=1 xim , n n P P i−m (Yi − Y n ) = (xim − xm )(Yi − ) n i=1 i=1 + n−m n P 2 i P − ng 2 + g xim (ei − e n ); = n i=1 i=m+1 Pn−m Pn ; g = (1=n) i=1 g(i=n); e n = (1=n) i=1 ei . where g(y) = y+ Similar to the proof of Theorem 3.3, on separating the cases n − m6log n and n − m¿log n, respectively, we have that 2 +1 P 2 i n−m 1 n−m − g 2 ¿c0 g ; (4.7) n i=1 n n
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
P 2 1 n−m g n i=1
2 +1 i n−m − g 2 6c1 n n
(4.8)
with positive constants c0 ; c1 . So, according to Theorem 3.3 together with (4.7) and (4.8), Pn i=m+1 (i − m) (Yi − Y n ) Pn P − bn an n ˆn ( i=m+1 (i − m)2 − (1=n)( i=m+1 (i − m) )2 )1=2 +1=2 , √ c0 n−m ¿an n n √ ˆn + OP (a2n ) c1 n which suces for the proof of (4.6). For the proof of (4.5), note that condition (4.2) implies +1=2 ! √ n − m n (as n → ∞) bn = O(a2n ) = o an n n n which, via the triangle inequality, suces for (4.5). 5. Concluding remarks (a) The asymptotics of Theorems 3.1–3.3 are not at all restricted to the location model of (1.1) with i.i.d. errors. There is a more general principle behind which allows for testing gradual changes in the mean of increments of certain stochastic processes by only taking advantage of a suitable weak invariance principle. This idea was pursued by Horvath and Steinebach (1999) in case of models with abrupt changes, and has been extended by Steinebach (1999) to the case of gradual changes. (b) Instead of considering the statistics T n and Tn of (1.6) and (1.8) which are motivated by an asymptotic log-likelihood ratio approach, one could also think of statistics with heavier weights in the normalizing denominators. This idea, developed by Huskova (1998) for the case = 1, can be extended to the case of general ¿ 0, and to other models of gradual changes satisfying a suitable invariance principle. Details of this will be discussed elsewhere. Acknowledgements The authors would like to thank the referees for valuable remarks. References Arnold, L., 1973. Stochastische Di erentialgleichungen. Oldenbourg, Munchen. Berman, S.M., 1992. Sojourns and Extremes of Stochastic Processes. Wadsworth, Belmont.
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Chu, C.-S.J., White, H., 1992. A direct test for changing trend. J. Bus. Econom. Statist. 10, 289–299. Csorg˝o, M., Horvath, L., 1997. Limit Theorems in Change-Point Analysis. Wiley, Chichester. Csorg˝o, M., Revesz, P., 1981. Strong Approximations in Probability and Statistics. Academic Press, New York. Horvath, L., Steinebach, J., 1999. Testing for changes in the mean or variance of a stochastic process under weak invariance. J. Statist. Plann. Inference, in press. Huskova, M., 1998. Remarks on test procedures for gradual changes. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability and Statistics. Elsevier, Amsterdam, pp. 577–584. Huskova, M., 1999. Gradual changes versus abrupt changes. J. Statist. Plann. Inference 76, 109–125. Jandhyala, V.K., MacNeill, I.B., 1997. Iterated partial sum sequences of regression residuals and tests for changepoints with continuity constraints. J. Roy. Statist. Soc. B 59, 147–156. Jaruskova, D., 1998. Testing appearance of linear trend. J. Statist. Plann. Inference 70, 263–276. Karatzas, I., Shreve, S.E., 1988. Brownian Motion and Stochastic Calculus. Springer, New York. Komlos, J., Major, P., Tusnady, G., 1975. An approximation of partial sums of independent r.v.’s and the sample d.f. I. Z. Wahrsch. Verw. Geb. 32, 111–131. Komlos, J., Major, P., Tusnady, G., 1976. An approximation of partial sums of independent r.v.’s and the sample d.f. II. Z. Wahrsch. Verw. Geb. 34, 33–58. Leadbetter, M.R., Lindgren, L., Rootzen, H., 1983. Extremes and Related Properties of Random Sequences and Processes. Springer, New York. Steinebach, J., 1999. Some remarks on the testing of smooth changes in the linear drift of a stochastic process. Preprint, University of Marburg.
www.elsevier.com/locate/jspi
Limit theorems for a class of tests of gradual changes a Department
b FB
Marie Huskovaa; ∗ , Josef Steinebachb
of Statistics, Charles University, Sokolovska 83, CZ-18600 Praha, Czech Republic Mathematik & Informatik, Philipps-Universitat, Hans-Meerwein-Str., D-35032 Marburg, F.R. Germany
Received 22 June 1999; received in revised form 21 December 1999; accepted 11 January 2000
Abstract We study a class of asymptotic tests for detecting various types of gradual changes in a location model. The results of Jaruskova (1998. J. Statist. Plann. Inference 70, 263–276) and Huskova (1998. In: Szyszkowicz, B. (Ed.), Asymptotic Methods in Probability and Statistics. Elsevier, Amsterdam, pp. 577–584.) are extended to other families of change alternatives. Under the null hypothesis of “no change”, limiting extreme value distributions of Gumbel type are derived for some asymptotic log-likelihood ratio test statistics. Consistency under certain alternatives is also c 2000 Elsevier Science B.V. All rights reserved. discussed. MSC: 62 F 05; 62 J 05 Keywords: Change-point test; Linear and nonlinear regression; Gradual change; Limiting extreme value distribution; Weighted embedding
1. Introduction Consider the following location model with a gradual change after an unknown time point m: i−m + ei ; i = 1; : : : ; n; (1.1) Yi = + n + where a+ = max(0; a); ; = n ; m = mn (6n) are unknown parameters, and e1 ; : : : ; en are i.i.d. random variables with Eei =0; var ei =2 ¿ 0, and E|ei |2+ ¡ ∞; i =1; : : : ; n; for some ¿ 0. The parameter ¿ 0 is supposed to be known. We are interested in testing the hypotheses H0 : m = n
vs:
H1 : m ¡ n; 6= 0;
This work was supported by the grant GACR-201=97=1163 and CES:J13=98:113200008. Corresponding author. E-mail addresses: [email protected] .cuni.cz (M. Huskova), [email protected] (J. Steinebach).
∗
c 2000 Elsevier Science B.V. All rights reserved. 0378-3758/00/$ - see front matter PII: S 0 3 7 8 - 3 7 5 8 ( 0 0 ) 0 0 0 9 4 - X
(1.2)
58
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
or the one-sided case H0 : m = n
H+ 1 : m ¡ n; ¿ 0:
vs:
(1.3)
Most of the testing procedures suggested so far deal with models of “abrupt” changes i.e. = 0 (cf. e.g. the monograph of Csorg˝o and Horvath, 1997). Jaruskova (1998) discussed the case of = 1 and derived some limiting extreme value distributions of Gumbel type for test statistics which are equivalent to the log-likelihood ratio test statistic if e1 ; : : : ; en are normally distributed. Huskova (1998), also for =1, proposed a new class of testing procedures by making use of heavier weights in the normalization. This results in limiting weighted sup-norm distributions, under H0 , and asymptotic normality, respectively, under certain local alternatives. Moreover, for ∈ (0; 1); Huskova (1999) discussed the problem of estimating the change-point m. A Bayesian type of statistics for polynomial changes in regression models has been studied by Jandhyala and MacNeill (1997). Here we study a class of test procedures based on the partial sums of weighted residuals: n P (1.4) Sk = (xik − xk )Yi ; k = 1; : : : ; n; i=1
where
xik = xik ( ; n) =
i−k n
+
n 1P xik ; xk = xk ( ; n) = n i=1
;
i; k = 1; : : : ; n; (1.5)
k = 1; : : : ; n:
The test statistic we are interested in is as follows: |Sk | 1 Pn ; max T n = ˆn 16k6n−1 ( i=1 (xik − xk )2 )1=2
(1.6)
where ˆn is an estimator of with the property ˆn − = oP ((log log n)−1 )
as n → ∞:
(1.7)
Here and in the sequel, log x = ln max(x; 1). Note that T n as given in (1.6) is based on an asymptotic log-likelihood ratio approach. Namely, it will turn out that the limiting extreme value distribution of T n coincides with that of its corresponding counterpart Tn in case and 2 are known, e.g. = 0 and 2 = 1. That is, it will be enough to discuss the asymptotics of Tn =
max
16k6n−1
|Sk | 2 1=2 i=1 xik )
(
(1.8)
k = 1; : : : ; n
(1.9)
Pn
with Sk =
n P i=1
xik Yi ;
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
59
which is equivalent to studying the log-likelihood ratio statistic for (1.1) and (1.2) if m is unknown and the {ei } are i.i.d. N(0; 1) normal random variables. This idea is well known and has also been pursued for the testing of possible changes in the trend of a regression model. Confer the “direct test” of Chu and White (1992), in which design points similar to (1.5) (with ¿1) are assumed (see also Csorg˝o and Horvath, 1997, Chapter 3:2:1). The resulting null distributions, however, are di erent from our main results (compare Theorems 3:2:1–3:2:3 in Csorg˝o and Horvath (1997) and Theorems 3.1–3.3 in Section 3 below). The paper is organized as follows. Section 2 contains some weak embeddings for the weighted partial sum processes n P [nt] − i Zi ; t ∈ [0; 1] (1.10) Sn (t) = g n i=1 and Sn (t) =
n P i=1
g
[nt] − i n
(Zi − Z n );
t ∈ [0; 1];
(1.11)
where g = g(y); y ∈ R; is a weight function, and {Zi }i=1; 2; ::: are i.i.d. with E Z1 = Pn 0; var Z1 = 1, and E|Z1 |2+ ¡ ∞ for some ¿ 0; Z n = (1=n) i=1 Zi . These weak invariance principles hold true with a rather general weight function g, and thus may be of independent interest. , to derive our main In Section 3, the above embeddings are applied, with g(y) = y+ results on the limiting distributions for the statistics Tn and T n of (1.8) and (1.6). There a key observation is that the limit behaviour of Tn under H0 is given by the exceedance probability of the limit process ) ( R1 (2 + 1)1=2 t (y − t) dW (y) : 06t ¡ 1 ; (1.12) (1 − t) +1=2 where {W (y): y¿0} denotes a standard Wiener process. The Gaussian process in (1.12) can be transformed into a stationary process, so that the results on the exceedence probability of a stationary Gaussian process on an increasing time interval can be applied. As a consequence, one gets limiting extreme value distributions of Gumbel type under H0 . It turns out that these asymptotics are di erent according to the three cases ¿ 12 ; = 12 , and 0 ¡ ¡ 12 . The limiting null distributions of {T n } are the same as those for {Tn } and can, in fact, be derived from the latter. Finally, in Section 4, we brie y discuss consistency of the proposed testing procedures. In a concluding remark, we mention some possible extensions of our main results which will be discussed elsewhere. 2. Embeddings of weighted partial sum processes Here we prove some weighted embeddings for the processes {Sn (t): t ∈ [0; 1]} and {S n (t): t ∈ [0; 1]} as de ned in (1.10) and (1.11) of Section 1. The weight function
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
g = g(y); y ∈ R; is assumed to satisfy the following assumptions: g is nondecreasing; square integrable and absolutely continuous on [0; 1] with derivative g0 ; g(y) = 0 (y60); g(y) ¿ 0 (y ¿ 0); lim sup R t t↓0
0
tg2 (t) g2 (y) dy
¡ ∞:
(2.1) (2.2)
Sometimes, assumption (2:2) will be replaced by tg(t) ¡ ∞: lim sup R t g(y) dy t↓0 0
(2.3)
Remark 2.1. (a) It is an easy consequence of Jensen’s inequality that, under (2.1), assumption (2:3) implies (2:2). Note that, for any 0 ¡ t61; Z t 2 Z 1 1 t 2 g (y) dy ¿ g(y) dy ; (2.4) t 0 t 0 wherein strict inequality follows from the fact that g cannot be constant in view of assumption (2:1). (b) Note that assumptions (2:2) and (2:3) are satis ed if g is regularly varying at 0 with index ¿ 0. Our main results in Section 3 will be based on the following weighted embedding of partial sums: Theorem 2.1. Let {Zi }1=1; 2::: be i.i.d. r.v.’s with EZ1 =0; var Z1 =1; and E|Z1 |2+ ¡ ∞ for some ¿ 0. Let the weight function g = g(y) satisfy conditions (2:1) and (2:2). Then there exists a sequence of Wiener processes {Wn (t): t ∈ [0; 1]}; n=1; 2; : : : ; such that as n → ∞; P Rt 0 n−1 g (y)W (t − y) dy g(([nt] − i)=n)Z n i i=1 − 0 Rt n sup t Pn−1 = OP (1) (2.5) 2 1=2 2 1=2 ( 0 g (y) dy) ( i=1 g (([nt] − i)=n) 1=n6t61 for any 06 6 12 − 1=(2 + ). Moreover; Z t 1 n−1 P [nt] − i 0 √ Zi − g g (y)Wn (t − y) dy sup n n i=1 06t61 0 =oP (n1=(2+)−1=2 ):
(2.6)
Remark 2.2. Using integration by parts (cf. e.g. Arnold (1973, (4.5.12) on p. 92) or Karatzas and Shreve (1988, (3.8) on p. 155)), we have, for t ∈ [0; 1]; Z t Z t g(t − y) dWn (y) = [g(t − y)Wn (y)]t0 + g0 (t − y)Wn (y) dy 0
0
Z =
t 0
0
g (t − y)Wn (y) dy =
Z 0
t
g0 (y)Wn (t − y) dy:
(2.7)
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
61
Remark 2.3. Note that the process {V (t): t ∈ [0; 1]} with Z V (t) =
t
0
g(t − y) dW (y);
(2.8)
where {W (y): y ∈ [0; 1]} is a Weiner process, de nes a zero mean Gaussian process with Z s∧t g(s − y)g(t − y) dy; (2.9) cov(V (s); V (t)) = 0
where s ∧ t = min(s; t); 06s; t61. Moreover, as n → ∞; |V (t)| = OP ((log log n)1=2 ): 1=2 1=n6t61 (var V (t)) sup
(2.10)
; y ∈ R; satis es conditions (2.1) – (2.3) for any Remark 2.4. The function g(y) = y+ ¿ 0.
Proof of Theorem 2.1. It is an immediate consequence of the Komlos et al. (1975,1976) strong approximations that on a suitable probability space there exists a Wiener process {W (y): y¿0} such that, as n → ∞, 1=(2+) k P 1 Zi − W (k) = OP (1); 16k6n k i=1
(2.11)
k P max Zi − W (k) = oP (n1=(2+) ): 16k6n
(2.12)
max
i=1
Also, by Theorem 1:2:1 of Csorg˝o and Revesz (1981), as T → ∞, for any Wiener process {W (y): y¿0}, sup
sup |W (t + y) − W (t)| = O((log T )1=2 ) a:s:
06t6T 06y61
(2.13)
If not mentioned otherwise the estimations below hold for n → ∞. A brief calculation shows that k−i n−1 k−1 P P P i−1 k −i i Zi = −g g Zj g n n n i=1 i=1 j=1 =
i−1 i −g W (k − i) g n n i=1 k−1 P i−1 i 1=(2+) −g (k − i) g + OP n n i=1 k−1 P
(2.14)
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
uniformly in k = 1; : : : ; n (by (2.11)). Now, for k = [nt], i−1 i −g W (k − i) g n n i=1
k−1 P
=
k−1 P
Z
i=1
Z =
i=n
(i−1)=n
(k−1)=n
0
g0 (y)(W (nt − ny) + W ([nt] − i) − W (nt − ny)) dy
k −1 (log k)1=2 ; g0 (y)W (n(t − y)) dy + OP g n
(2.15)
uniformly in k =1; : : : ; n (by (2.13)). On combining (2.14) and (2.15), we get uniformly in t ∈ [1=n; 1];
n−1 P
g
i=1
[nt] − i n
Z =
Zi
([nt]−1)=n
0
[nt] − 1 1=(2+) (nt) : g (y)W (n(t − y)) dy + OP g n 0
(2.16)
Next we check that, uniformly in t ∈ [1=n; 1], Z
([nt]−1)=n 0
g0 (y)W (n(t − y)) dy =
Z
t
0
g0 (y)W (n(t − y)) dy + OP (g(t))
(2.17)
and P 2 1 [nt]−1 g n i=1
Z t 1 i = : g2 (y) dy + O g2 (t) n n 0
(2.18)
For the proof of (2.17), note that Z 06
t
Z
0
([nt]−1)=n
g (y)|W (n(t − y))| dy6
t
([nt]−1)=n
g0 (y) sup |W (x)| dy = OP (g(t)): x∈[0;2]
On the other hand, P 2 1 [nt]−1 g n i=1 =
=
Z t i − g2 (y) dy n 0
[nt]−1 P i=1 [nt]−1 P i=1
Z
i=n
(i−1)=n
Z
i=n
(i−1)=n
g2
i 1 − g2 (y) dy + O g2 (t) n n
Z
i=n
y
! 0
2g (x)g(x) d x
1 dy + O g2 (t) n
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
=
Z
[nt]−1 P i=1
1 6 n
Z
i=n
(i−1)=n t
0
2g0 (x)g(x)
Z
x
(i−1)=n
1 2g (x)g(x) d x + O g (t) n 0
dy
2
63
1 d x + O g2 (t) n
1 = O g (t) n 2
:
√ A combination of (2.14) – (2.18) yields, uniformly in t ∈ [1=n; 1]; with Wn (y) = (1= n) W (ny) Rt 0 Pn−1 g (y)Wn (t − y) dy i=1 g(([nt] − i)=n)Zi − 0 Rt Pn−1 2 ( 0 g2 (y) dy)1=2 ( i=1 g (([nt] − i)=n)1=2 =
n−1=2 P[nt]−1 2 ((1=n) i=1 g (i=n))1=2 Z +
t 0
= OP
+ OP
n−1 P i=1
g(( [nt] − i)=n)Zi − n1=2
Z 0
t
g0 (y)Wn (t − y)dy
) ( Rt P[nt]−1 ( 0 g2 (y) dy)1=2 − ((1=n) i=1 g2 (i=n))1=2 g (y)Wn (t − y) dy Rt Pn−1 ( 0 g2 (y) dy)1=2 ((1=n) i=1 g2 (i=n))1=2 0
t 1=2 g(t)((nt)1=(2+) + g(t)=n) Rt (nt)1=2 ( 0 g2 (y) dy)1=2
!
tg2 (t) t 1=2 g(t)(log log(nt))1=2 Rt Rt ( 0 g2 (y) dy)1=2 (nt) 0 g2 (y) dy
! :
(2.19)
For the second rate above note that, by the law of iterated logarithm, uniformly in t ∈ [1=n; 1]; sup |W (n(t − y))| = OP ((nt log log(nt))1=2 ) as n → ∞:
06y6t
(2.20)
In view of conditions (2.1) and (2.2), the rate in (2.19) amounts to (log log(nt))1=2 = OP ((nt)(1=(2+))−(1=2) ) OP (nt)(1=(2+))−(1=2) + (nt) which immediately implies (2.5) for any 06 6 12 − 1=(2 + ). Making use of (2.12) instead of (2.11) in the above proof, assertion (2.6) is an immediate consequence of (2.14) – (2.18). Rt Proof of Remark 2.3. Since var V (t) = 0 g2 (y) dy = O(tg2 (t)) for t ↓ 0; and V (t) = Rt 0 g (y)W (t − y) dy, the law of iterated logarithm for the Wiener process gives (2.10) 0 by a similar argument as in (2.20). Next we prove an analogue of Theorem 2.1 for the centered weighted sums: Theorem 2.2. Let the assumptions of Theorem 2:1 be satis ed; but with condition (2:2) on the weight function g replaced by (2:3). Then there exists a sequence
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
{Bn (t): t ∈ [0; 1]}; n = 1; 2; : : : ; of Brownian bridges such that; as n → ∞; Pn−1 i=1 g(([nt] − i)=n)(Zi − Z n ) sup t Pn−1 n Pn−1 1 2 2 1=2 ( 1=n6t61 i=1 g (([nt] − i)=n) − n ( i=1 g(([nt] − i)=n) ) Rt 0 g (y)Bn (t − y) dy 0 − Rt Rt = OP (1) ( g2 (y) dy − ( g(y) dy)2 )1=2 0
(2.21)
0
for any 06 6 12 − 1=(2 + ). Moreover; Z t 1 n−1 P [nt] − i 0 (Zi − Z n ) − g g (y)Bn (t − y) dy sup √ n n 06t61 i=1 0 =OP (n(1=(2+))−(1=2) ):
(2.22)
Remark 2.5. Note that 2 n−1 n−1 P P 2 k −i P 1 n−1 k −i k −i (Zi − Z n ) = − g g g var n n n i=1 n i=1 i=1 and that the process {V (t): t ∈ [0; 1]} with Z t g0 (y)B(t − y) dy; V (t) =
(2.23)
0
where {B(y): y ∈ [0; 1]} is a Brownian bridge, de nes a zero mean Gaussian process with Z s Z t Z s∧t g(s − y)g(t − y) dy − g(s − y) dy g(t − y) dy; cov(V (s); V (t)) = 0
0
0
(2.24)
s; t ∈ [0; 1]. Moreover, Z t g0 (t − s)B(y) dy V (t) = 0
= [g(t − y)B(y)]t0 + Z =
0
t
Z
t
0
g(t − s) dB(y)
g(t − y) dB(y):
(2.25)
Proof of Theorem 2.2. The proof follows along the lines of the proof of Theorem 2.1. Details will be omitted. Note that, by condition (2.3) and Remark 2.1, Z t Z t g2 (y) dy tg2 (t); g(y) dy tg(t); 0
0
where at bt means that at =bt is bounded away from 0 and ∞ as t ↓ 0. So, Z t 2 Z t Z t g2 (y) dy − g(y) dy = g2 (y) dy(1 + o(1)); t ↓ 0: 0
0
0
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
65
Moreover, n−1 P i=1
g
2
[nt] − i n
1 − n
n−1 2 n−1 P P 2 [nt] − i [nt] − i (1 + o(1)) g = g n n i=1 i=1
as n → ∞. This suces for the corresponding modi cation of the proof of Theorem 2.1. The following corollaries of Theorems 2.1 and 2.2 will be used later on. Corollary 2.1. Under the assumptions of Theorem 2:1; as n → ∞; Pn−1 | i=1 g((k − i)=n)Zi | max P log n6k6n ( n−1 g2 ((k − i)=n))1=2 i=1
−
|
sup
log n=n6t61
0 g (y)W (t − y) dy| n 0 Rt = oP ((log log n)−1=2 ): 2 1=2 ( 0 g (y) dy)
Rt
(2.26)
Proof. It follows immediately from (2.5) in Theorem 2.1 that the left-hand side of (2.26) is of order OP ((log n)− ) for some ¿ 0. This proves (2.26). Corollary 2.2. Under the assumptions of Theorem 2:2; as n → ∞; Pn−1 | i=1 g((k − i)=n)(Zi − Z n )| max P P log n6k6n ( n−1 g2 ((k − i)=n) − (1=n)( n−1 g((k − i)=n))2 )1=2 i=1
i=1
g0 (y)Bn (t − y) dy| − sup Rt Rt = oP ((log log n)1=2 ): 2 2 1=2 log n=n6t61 ( 0 g (y) dy − ( 0 g(y) dy) ) |
Rt 0
(2.27)
Proof. It follows immediately from (2.21) in Theorem 2.2. Making use of Theorems 2.1, 2.2 and Corollaries 2.1, 2.2, we are now in a position to derive our main results.
3. Limiting extreme value distributions Our main results are based on the following theorem. Theorem 3.1. Let Z1 ; Z2 ; : : : be i.i.d. r.v.’s with EZ1 = 0; EZ12 = 1; and E|Z1 |2+ ¡ ∞ for some ¿ 0. Then ! Pk−1 (k − i) Z i x i=1 (3.1) lim P an max Pk−1 2 1=2 6x + bn = exp(−e ) n→∞ 26k6n ( i ) i=1
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
and lim P an max
n→∞
|
!
Pk−1
26k6n
i=1 (k − i) Zi | Pk−1 2 1=2 6x ( i=1 i )
+ bn
= exp(−2ex )
(3.2)
for all x; where an = (2log log n)1=2 ; and in case (i) ¿ 12 : 1 bn = 2log log n + log 4
2 + 1 2 − 1
1=2 ! ;
(ii) = 12 : 1 bn = 2 log log n + log log log log n − log(4); 2 (iii) 0 ¡ ¡ 12 : 1 − 2 C 1=(2 +1) H2 +1 log log log n + log bn = 2 log log n + 2(2 + 1) 1=2 22 =(2 +1)
! ;
with H2 +1 as in Remark 12:2:10 of Leadbetter et al. (1983); and Z ∞ y ((y + 1) − y − y −1 ) dy: C = −(2 + 1) 0
Under H0 ; extreme value limiting distributions for {Tn } and {T n } of (1.8) and (1.6), with weights xik as in (1.5), are now immediate from Theorem 3.1. Note that xik can be replaced by x˜ik = (i − k) + (i; k = 1; : : : ; n); since the denominator 1=n cancels out by the normalization. Theorem 3.2. Let Y1 ; : : : ; Yn satisfy the location model of (1:1) for n = 1; 2; : : : . Then; under H0 : m = n; Pn i=1 (i − k)+ (Yi − ) Pn 6x + bn = exp(−e−x ) (3.3) lim P an max 2 1=2 n→∞ k=1;:::; n−1 ( i=1 (i − k)+ ) and
lim P an
n→∞
max
k=1;:::; n−1
|
Pn i=1 (i − k)+ (Yi − )| Pn 6x + bn = exp(−2e−x ) 1=2 ( i=1 (i − k)2 +)
(3.4)
for all x; where an and bn = bn ( ) are as in Theorem 3:1 according to the cases ¿ 12 ; = 12 ; 0 ¡ ¡ 12 . If is replaced by Yn and by a suitable estimator ˆn satisfying (1.7), then the asymptotics of Theorem 3.1 retain, i.e.
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
67
Theorem 3.3. Under the assumptions of Theorem 3:2; Pn i=1 (i − k)+ (Yi − Y n ) Pn P 6x + b lim P an max n n 2 2 1=2 n→∞ k=1;:::; n−1 ˆn ( i=1 (i − k)+ − (1=n)( i=1 (i − k)+ ) ) = exp(−e−x ) and
(3.5)
Pn | i=1 (i − k) + (Yi − Y n )| Pn Pn 6x + bn lim P an max 2 2 1=2 n→∞ k=1;:::; n−1 ˆn ( i=1 (i − k)+ − (1=n)( i=1 (i − k)+ ) )
=exp(−2e−x );
(3.6)
for all x; where an and bn = bn ( ) are as in Theorem 3:1 according to ¿ 1 1 2; 0 ¡ ¡ 2.
1 2;
=
Remark 3.1. In case = 1, the results of Theorems 3.2 and 3.3 have been obtained by Jaruskova (1998). For the proof of Theorem 3.1 we make use of the following extreme value asymptotics: Proposition 3.1. Consider a zero mean; unit variance; stationary Gaussian process {X (t): t¿0} with autocorrelation r( ) = EX (t)X (t + ); t; ¿0. Assume that lim r( ) log = 0;
(3.7)
→∞
r( ) = 1 − C + o( ) Then
as ↓ 0:
lim P aT sup X (t)6x + bT
T →∞
06t6T
(3.8) = exp(−e−x )
(3.9)
for all x; where aT = (2 log T )1=2 ;
(3.10)
2− log log T + log(C 1= H (2)−1=2 2(2− )=(2 ) ); (3.11) 2 √ H as in Remark 12:2:10 of Leadbetter et al. (1983); H1 = 1; H2 = 1= . bT = 2 log T +
Proof. Confer Leadbetter et al. (1983, Theorem 12:3:5). We need yet another auxiliary result from extreme value theory: Proposition 3.2. Consider {X (t): t¿0} as in Proposition 3:1; and let 1 − r( ) be regularly varying of index 2; as ↓ 0. De ne v = vT to be the largest solution of the equation 1 =1 (3.12) (2 log T ) 1 − r v
68
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
and set u = uT = Then;
!1=2 √ Tv 2 2 log p : 2 log T
(3.13)
√ lim P uT sup X (t)6x + uT2 = exp(−e−x = )
T →∞
06t6T
(3.14)
for all x. Proof. This is a special case of Berman (1992, Theorem 10.6.1). Note that, in Berman √ (1992, (3.3.24)), ! = 1= : Proof of Theorem 3.1. We rst note that, as n → ∞, Pk−1 Pk−i Pk−1 | i=1 (i − (i − 1) ) j=1 Zj | | i=1 (k − i) Zi | = max max Pk−1 Pk−1 26k6log n 26k6log n ( i=1 i2 )1=2 ( i=1 i2 )1=2 k (k log log k)1=2 = OP max 26k6log n k +1=2 = OP ((log log log n)1=2 ) = oP ((log log n)1=2 ) which means that it suces to investigate the max over k ∈ [log n; n] only. In view of Corollary 2.1, the corresponding asymptotics can be derived from those of Rt Rt | 0 y −1 Wn (t − y) dy| | 0 y −1 Wn (t − y) dy| = sup (2 + 1)1=2 : sup Rt 2 +1 )1=2 2 1=2 (t ( 0 y dy) log n=n6t61 log n=n6t61 This is equivalent to investigating Rs | 0 x −1 W (s − x) d x| (2 + 1)1=2 sup (s2 +1 )1=2 log n6s6n by substituting s = nt; x = ny. We now put s = et , and consider the process {X (t): t¿0} given by X (t) = =
(2 + 1)1=2 (2 + 1)1=2
R et
x −1 W (et 0 et( +1=2)
R et
(et − 0 et( +1=2)
− x) d x
y) dW (y)
:
(3.15)
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
69
Note that {X (t): t¿0} is a zero mean Gaussian process with covariance function R es s (e − y) (et − y) dy cov(X (s); X (t)) = (2 + 1) 0 s( +1=2) t( +1=2) e e Z 1 = (2 + 1)e(s−t)=2 (1 − z) (1 − zes−t ) d z; 06s6t: 0
Clearly, {X (t): t¿0} is stationary with unit variance, and, in order to get the limiting extreme value distribution, it will be enough to check the behaviour of r(t), where r(t) − 1 = cov(X (s); X (s + t)) − 1 Z 1 (1 − z) (1 − ze−t ) d z − 1 = (2 + 1)e−t=2 0
(
= (2 + 1) e−t=2
+ (e
−t=2
Z − 1)
0
Z 0 1
1
(1 − z) ((1 − ze−t ) − (1 − z) ) d z ) 2
(1 − z) d z
:
(3.16)
Obviously, since r(t) = O(e−t=2 ) as t → ∞, condition (3.7) is satis ed for any ¿ 0. Concerning the asymptotics for t → 0, we prove the following lemma. Lemma 3.1. For r(t) as in (3:16); we have; as t → 0; if (i) ¿ 12 : r(t) = 1 −
2 +1 2 2 8(2 −1) t + o(t ); 1+o(1) (ii) = 12 : r(t) = 1 − 4 t 2 log( 1t ); (iii) 0 ¡ ¡ 12 : r(t)R= 1 − C t 2 +1 + o(t 2 +1 ), ∞ with C = −(2 + 1) 0 y ((y + 1) − y − y −1 ) dy.
Proof. (i) ¿ 12 : On setting t2 + o(t 2 ) 2 a Taylor series expansion gives = 1 − e−t = t −
(t ↓ 0);
(1 − ze−t ) − (1 − z) = (1 − z + z) − (1 − z) = (1 − z) −1 z + ( − 1)
2 z 2 −2 ; (1 − z + z) 2
R1 where 0 ¡ ¡ 61. Since, for ¿ 12 ; 0 (1 − z)2 −2 z 2 d z = (2 − 1) (3)= (2 + 2), we obtain, by the dominated convergence theorem, as t ↓ 0, Z 1 (1 − z) ((1 − ze−t ) − (1 − z) ) d z 0
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M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
Z =
1
0
(1 − z)2 −1 z d z +
( − 1) 2 2
Z
1
0
(1 − z)2 −2 z 2 d z(1 + o(1))
−1 1 + 2 + o(2 ): 2(2 + 1) 2(2 − 1)(2 + 1) R1 So, with (2 + 1) 0 (1 − z)2 d z = 1, t2 t −1 1 2 2 2 t − + o(t ) +t r(t) − 1 = 1 − + o(t) + o(t ) 2 2 2 2(2 − 1) =
t2 t − + + o(t 2 ) 2 8 −1 1 1 1 − + − + o(t 2 ) = −t 2 4 2(2 − 1) 4 8 2 + 1 + o(t 2 ): 8(2 − 1)
= −t 2
Note that, for = 1; (2 + 1)=8(2 − 1) = 3=8, which coincides with the results of Jaruskova (1998). (iii) 0 ¡ ¡ 12 : With as before, and on substituting 1 − z = y; −d z = dy, we have Z 1 (1 − z) ((1 − z + z) − (1 − z) ) d z 0
= 2 +1 = 2 +1 Z +
Z
1=
0
(Z
y ((y + 1 − y) − y ) dy
1=
0 1=
0
y ((y + 1 − y) − y − y −1 (1 − y)) dy )
y
2 −1
(1 − y) dy
= 2 +1 (I1 + I2 ):
Now, for 0 ¡ ¡ 1=2, 0¿(y + 1 − y) − y − y −1 (1 − y)¿ and
Z
1
0
y
2 −1
Z dy ¡ ∞;
1
∞
( − 1) −2 y 2
y2 −2 dy ¡ ∞:
Therefore, by dominated convergence, Z ∞ y ((y + 1) − y − y −1 ) dy: lim I1 = →0
0
M. Huskova, J. Steinebach / Journal of Statistical Planning and Inference 89 (2000) 57–77
71
Moreover,
y2 +1 y2 − I2 = 2 2 + 1
1= 0
2 1 1 : = 2(2 + 1)
So, r(t) − 1 = −C t 2 +1 (1 + o(1)) +
t t 1 1 − + O(t 2 ) (t + O(t 2 )) + − + O(t 2 ) 2 2 2
= −C t 2 +1 (1 + o(1)) + O(t 2 ) = −C t 2 +1 + o(t 2 +1 ) since 2 + 1 ¡ 2. (ii) = 12 : Consider I1 and I2 as in case (iii). Here, I2 =
1 4
and
Z I1 = O(1) +
1
= O(1) + I˜1
1=
y
1=2
1=2
(y + 1 − y)
−y
1=2
1 − y−1=2 (1 − y) 2
dy
(t ↓ 0):
First note that, by the same estimations as above, Z 1 (− 1 ) 1= 1 1 1 ˜I 1 ¿ 2 2 : ds = − ln 2 s 8 1 On the other hand, with y0 such that 16y0 61=; (y=y + 1)3=2 ¿1 − for all y¿y0 , Z (1 − ) 1= (1 − y)2 dy I˜1 6 − 8 y y0 1= 2 y2 (1 − ) ln y − 2y + =− 8 2 y0 1 (1 − ) ln (1 + o(1)) =− 8
(t ↓ 0):
Since ¿ 0 can be chosen arbitrarily small, this results in t 1 1 + + − + O(t 2 ) r(t) − 1 = − 2 ln 4 2 2 1 (1 + o(1)) 2 (t ↓ 0): t ln =− 4 t This proves Lemma 3.1.
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We are now in a position to complete the proof of Theorem 3.1. It is enough to consider MT = max X (t) 06t6T
and
M˜ T = max |X (t)| 06t6T
with X (t) from (3.15), and T = log n. In cases (i) and (iii), the limiting extreme value distributions of {MT } and {M˜ T } are now immediate from Proposition 3.1 and Lemma 3.1. The asymptotic of {M˜ T } follows from that of {MT } by symmetry and asymptotic independence of max and min in case of the Gaussian process {X (t)}. The case = 12 can be treated by Proposition 3.2 together with Lemma 3.1(ii). Recall v = vT ; u = uT from (3.12) and (3.13). A simple calculation shows that, as T → ∞, (1 + o(1)) log T log log T: 4 On choosing 1=2 1 log T log log T ; v˜ = 4 ! √ T v= ˜ 2 T (log log T )1=2 √ u˜ 2 = 2 log = 2 log (2 log T )1=2 4 v2 =
= 2 log T + O(log log log T ) = (1 + o(1))2 log T; we have u − u˜ =
√
u2
p 1 1 v 2 =o − u˜ = O log u˜ v˜ u˜
and by Theorem 10:6:1 of Berman (1992), u˜T (MT − u˜T ) = u˜T (MT − uT ) + OP (u˜T (uT − u˜T )) = (1 + o(1))uT (MT − uT ) + oP (1); since, as T → ∞,
√ P(uT (MT − uT )6x) → exp(−e−x = )
for all x. So, the same asymptotic applies to u˜ T (MT − u˜ T ). Moreover, (2 log T )1=2 (MT − u˜ T ) = (2 log T )1=2 MT − u˜ T (2 log T )1=2 and by an expansion of u˜ T , 1=2 (log log T )1=2 √ u˜ T (2 log T )1=2 = (2 log T )2 + 4 log T log 4 √ 1 = 2 log T + log log log T − log(4 ) + o(1) 2 = b˜˜T + o(1):
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So, with a˜T (2 log T )1=2 and b˜˜T as above, √ P(a˜ M 6x + b˜˜ ) → exp(−e−x = ) T
T
T
for all x, which is equivalent to P(a˜T MT 6x + b˜T ) → exp(−e−x ) √ for all x, if b˜T = b˜˜T + log( ). This completes the proof of Theorem 3.1. Note that MT = max06t6T X (t) can be replaced by maxlog T 6t6T X (t), since X (t) = OP ((log log T )1=2 ) = oP ((log T )1=2 ) (T → ∞):
max
06t6 log T
Proof of Theorem 3.2. On replacing i = n + 1 − j; k = n + 1 − ‘, assertion (3.3) is immediate from Theorem 3.1, since D
{Zj }j=1; :::; n :={(Yn+1−j − )=}j=1; :::; n ={(Yj − )=}j=1; :::; n satis es the required assumptions. Proof of Theorem 3.3. Since, by condition (1.7), 1 1 − = oP ((log log n)−1 ) (n → ∞); ˆn it suces to prove (3.5) with ˆn replaced by . Assume w.l.o.g., = 0 and 2 = 1, and set i = n + 1 − j; k = n + 1 − ‘; Yn+1−j = Zj , i.e. we consider P‘−1 j=1 (‘ − j) (Zj − Z n ) : max P‘−1 P ‘−1 2 1=2 2 ‘=2;:::; n ( j=1 j − (1=n)( j=1 j ) ) We have ‘−1 P j=1 ‘−1 P j=1
j 6
‘ +1 ; +1
j 2 6
‘−1 P j=1
‘2 +1 ; 2 + 1
j ¿
‘−1 P j=1
(‘ − 1) +1 ; +1
j 2 ¿
(‘ − 1)2 +1 ; 2 + 1
so that, by Jensen’s inequality, for ‘6n=log n, as n → ∞, !2 ‘−1 ‘−1 P 2 ‘−1 P 2 1 ‘−1 P P 2 1 : j ¿ j − j ¿ j 1+O n j=1 log n j=1 j=1 j=1 Moreover, if ‘6n=log n, for some c0 ¿ 0, P‘−1 j=1 (‘ − j) 6c0 ‘1=2 : P‘−1 P‘−1 ( j=1 j 2 − (1=n)( j=1 j )2 )1=2 On setting S‘ =
‘−1 P j=1
(‘ − j) Zj ;
S‘ =
‘−1 P j=1
(‘ − j) (Zj − Z n );
(3.17)
(3.18)
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we obtain
1=2 P‘−1 var S‘ S‘ S‘ j=1 (‘ − j) Z n = − : (var S‘ )1=2 (var S‘ )1=2 var S‘ (var S‘ )1=2 √ Since |Z n | = OP (1= n); (3.17) – (3.19) imply S‘ 26‘6n=log n (var S ‘ )1=2 S‘ 1 1 + OP = max 1+O : 26‘6n=log n (var S‘ )1=2 log n (log n)1=2
(3.19)
max
(3.20)
On the other hand, uniformly for n=log n6‘6n; as n → ∞; ‘−1 P 2 log n ‘2 +1 1+O ; j = 2 + 1 n j=1 ‘ +1 j = +1 j=1
‘−1 P
log n 1+O n
so that 16
var (S‘ ) = O(1); var (S‘ )
and max
n=log n6‘6n
|
(3.21)
P‘−1
− j) Z n | = OP (1): (var S‘ )1=2
j=1 (‘
(3.22)
Finally, since max
log(n=log n)6t6log n
D
|X (t)| =
max
06t6log log n
|X (t)| = OP (log log log n);
also |S‘ | = OP (log log log n); n=log n6‘6n (var S‘ )1=2
(3.23)
|S‘ | = OP (log log log n): n=log n6‘6n (var S ‘ )1=2
(3.24)
max
and max
A combination of (3.19) – (3.24) completes the proof of Theorem 3.3. Remark 3.2. In view of Theorem 3.3, asymptotic critical values for the testing of H0 vs. H1 , or H0 vs. H1+ , respectively, can now be chosen as appropriate upper quantiles of the two-sided or one-sided Gumbel distribution. The latter tests are consistent as will be outlined next.
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4. Consistency In this section, we brie y discuss consistency of the asymptotic tests suggested above. It holds that, for the location model of (1.1), under H1 , as n → ∞, Pn | i=1 (i − k) + (Yi − )| P Pn − bn → +∞ (4.1) an max 1=2 k=1;:::; n−1 ( i=1 (i − k)2 ) + with an and bn as in Theorem 3.1, provided +1=2 , p √ n − mn log log n → +∞: |n | n n Moreover, under H1+ , as n → ∞, Pn P i=1 (i − k)+ (Yi − ) Pn − bn → +∞; an max 2 1=2 k=1;:::; n−1 ( (i − k) ) + i=1 provided √
n n
n − mn n
+1=2 ,
p log log n → +∞:
We also have, under H1 , as n → ∞, Pn | i=1 (i − k) + (Yi − Yn )| P Pn Pn − bn → +∞; an max 2 2 1=2 k=1;:::; n−1 ˆn ( (i − k) − (1=n)( (i − k) ) ) + + i=1 i=1 provided ˆn = OP (1) and (4.2) holds, and, under H1+ , as n → ∞, Pn P i=1 (i − k)+ (Yi − Yn ) Pn Pn − bn → +∞; an max 2 2 1=2 k=1;:::; n−1 ˆn ( (i − k) − (1=n)( (i − k) ) ) + + i=1 i=1
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
provided ˆn = OP (1) and (4.4) holds. This proves consistency of the asymptotic tests rejecting H0 for large values of the test statistics, if critical values are chosen from the asymptotic distributions in Theorems 3.2 and 3.3, respectively. We prove (4.5) and (4.6), for example. The other statements can be proved in a Pn similar vein. Note that, with m = mn ; = n ; xim = ((i − m)=n) + ; xm = (1=n) i=1 xim , n n P P i−m (Yi − Y n ) = (xim − xm )(Yi − ) n i=1 i=1 + n−m n P 2 i P − ng 2 + g xim (ei − e n ); = n i=1 i=m+1 Pn−m Pn ; g = (1=n) i=1 g(i=n); e n = (1=n) i=1 ei . where g(y) = y+ Similar to the proof of Theorem 3.3, on separating the cases n − m6log n and n − m¿log n, respectively, we have that 2 +1 P 2 i n−m 1 n−m − g 2 ¿c0 g ; (4.7) n i=1 n n
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P 2 1 n−m g n i=1
2 +1 i n−m − g 2 6c1 n n
(4.8)
with positive constants c0 ; c1 . So, according to Theorem 3.3 together with (4.7) and (4.8), Pn i=m+1 (i − m) (Yi − Y n ) Pn P − bn an n ˆn ( i=m+1 (i − m)2 − (1=n)( i=m+1 (i − m) )2 )1=2 +1=2 , √ c0 n−m ¿an n n √ ˆn + OP (a2n ) c1 n which suces for the proof of (4.6). For the proof of (4.5), note that condition (4.2) implies +1=2 ! √ n − m n (as n → ∞) bn = O(a2n ) = o an n n n which, via the triangle inequality, suces for (4.5). 5. Concluding remarks (a) The asymptotics of Theorems 3.1–3.3 are not at all restricted to the location model of (1.1) with i.i.d. errors. There is a more general principle behind which allows for testing gradual changes in the mean of increments of certain stochastic processes by only taking advantage of a suitable weak invariance principle. This idea was pursued by Horvath and Steinebach (1999) in case of models with abrupt changes, and has been extended by Steinebach (1999) to the case of gradual changes. (b) Instead of considering the statistics T n and Tn of (1.6) and (1.8) which are motivated by an asymptotic log-likelihood ratio approach, one could also think of statistics with heavier weights in the normalizing denominators. This idea, developed by Huskova (1998) for the case = 1, can be extended to the case of general ¿ 0, and to other models of gradual changes satisfying a suitable invariance principle. Details of this will be discussed elsewhere. Acknowledgements The authors would like to thank the referees for valuable remarks. References Arnold, L., 1973. Stochastische Di erentialgleichungen. Oldenbourg, Munchen. Berman, S.M., 1992. Sojourns and Extremes of Stochastic Processes. Wadsworth, Belmont.
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