Asymptotic theory for bent-cable regression the basic case

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1 Journal of Statistical Planning and Inference 27 (2005) Asymptotic theory for bent-cable regression the basic case Grace Chiu a;b;, Richard Lockhart a, Richard Routledge a a Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, British Columbia, Canada V5A S6 b Pacic Institute for the Mathematical Sciences, Simon Fraser University, Burnaby, British Columbia, Canada V5A S6 Accepted 6 September 2003 Abstract We use what we call the bent-cable model to describe potential change-point phenomena. The class of bent cables includes the commonly used broken stick (a bent cable without a bend segment). Theory for least-squares (LS) estimation is developed for the basic bent cable, whose incoming and outgoing linear phases have slopes 0 and, respectively, and are joined smoothly by a quadratic bend. Conditions on the design are given to ensure regularity of the estimation problem, despite non-dierentiability of the model s rst partial derivatives (with respect to the covariate and model parameters). Under such conditions, we show that the LS estimators (i) are consistent, regardless of a zero or positive true bend width; and (ii) asymptotically follow a bivariate normal distribution, if the underlying cable has all three segments. In the latter case, we show that the deviance statistic has an asymptotic chi-squared distribution with two degrees of freedom. c 2003 Elsevier B.V. All rights reserved. MSC: 62F2 (62J02) Keywords: Change points; Segmented regression; Least-squares estimation; Asymptotic theory This research has been funded by the Natural Sciences and Engineering Research Council of Canada through a Postgraduate Scholarship to G. Chiu during her graduate studies at SFU, and Research Grants to R. Lockhart and R. Routledge. Corresponding author. addresses: sgchiu@stat.sfu.ca, grace@pims.math.ca (G. Chiu), lockhart@stat.sfu.ca (R. Lockhart), routledg@stat.sfu.ca (R. Routledge) /$ - see front matter c 2003 Elsevier B.V. All rights reserved. doi:0.06/j.jspi

2 44 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) Introduction Given known design points x ;:::;x n, we consider random responses Y ;:::;Y n generated by the regression model where Y i = q(x i ; 0 ; 0 )+ i ; i=;:::;n; () (x + )2 q(x; ; )= { x 6 } +(x ){x } (2) 4 is referred to as the basic bent cable (Fig. ), and i s are i.i.d. random errors with mean 0 and known, constant standard deviation. We write 0 =( 0 ; 0 ). Least-squares (LS) estimation of 0 =( ;M] [0; ) on the open regression domain X = R is considered. Here, M is some nite positive upper bound (large) for the candidate -values. Zero is the natural lower bound for candidate -values. Any basic bent cable q(x; ) for is a candidate model. In this article, we prove, given a set of conditions on the location of the design points x ;:::;x n (Section 3), that the least-squares estimators (LSEs) for 0 and 0 are consistent when 0 0, and asymptotically follow a bivariate normal distribution when 0 0. Asymptotic distributional properties for the case of 0 = 0 appear in Chiu et al. (2002a). A bent cable with free slope parameters is required in practice. The full bent-cable model can be written as f(x; 0 ; ; 2 ;;)= 0 + x + 2 q(x; ; ). This article is intended to provide a framework for the complex estimation theory associated with the full model. Seber and Wild (989, Chapter 9) have suggested employing the class of bent-cable models which includes the piecewise-linear broken-stick model when = 0 in situations where both smooth and sharp transitions are plausible. However, modeling change phenomena by the broken stick remains common (Barrowman and Myers, 2000; Naylor and Su, 998; Neuman et al., 200). Numerical instability due to the non-dierentiability of this model prompted Tishler and Zang (98) to develop (2). Their introduction of a phoney bend of xed, non-trivial width to replace the kink at was a computational tactic. Upon numerical convergence, would be ignored. However, when no law of nature or auxiliary knowledge is available to support an abruptness notion, a broken-stick t would encourage the investigator to look for sources of change associated with the sole value of ˆ. In contrast, the bent cable incorporates as part of the parametric model. It generalizes the broken stick by removing the a priori assumption of a sharp threshold, allowing for a possibly gradual transition. A bent-cable t would point to one or more sources of change whose inuence took hold gradually over a certain covariate range. Thus, it helps to avoid data misinterpretation due to possible over-simplication of the nature of change. We call it the bent cable due to the smooth bend as opposed to a sharp break in a snapped stick. The performance of bent-cable regression for assessing the abruptness of change is discussed in Chiu et al. (2002b). In practice, estimation of may be required. Chiu (2002) shows that the results of this article extend to LS estimation assuming unknown, and that the LSE of is consistent.

3 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) Fig.. The basic bent cable q(x; ; ), with the dotted quadratic bend. (The solid line is the broken stick.) 2. Theory for segmented models Whether is zero or positive, the second partial derivatives of (2) fail to exist where the segments meet. This irregularity prevents the direct application of asymptotics readily available for LS problems that are regular. Feder (975) and Ivanov (997), among others, have discussed asymptotics for non-linear regression involving segmented models of general forms with unknown join points. Similar regression problems include the consideration of broken-stick models (Bhattacharya, 990; Huskova, 998) and general two-phase linear-nonlinear or other multiphase non-linear models (Gallant, 974, 975; Huskova and Steinebach, 2000; Jaruskova, 998a, b, 200; Rukhin and Vajda, 997). Feder s principal assumption is continuity of the underlying function without extra smoothness constraints. If the underlying model has an odd order of smoothness (number of continuous derivatives plus one), then the asymptotics are radically dierent compared to those for an even order. Gallant considers once-dierentiable candidate and underlying models of common functional form. In the case of a once-dierentiable quadratic quadratic model, consistency and asymptotic normality of the LSE for the unknown joint is established by, among other requirements, (i) taking the design points to be near replicates of a basic design with ve distinct covariate values (see Gallant,

4 46 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) , p. 26; Gallant, 974, pp. 6 7), and (ii) restricting the search for candidate knots to within a compact subset trapped between two consecutive covariate values from (i) (see Gallant, 975, p. 26). Gallant, Ivanov, and Rukhin and Vajda all establish consistency by assuming a bounded or compact parameter space. Ivanov adds a somewhat unintuitive condition which relates the candidate model to the response error variability (see Ivanov, 997, p. 30, expression (3.3)). Rukhin and Vajda simply exclude the LSE from their M-estimators (see condition (C3) in their article). To establish asymptotic normality, Rukhin and Vajda and Ivanov assume twice-dierentiability of the regression function, although Rukhin and Vajda cite a tactic for relaxing such an assumption (see Vaart and Wellner, 996, Chapter 3). Huskova, Huskova and Steinebach, Jaruskova, and Rukhin and Vajda all consider evenly spaced regressors, while Bhattacharya considers a more general design. These authors have all shown standard LS asymptotics despite a lack of higher-order derivatives of the regression function. However, their results are not directly applicable to our specialized problem. We consider underlying and candidate models within the class of bent cables. General and structurally simple design conditions are provided and proved to suce for establishing regularity. 3. Conditions for the basic bent cable Besides an unspecied upper bound on the -space, our only regularity conditions are placed on the design. Given 0 and sequence n 0, rst dene c 0 () = lim inf { xi }; n n c () = lim inf n c + () = lim inf n n n { xi }; {xi }; n ( n )= n { xi n }: The regularity conditions on x ;:::;x n are [A ]If 0 =0; then 0; c 0 () 0. [A 2 ]If 0 0; then 0 such that c c ( ) 0. [B] For 0 0; 0 such that c+ c + ( ) 0. [C] If 0 0; then n 0; n ( n ) 0. [D] If 0 0; then x i 0 ± 0 i =;:::;n. The practical value in these conditions is that they indicate a general design (not necessarily equidistant) which ensures that data are collected at appropriate x-locations for reliable LS estimation of 0. While 0 is unknown, the investigator s expertise in the subject matter should suggest a range of design points that easily satises conditions

5 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) [A] and [B]. Satisfying condition [C] in practice is not dicult as the precise locations of 0 ± 0 are unknown. For a continuous covariate, it is reasonable to rule out exact equality that would violate Condition [D]. At the cost of some notational complexity, [D] could be entirely eliminated. 4. Method and results For normally distributed i s in (), the log-likelihood as a function of is n ( )= 2 2 S n( ) + constant; where S n ( )= n i= [Y i q(x i ; )] 2. In this case, maximum likelihood (ML) and LS theories coincide. In the absence of normality, we maximize n but do not refer to it as the ML function. Maximization of n is equivalent to minimization of S n (i.e. LS estimation). The results in this section imply that ML and LS asymptotics coincide even without the normality of the i s in the latter case. When a sample yields multiple maximizers of n, we take the LSE ˆ n to be the one selected sequentially as follows: (i) pick out the one(s) with the least vector norm; (ii) keep the one(s) with the least ; (iii) if necessary, select that with the least. The following notation is used in stating the results: U n ( ( ); U n( ( ); U n( )=(U n ( ); U n ( )) T ; V n ( )= U n ( ) (wherever dened); I n ( ) = Cov 0 [U n ( )]: Note that I n and U n are analogous to the Fisher Information and the score function in ML estimation. With conditions [A] [D], our only irregularity lies in a Hessian, V n, that is not well-dened everywhere. However, [D] preserves regularity at 0, and is used to establish the uniform convergence (in probability) of V n and I n in a shrinking neighborhood of 0 (Lemma 3). Regular LS asymptotics follow. Condensed proofs for the following key results appear in Section 6. Motivational and mathematical details are in Chiu (2002). Lemma (Identiability). Take and from [A] and [B]. Suppose w [ ; ). Furthermore, () if 0 0 and w 0 [ ; ], then q(w i ; 0 )=q(w i ; ) for i =; 2 implies = 0 ;(2) if 0 =0 and w 0 [ 0 =8; 0 +=8], where (0; =2), then q(w i ; 0 )=q(w i ; ) for i =; 2 implies 0 6. Remark. This lemma is fundamental to establishing consistency. Theorem (Consistency): Consider LS estimation for model () as dened in Section. Under conditions [A] and [B], ˆ n P 0 as n.

6 48 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) Before stating our next theorem, we provide a fact about the square-root matrix to avoid ambiguity. For a symmetric positive-denite matrix A, there is a unique symmetric positive-denite matrix B such that A = B 2 = B T B (Golub and Van Loan, 996, Exercise P.2.4.). We write B = A =2, the square root of A. Theorem 2 (Asymptotic Normality). Consider LS estimation for model () as dened in Section. Under conditions [A] [D] and 0 0,. (=n)i n ( 0 ) is positive denite for all suciently large n; 2. n[(=n)i n ( 0 )] =2 ( ˆ n 0 ) converges in distribution to a standard bivariate normal random variable; 3. P 0 {(=n)i n ( ˆ n ) is positive denite} ; 4. n[(=n)i n ( ˆ n )] =2 ( ˆ n 0 ) converges in distribution to a standard bivariate normal random variable; 5. G n 2[ n ( ˆ n ) n ( 0 )], the deviance statistic, converges in distribution to a 2 random variable with 2 degrees of freedom. 5. Auxiliary lemmas for proving Theorem 2 First, note that the ith summands of U n ( ) and U n ( ) are, respectively, U ;i ( )= 2 [ i+d 0; (x i (x i where d ; 2 (x)=q(x; ) q(x; 2 (x (x i = x i ( ) 2 [ = 4 x i ; U ;i ( )= 2 [ i+d 0; (x i (x i ) { x i 6 } {x i + }; (3) 2] { x i 6 }: (4) Thus, for each i, both U ;i and U ;i are continuous surfaces over the (; )-plane, but they have folds along the rays dened by R i+ = { : = x i }; R i = { : = x i }: Summing these surfaces over i produces continuous U n and U n surfaces, each with n pairs of folds indexed by the data x ;:::;x n (Chiu, 2002, Fig. 4.). As a result, the matrix V n is well-dened everywhere on except along the R i± s. To avoid technical diculties we dene a directional Hessian: V j;i ( ) = lim h U j;i( + h; ); V j;i ( ) = lim h U j;i(; + h)

7 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) for all i =;:::;n and j = ;. Of course, these V i + s are merely regular derivatives when evaluated at some R k± ; k =;:::;n. Now, we can replace V n by [ n V + V + ;i ( ) V ;i + n ( )= ( ) ] i= V ;i + ( ) V ;i + ( ) ; which is well-dened on, and coincides with V n ( ) except along the R k± s. Lemma 2. For all, we have [ T U n ( )=U n ( 0 )+ Vn( ;t)dt] ( 0 ); (5) 0 where, for all t [0; ] and, [ n V + V ;i ( 0 + t( 0 ); 0 ) V ;i + n( ;t)= ( ] 0 + t( 0 ); 0 ) V ;i + (; 0 + t( 0 )) V ;i + (; : 0 + t( 0 )) i= Remark 2. Had the Hessian been continuous, all its components in the integrand would have been simply ( 0 +t( 0 ); 0 +t( 0 ))= 0 +t( 0 ). As V + n is discontinuous, slightly dierent arguments are used, as given by V n. Lemma 3. Given are conditions [A] [D], and a sequence n 0. Then, j; k = ; + V + n;jk ( ) P 0 as n ; (6) I n;jk ( 0 ) sup n where r = { : 0 6 r} and V + n;jk similarly for I n;jk. denotes the (j; k)th component of V+ n, and Lemma 4 (Corollary to Theorem ): Under conditions [A] and [B], there exists a decreasing sequence n 0 such that P 0 { ˆ n 0 6 n } as n. Remark 3. In proving Theorem 2, we concentrate on the behavior of the U n surface over n. 6. Proofs In addition to the notation from Sections 4 and 5, write T n ( )= n d 0; (x i ) 2 ; n S n ( )= n i + d 0; (x i ) 2 ; S n ( )= n S n( ); H n ( )=E 0 [ S n ( )]:

8 50 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) Proof of Lemma. See Chiu (2002). Proof of Theorem. This proof of consistency is mostly standard, with the exceptions that the parameter space is unbounded, and that the boundary value of 0 =0 is handled properly. Below, we highlight the crucial elements of the proof. Claim. There exists M 0 such that P 0 { ˆ n 0 M } 0. Proof. By conditions [A] and [B], there are M ;M 2 ;N 0 such that n N inf M 2 [ M ;M] inf M 0 n T n( ) 6 2 : This lower bound and the Strong Law of Large Numbers can then be applied to show that inf M S n ( ) a:s: S n ( ˆ n ) for all n N, where M = max{m ;M 2 }. Claim 2. For each (0; =2), there is an 0 such that lim inf n inf D {H n ( ) H n ( 0 )}, where D = { : M }. Proof. Dene T (w 0 ;w ; )= d 0; (w 0 ) 2 + d 0; (w ) 2 and { [0 C ;R = (w 0 ;w ): w 0 + ; ] if [ 0 =8; 0 + =8] if 0 =0 w [ ;R] for some R ( M ; ). By Lemma, a compactness argument can show that ;R inf {T (w 0 ;w ; ): D ; (w 0 ;w ) C ;R } 0. Next, pick (0;=4) and dene c = min{c ;c 0 ( );c +} 0. Then, choose from the dataset {x ;:::;x n } pairs of design points (x 0j ;x j ) C lim K C ;K. Conditions [A] and [B] imply that there are at least nc such pairs for all n N for some N. Moreover, it can be shown that ;K = ;R for all K R. Take = ;R c. Then, T n ( )= n n d 0; (x i ) 2 nc T (x 0j ;x j ; ) n n i= j= nc j= ; { } inf T ;R c = : C This lower bound is positive and -free. Then, inf D T n ( ) for all n N for the given D. Since T n ( )=H n ( ) H n ( 0 ), the claim follows. Claim 3. For all 0; P 0 {sup M S n ( ) H n ( ) 6 }. Proof. We employ a standard chaining argument, using the fact that the basic bent cable q(x i ; ) satises the Lipschitz condition q(x; ) q(x; 2 ) = d ; 2 (x) 6 B r 2 ; 2 r x R; (7)

9 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) where r 6 M, and B r is a positive constant that can be derived using (3) and (4). The remainder of the proof is straightforward. Proof of Lemma 2. Each summand of U n is piecewise continuously dierentiable as a function of either or. Thus, for example, integrating V + n; over gives U n (; ) U n ( 0 ;)= 0 n i= =( 0 ) 0 V ;i + (s; )ds n V ;i + ( 0 + t( 0 );)dt: i= The algebra is similar for integrating V n; + over and for other V + n;jk s. Then, since U nj ( )=U nj ( 0 ; 0 )+[U nj (; 0 ) U nj ( 0 ; 0 )]+[U nj (; ) U nj (; 0 )] for all j = ;, the lemma follows by routine algebra. Proof of Lemma 3. We examine the components of V + n and I n. To simplify the algebra, write i = {x i + }; 2i = x i ( ) { x i 6 }; 2 [ 3i = x i 2] 4 { x i 6 }; 4i = x i all of which are functions of (but the argument is suppressed in the notation). (x i )=@ = i + 2i (x i )=@ = 3i, and the directional derivatives for the summands of U n are V ;i + ( )= I ;i [ i + d 0; (x i )]{ x i 6 + }; V + ;i ( )= I ;i + V + ;i ( )= I ;i + V + ;i ( )= I ;i + 4i [ i + d 0; (x i )]{ x i 6 + }; 4i [ i + d 0; (x i )]{ x i 6 }; 2 4i [ i + d 0; (x i )]{ x i 6 }; where the summands of the components of I n are I ;i ( )= i + 2 2i 2 ; I ;i ( )= 2i 3i 2 = I ;i ( ); I ;i ( )= 2 3i 2 : (8)

10 52 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) By conditions [A] and [B], one can show that there are N and M jk m jk 0; j;k=;, such that n N implies m jk 6 n I n;jk( 0 ) 6 M jk j; k = ; : (9) That is, for all j; k = ; ; I n;jk ( 0 ) is bounded between two non-trivial multiples of n for all suciently large n. Thus, to prove the lemma, it suces to show sup n;jk ( ) =o(n) (0) n sup n n;jk ( ) =o p (n) () for each pair of (j; k), where n;jk ( ) =I n;jk ( ) I n;jk ( 0 ) and n;jk ( ) =V + n;jk ( ) + I n;jk ( ). There are three cases: (a) j = k =, (b) j = k =, and (c) j k. We do only Case (a). First, consider (0). Given any n, apply (8) toget 2 n; ( ) 6 2 i 2 i; i 2 2i;0 ; (2) where ki;0 (k =; 2; 3) is the value of ki with replaced by 0. For the second sum in (2), partition the index set {;:::;n} into the four sets I {i : x i ; x i 0 0 }; I 2 {i : x i 6 ; x i }; I 3 {i : x i 6 ; x i 0 0 }, and I 4 {i : x i ; x i }. Then, use (7) to show that sup n 2 2i 2 2i;0 =O( n) over I and I 2. For i I 3 I 4, note that 2 2i 2 2i;0 6. Now, dene K + n =[ n ; n ] and K n =[ 0 0 n ; n ], which shrink to the join points 0 ± 0 as n. Then, apply condition [C] to see that for all suciently large n, n 2i 2 2i;0 2 = no( n )+ i= 6 no() + i I 3 I 4 2 2i 2 2i;0 i: x i K ± n 6 o(n)+n n ( n )=o(n): This bound is -free. A similar argument applies to the rst sum in (2), with a suitable partition of the index set. Therefore, n; ( ) is uniformly bounded by o(n) over n. (For Cases (b) and (c), not having accumulation at the joints 0 ± 0 is crucial in keeping the matrices I n ( ) and I n ( 0 ) close everywhere on n.) To show (), Case (a), rst, dene a n; ( )=(2) d 0; (x i ){ x i 6 + } and B n; ( ) =(2) i { x i 6 + } so that 2 n; ( ) =a n; ( ) + B n; ( ). By (7), one can show that sup n a n; ( ) =o(n). Now, relabel the data so that x 6 x 2 6 6x n, and dene the martingale M m = m i= i; 6 m 6 n. For each n, there are integers s and t; 6 s 6 t 6 n, such that n i= i{ x i 6 +

11 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) } = M t M s. Hence, B n; ( )= M t M s 2 6 M t + M s n max 6m6n M m This -free bound is O p ( n) by the Doob Kolmogorovinequality (see Breiman, 968, p. 89, Problem 2). Thus, sup n n; ( ) 6 o(n)+o p ( n)=o p (n). Note that for Cases (b) and (c), each summand of B n; ( ) orb n; ( ) has the form i! i ( ), and the martingale tactic above does not apply. Instead, the reader can verify that a routine chaining argument suces in these cases. Proof of Theorem 2. Part. Denote the eigenvalues of the covariance matrix n I n ( 0 ) by n and n2, where 0 6 n 6 n2. With the upper bounds of (9), it remains to show integer N and 0 s:t: n N n : (3) One can prove (3) by relating n and n2 to the trace and determinant of n I n ( 0 ), then applying conditions [A] and [B]. We omit the details. Part 2. By eigenvalue properties, Lemma 3, and Part of the theorem, P 0 { n V+ n ( ) is negative denite for all n } : (4) : Now, we introduce a lemma about the concavity of a once-dierentiable function. Its rst assertion is due to Theorem in Stoer and Witzgall (970), and the second, the denition of concavity. We omit the proof. Lemma 5. () Let N be a subset of [0; ] consisting of isolated points. Suppose that a dierentiable function f :[0; ] R has a continuous second derivative, f, in [0; ] \ N. Then, f is strictly concave on [0; ] if f (t) 0 for all but those isolated values of t N. (2) A dierentiable function g : n R is strictly concave if and only if h(t) g( + t( 2 )) is concave on [0; ] for each ; 2 n, where 2. For our problem, we study h n (t) n ( + t( 2 )) for ; 2 n ; 2 ; t [0; ]. In what follows, we restrict our attention to the event dened in (4). First, consider and 2 which do not both lie on R k± for any k. They dene a line segment along which n is piecewise twice dierentiable. By the chain rule, [ ] n h n (t)=( 2 ) T n V n( + t( 2 )) ( 2 ) (5) [ ] =( 2 ) T n V+ n ( + t( 2 )) ( 2 ) 0 (6) for all but isolated values of t [0; ]. By Lemma 5, it follows that n n, hence, n, is concave along this line segment.

12 54 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) We are left to examine the case where and 2 both lie on a ray. By symmetry, we need only to consider, say, the 45 R k+ s. Here, the chain rule cannot be applied to yield (5), since V n is not dened at + t( 2 ) for any t [0; ]. Instead, one can expand h n (t), dierentiate it twice, and verify, after some lengthy algebra, that n h n (t) indeed equals the expression in (6) for all but isolated values of t [0; ]. These isolated values correspond to the intersections of the line segment (joining and 2 on R k+ ) with the 35 R i s. Thus, we have shown that every cross-section, and hence, the entire surface, of n over n is strictly concave on the event dened in (4). By consistency, P 0 { ˆ n is the unique maximizer of n over n } : Now, this unique ˆ n can be substituted into the Taylor-type expansion of Lemma 2. Finally, asymptotic normality of ˆ n results from standard matrix manipulation by noting that the summands of U n ( 0 ) satisfy the Lindeberg Condition, and thus Part and the Lindeberg Feller Central Limit Theorem ensure that [I n ( 0 )] =2 U n ( 0 )is asymptotically standard normal. Parts 3 and 4. The results follow, respectively, from Parts and 2 due to consistency and (0). Part 5. Since n is once-dierentiable, we can write n ( = n ( 0 )+( 0 ) T U n ( 0 + t( 0 )) dt: (7) By Lemma 2, we have, for all t [0; ], 0 [ T U n ( 0 + t( 0 )) = U n ( 0 )+ V n( 0 + t( 0 );s)ds] t( 0 ): (8) Substitute (8) into (7) and apply Lemma 3 and Part 2 to complete the proof Concluding remarks We have given some structurally simple conditions on the design. These provide a practical guideline for data collection when considering basic bent-cable regression. We have shown that these few regularity conditions suce to compensate for the intrinsic irregularity of the problem due to non-dierentiability of the model s rst partial derivatives. In particular, they ensure that the I n ( 0 ) matrix is asymptotically well-behaved, in the sense that the amount of information about the unknown parameter 0 contained in all suciently large datasets is no less than a non-trivial multiple of the sample size. Furthermore, as V + n ( ) is the negative (directional) Hessian of n, its uniform closeness to I n ( 0 )on n ensures that its discontinuities are asymptotically negligible, leading to an n -surface that is uniformly well-behaved

13 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) on this ever-decreasing neighborhood of 0. This is the basis of the regularity of our problem, despite U n surfaces that are folded. Thus, standard asymptotic results apply. With slightly modied conditions, our results here can be extended to the full bent-cable model with non-normal errors and unknown constant variance. The details are to appear in a future article and are currently available in Chiu (2002). Acknowledgements We thank the Co-ordinating Editor and referees for providing valuable suggestions and extra references. References Barrowman, N.J., Myers, R.A., Still more spawner-recruitment curves: the hockey stick and its generalizations. Can. J. Fish. Aquat. Sci. 57, Bhattacharya, P.K., 990. Weak convergence of the log likelihood process in the two-phase linear regression problem. Proceedings of the R.C. Bose Symposium on Probability, Statistics and Design of Experiments, Wiley Eastern, New Delhi. Breiman, L., 968. Probability. Addison-Wesley, Reading, MA. Chiu, G.S., Bent-cable regression for assessing abruptness of change. Ph.D. Thesis, Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC. Chiu, G., Lockhart, R., Routledge, R., 2002a. Bent-cable asymptotics when the bend is missing. Statist. Probab. Letters 56, 9 6. Chiu, G., Routledge, R., Lockhart, R., 2002b. Using the bent cable to assess an abrupt change in species abundance and other phenomena. Winning student paper. WNAR/IMS 2002 Conference, Los Angeles, CA. Feder, P.I., 975. On asymptotic distribution theory in segmented regression problems identied case. Ann. Statist. 3, Gallant, A.R., 974. The theory of nonlinear regression as it relates to segmented polynomial regressions with estimated join points. Mimeograph Series No Institute of Statistics, North Carolina State University, Rayleigh, NC. Gallant, A.R., 975. Inference for nonlinear models. Mimeograph Series No Institute of Statistics, North Carolina State University, Rayleigh, NC. Golub, G.H., Van Loan, C.F., 996. Matrix Computations. Johns Hopkins University Press, Baltimore, MD. Huskova, M., 998. Estimators in the location model with gradual changes. Comment. Math. Univ. Carolinae 39, Huskova, M., Steinebach, J., Limit theorems for a class of tests of gradual changes. J. Statist. Plann. Inference 89, Ivanov, A.V., 997. Asymptotic Theory of Nonlinear Regression. Kluwer Academic Publishers, Dordrecht. Jaruskova, D., 998a. Testing appearance of linear trend. J. Statist. Plann. Inference 70, Jaruskova, D., 998b. Change-point estimator in gradually changing sequences. Comment. Math. Univ. Carolinae 39, Jaruskova, D., 200. Change-point estimator in continuous quadratic regression. Comment. Math. Univ. Carolinae 42, Naylor, R.E.L., Su, J., 998. Plant development of triticale cv. Lasko at dierent sowing dates. J. Agric. Sci. 30, Neuman, M.J., Witting, D.A., Able, K.W., 200. Relationships between otolith microstructure, otolith growth, somatic growth and ontogenetic transitions in two cohorts of windowpane. J. Fish Biol. 58,

14 56 G. Chiu et al. / Journal of Statistical Planning and Inference 27 (2005) Rukhin, A.L., Vajda, I., 997. Change-point estimation as a nonlinear regression problem. Statistics 30, Seber, G.A.F., Wild, C.J., 989. Nonlinear Regression. Wiley, New York. Stoer, J., Witzgall, C., 970. Convexity and Optimization in Finite Dimensions, I.. Springer, Berlin. Tishler, A., Zang, I., 98. A new maximum likelihood algorithm for piecewise regression. J. Amer. Statist. Assoc. 76, Vaart, A.W.van der., Wellner, J.A., 996. Weak Convergence and Empirical Processes. Springer, New York.