Goodness-of-Fit Test for Monotone Functions

Scandinavian Journal of Statistics

doi: 10.1111/j.1467-9469.2010.00688.x © 2010 Board of the Foundation of the Scandinavian Journal of Statistics. Published by Blackwell Publishing Ltd.

Goodness-of-Fit Test for Monotone Functions ´ CECILE DUROT Département de Mathématiques, Université Paris-Sud

LAURENCE REBOUL Département de Mathématiques, Université de Poitiers ABSTRACT. In this article, we develop a test for the null hypothesis that a real-valued function belongs to a given parametric set against the non-parametric alternative that it is monotone, say decreasing. The method is described in a general model that covers the monotone density model, the monotone regression and the right-censoring model with monotone hazard rate. The criterion for testing is an L p -distance between a Grenander-type non-parametric estimator and a parametric estimator computed under the null hypothesis. A normalized version of this distance is shown to have an asymptotic normal distribution under the null, whence a test can be developed. Moreover, a bootstrap procedure is shown to be consistent to calibrate the test. Key words: bootstrap, composite hypothesis, least concave majorant, monotone density, monotone hazard rate, monotone regression, non-parametric alternatives

1. Introduction Monotonicity is a shape restriction that emerges naturally in many application areas. In economics for instance, empirical evidence often suggests a monotonic relationship between a covariate and a response variable, as it is the case of the presumed decreasing (respectively increasing) relationship between price and demand (respectively supply). In reliability or biological studies, the behaviour of a system or human lifetime on a specific time period can often be appropriately described by a distribution with a monotone hazard rate. In such situations, one may hope to obtain more accurate modelling by using statistical methods specifically designed for monotone functions. Consider the problem of estimating a function under the constraint that it is monotone, say decreasing (the increasing case can be treated likewise). Adopting a non-parametric approach, one can build in various models a non-parametric estimator under the monotonicity constraint. Alternatively, one can postulate a parametric model that respects the monotonicity constraint, and thus consider a parametric estimator. In that case, the presumed model should be checked by an appropriate goodness-of-fit test. It is the aim of this article to build such a test: we aim to test that belongs to a given parametric family that respects the monotonicity constraint, against the alternative that is decreasing. A pioneer paper about non-parametric inference under monotonicity constraint is that of Grenander (1956), where it is shown that the non-parametric maximum likelihood estimator of a non-increasing density, based on an i.i.d. sample, is the slope of the least concave majorant of the empirical distribution function. Then, Brunk (1970) proves that the non-parametric least-squares estimator of a non-increasing regression mean is a Grenandertype estimator, in the sense that it is the slope of the least concave majorant of an estimator of the primitive of the function to estimate. Huang & Wellner (1995) study a Grenander-type estimator for a monotone hazard rate in a right-censoring model. Reboul (2005) and Durot (2007) study such estimators in a general model that covers the cases of density, regression

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and right-censoring models: Reboul (2005) provides a non-asymptotic control for the L1 -risk and proves that the estimator is spatially adaptive, whereas Durot (2007) gives the limit distribution of the Lp -error as the number of observations tends to infinity. On the problem of testing a parametric null hypothesis against a non-parametric alternative, many papers are inspired by the smooth test of Neyman (1937); see among others Eubank & Hart (1992), Ledwina (1994), Fan & Huang (2001) Fan (1996), Aerts et al. (1999), ˜ (1998). There, the alternative is modelled as the product Gray & Pierce (1985) and Pena between the null parametric function and a series expansion along an orthonormal family of functions. The number of selected functions in the expansion turns out to be a smoothing parameter, and testing the null hypothesis reduces to test the nullity of the coefficients in the expansion.This can be done thanks to likelihood ratio or score test statistics with a datadriven calibration of the smoothing parameter; see Ledwina (1994) or Fan (1996). Another popular approach for testing a parametric model versus a non-parametric alternative is to reject the null hypothesis if a non-parametric estimator is too far from a parametric estimator computed under the null hypothesis; see among others Hardle & Mammen (1993), Alcalá et al. (1999), Stute & González-Manteiga (1996), Hart (1997), Liero et al. (1998), Liero & Läuter (2006). In the aforementioned papers, the alternative is not subjected to a monotonicity constraint. If the function under interest is known to be monotone, one can even so use these tests and omit the monotonicity assumption, but it seems more relevant to use statistical methods specifically designed for monotone functions. In this spirit, Durot & Tocquet (2001) carried out a goodness-of-fit test for a simple null hypothesis in a regression model with monotone regression mean: they reject the null hypothesis if the L1 -distance between the null hypothesis and the Brunk estimator is too large. Ducharme & Fontez (2004) adapt the smooth test of Neyman to a regression setting where the regression mean is assumed positive and increasing. In this article, we propose a goodness-of-fit test that generalizes the method of Durot & Tocquet (2001) in two ways. First, we allow composite null hypotheses; second, we address the testing problem in a fairly general setting that covers the density, regression and rightcensoring models. The criterion for testing is an Lp -distance between a Grenander-type estimator and a parametric estimator under the null hypothesis. A normalized version of this distance is shown to have an asymptotically normal distribution under the null, whence a goodness-of-fit test can be developed. To overcome the difficulty of estimating the parameters on which the asymptotic normal distribution depends, we establish the consistency of a bootstrap procedure to calibrate the test. The rest of the article is organized as follows. In section 2, we present the test statistic and study its asymptotic distribution under the null hypothesis. Calibrations are described in section 3. In section 4, we detail our method in the aforementioned models. A simulation study is reported in section 5. The proofs are deferred to the Appendix. 2. The test statistic Assume one observes X (n) the distribution of which depends on a monotone function : [0, 1] → R and possibly on a nuisance parameter ∈ H, where H is a set that could be of infinite dimension. Typically, X (n) consists of n independent but not necessarily identically distributed random variables. The nuisance parameter could be, for instance, the distribution of a censoring variable. Assume that is defined on a compact interval and without loss of generality, assume (possibly changing scale and origin) that this interval is [0, 1] and (possibly changing into −) that is decreasing. We wish to test the parametric null hypothesis © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

Goodness-of-fit test for monotone functions

Scand J Statist

H0 : ∈ { , ∈ }

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(1)

against the non-parametric alternative that is decreasing, where ⊂ Rr is a given set and for every , is a given decreasing function on [0, 1]. The criterion we consider for testing relies on the Grenander-type estimator introduced in Durot (2007). Precisely, we assume in the sequel that we have at hand a cadlag step estimator n : [0, 1] → R of the cumulative function (t) =

t

(u) du,

t ∈ [0, 1]

0

and we define ˆn as the left-hand slope of the least concave majorant of n , with ˆ n (0) = limt↓0 ˆn (t). The Grenander-type estimator ˆn can easily be computed thanks to simple algorithms such as the Pool-Adjacent-Violators-Algorithm (PAVA); see Barlow et al. (1972). It is entirely data-driven and from Reboul (2005), it is known to be spatially adaptive in various models. Our test statistic is Spn =

1

|ˆn (t) − ˆn (t)|p dt,

0

(2)

where ˆn is a suitable estimator of under H0 and p is a fixed real. It should be mentioned that in the particular case of a simple null hypothesis (of the form H0 : = 0 ), under H0 , Spn is the Lp -error of ˆn so its asymptotic distribution is given by theorem 2 of Durot (2007); our main task here is to generalize the method of Durot (2007) to the case of a possibly composite null hypothesis. We need some notation to describe the asymptotic distribution of Spn under H0 . Let W be a standard two-sided Brownian motion on R, X (a) = argmax{−(u − a)2 + W (u)},

a ∈ R,

u∈R

p = E|X (0)|p and ∞ kp = cov(|X (0)|p , |X (a) − a|p ) da. 0

Note that p and kp are both well defined and finite. Let · denote the Euclidean norm on Rr . For every ∈ , let P and E denote the underlying probability and expectation when = , where we recall that is a nuisance parameter. Note that for typographical convenience, we omit the dependence with respect to n in the notation. Let Mn = n − , where t (t) = 0 (u) du, t ∈ [0, 1]. Finally, for every f : [0, 1] → R, we set (when it exists) f˙ (t) =

∂ ∂ f (t), . . ., f (t) , ∂1 ∂r

where i is the ith component of , i = 1, . . ., r. We then make the following assumptions. (A) For every ∈ , is decreasing and differentiable on [0, 1] with inf | (t)| > 0, t

and there are C > 0 and s ∈ (3/4, 1] such that for all t, u, | (t) − (u)| C|t − u|s . © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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(B) Let Bn be either a Brownian bridge or a Brownian motion. For all and , there exist q > 12, C > 0, L : [0, 1] → R and versions of Mn and Bn such that P n1−1/q sup Mn (t) − n−1/2 Bn ◦ L (t) > x Cx−q t∈[0, 1]

for all x ∈ (0, n], and sup E

2

Mn (u) − Mn (t)

u∈[0, 1], x/2|t−u|x

Cx n

for all x > 0 and t ∈ {0, 1}. Moreover, L is increasing and twice differentiable on [0, 1] with L (0) = 0, supt |L (t)| < ∞ and inft L (t) > 0. ˆ ˆ (C) n (0) and n (1) are stochastically bounded under H0 . (D) ˆn = + OP (n−1/2 ) for every and . (E) For every t ∈ [0, 1], the functions → (t) and → (t) are differentiable on the convex hull of and for every , the function t → ˙ (t) is bounded on [0, 1]. Moreover, for every and > 0 there exists > 0 such that sup

sup ˙ (t) − ˙ (t) ,

− t∈[0, 1]

and for every , there exist > 0 and c > 0 such that sup

sup ˙ (t) c.

− t∈[0, 1]

Assumptions (A), (B) and (C) are an adaptation to our testing problem of the assumptions in Durot’s theorem 2: (A) is a smoothness assumption on under H0 whereas (B) ensures that Mn can be approximated in distribution by either a Brownian motion or a Brownian bridge with possibly non-standard variance function. Note that the constants involved in these assumptions may depend on and but for typographical convenience, this does not appear in the notation. Similar to Durot’s theorem 2, the assumptions that s > 3/4 and q > 12 could probably be weakened at the price of more technicalities. A sufficient condition for assumption (C) to hold is given by lemma 1 in Durot (2007). Assumptions (A), (B) and (C) allow computing the asymptotic distribution of the Lp -error of ˆn under H0 , whereas assumptions (D) and (E) are designed to ensure that ˆn properly estimates under H0 . Under these assumptions, we obtain the asymptotic distribution of the test statistic under H0 . Theorem 1 Assume (A)–(E) and let l = L . Then for every p ∈ [1, 5/2), and , 1 |4 (t)l (t)|p/3 dt n1/6 np/3 Spn − p 0

converges in distribution under P as n → ∞ to the Gaussian law with mean zero and variance 1 |4 (t)l (t)|2(p−1)/3 l (t) dt. (3) 2p = 8kp 0

Let us comment on the choice of p. The constraint p ∈ [1, 5/2) comes from the fact that for a large p, the contribution of the boundaries dominates, which has the effect of changing the asymptotic behaviour of Spn . The choice p = 1 is the simplest one because 1 and 8k1 are known to approximately equal 0.41 and 0.17, respectively (see Groeneboom, 1985), and the asymptotic variance reduces to 21 = 8k1 L (1) in this case. We have observed on a simulation study not reported here that the performances of the test defined with p = 2 are quite similar © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.


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to the ones of the test defined with p = 1. Thus, hereafter we are mainly concerned with the choice p = 1. To conclude this section let us briefly discuss a possible version of the method in semiparametric models with covariates. Assume we observe X (n) the distribution of which depends on a decreasing function and on a nuisance parameter = ( , ), where ∈ Rp describes the effect of a covariate and belongs to a possibly infinite dimensional space. For instance, in the Cox model, X (n) is an n-sample of the pair (T , Z) where T is a lifetime, Z ∈ Rp is a covariate with unknown distribution and the conditional hazard rate of T given Z is t → exp( T Z)(t). In typical situations, we have at hand a cadlag step estimator n, of in the sub-model √ where is known, an estimator ˆn of which converges at the n-rate to a centred Gaussian √ distribution and an estimator ˆn which converges to at the n-rate under the null hypothesis (1). In this setting, define n to be n, ˆn and ˆn to be the Grenander-type estimator based on n , and define Spn by (2). It is expected that under suitable smoothness assumptions, an analogous assumption to (B) holds so that under H0 , Spn is asymptotically Gaussian with a similar asymptotic behaviour as in theorem 1. 3. Calibration 3.1. Plug-in calibration For the sake of simplicity, we focus here on the case p = 1. One obtains a test with prescribed asymptotic level as an immediate consequence of theorem 1, by simply plugging-in suitable estimators of the unknown quantities in the limit distribution. Corollary 1 Let ∈ (0, 1) and q be the -upper percentile point of the standard Gaussian law. Assume (A)–(E) and let l = L . Let lˆn and Lˆ n be estimators such that for every and , n1/6 sup |lˆn (t) − l (t)| and |Lˆ n (1) − L (1)|

(4)

t∈(0, 1)

stochastically converge to zero under P . Then the test with critical region

1 1/3 1/3 −1/6 n S1n > 1 |4ˆn (t)lˆn (t)| dt + n q 8k1 Lˆ n (1)

(5)

0

has asymptotic level . In principle, lˆn and Lˆ n could be any estimator satisfying (4). One can consider for instance estimators of the form Lˆ n = Lˆn ˆn and lˆn = lˆn ˆn where ˆn estimates . 3.2. Bootstrap calibration From the point of view of implementing the test with the aforegiven calibration, one needs to estimate several parameters on which the asymptotic distribution of the test statistic under H0 depends. Besides this difficulty, the Gaussian approximation may be misleading for moderate sample sizes as already mentioned by Durot & Tocquet (2001) in the particular case they consider. To overcome these difficulties, we establish in this section the validity of a bootstrap procedure to calibrate the test. © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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Hereafter, P denotes the distribution of X (n) under P , P denotes the true distribution of X , (X , A) denotes the measurable space on which X (n) takes values and ˆn denotes an estimator of . We can write (ˆn , ˆn ) = n (X (n) ) for a given measurable function n . Assume that for every A ∈ A, x → Pn (x) (A) is A-measurable. Then, we can consider a measurable space ∗ equipped with a probability measure Q, and a measurable pair (X (n) , X (n)∗ ) : ∗ → X × X , such that for every B ∈ A ⊗ A, Q((X (n) , X (n)∗ ) ∈ B) = 1B (x, y) dPn (x) (y) dP(x). (n)

We call X (n)∗ the bootstrap observation; its conditional distribution given X (n) is Pˆn ˆn . As is customary, we denote by P∗ the probability measure defined on ∗ by P∗ ((X (n) , X (n)∗ ) ∈ B) = Q((X (n) , X (n)∗ ) ∈ B | X (n) ) for every B ∈ A ⊗ A and we denote by E∗ the corresponding expectation. Let ∗n , ˆ∗n and ˆ∗n be computed in the same manner as n , ˆn and ˆn but with X (n) replaced by X (n)∗ , and define the bootstrap version of Spn by ∗ = Spn

1 p ˆ∗ n (t) − ˆ∗n (t) dt.

(6)

0

∗ We propose to reject H0 if Spn exceeds the -upper percentile point qpn of the conditional ∗ ∗ distribution of Spn . It should be noticed that for practical purposes, qpn can be suitably estimated by Monte Carlo simulations. To show that this bootstrap calibration is consistent, we define Mn∗ = ∗n − ˆn and we make the following assumptions, which are bootstrap versions of assumptions (B), (C) and (D):

(B*) Let Bn∗ be either a Brownian bridge or a Brownian motion under P∗ . For all and , with P -probability tending to one, there exist a constant C > 0, an X (n) -measurable Ln∗ : [0, 1] → R and versions of Mn∗ and Bn∗ such that P∗ n1−1/q sup Mn∗ (t) − n−1/2 Bn∗ ◦ Ln∗ (t) > x Cx−q t∈[0, 1]

for all x ∈ (0, n], and E∗

sup

(Mn∗ (u) − Mn∗ (t))2

u∈[0, 1], x/2|t−u|x

Cx n

for all x > 0 and t ∈ {0, 1}. Moreover, Ln∗ is continuously differentiable with first derivative ln∗ and, setting l = L , there exists so > 3/4 such that sup |l (t) − ln∗ (t)| = OP (n−so /3 ).

(7)

t∈(0, 1)

(C*) For all and , there exists an X (n) -measurable set on which ˆ∗n (0) and ˆ∗n (1) are of the order OP∗ (1), and which P -probability tends to one as n → ∞. (D*) For all , ˆ∗n = ˆn + OP∗ (n−1/2 ) on the set {ˆn − n−1/2 log n}. Theorem 2 Assume p ∈ [1, 5/2), (A)–(E), (B*)–(D*). Let l = L . Then in P -probability, the conditional distribution of © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.


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n

1/6

n

p/3

∗ Spn − p

0

1

7

|4 (t)l (t)|p/3 dt

given X (n) converges as n → ∞ to the Gaussian law with mean zero and variance given by (3). As the Gaussian limit distribution is continuous, an immediate consequence of theorems 1 and 2 is that the bootstrap calibration is consistent. Corollary 2 ∗ be the -upper percentile point Assume p ∈ [1, 5/2), (A)–(E), (B*)–(D*). Let ∈ (0, 1) and qpn ∗ (n) ∗ of the conditional distribution of Spn given X . Then the test with critical region {Spn > qpn } has asymptotic level . Consistency of the bootstrap deserves comments. Indeed, the naive bootstrap, which consists in drawing X (n)∗ in such a way that the bootstrap version of is n , fails in our setting, owing to the lack of smoothness of n . For instance, in the density model where X (n) is an n-sample (X1 , . . ., Xn ) from a distribution with monotone density and ˆn is the Grenander estimator, the naive bootstrap amounts to draw X (n)∗ as an n-sample from the distribution that puts mass 1/n at each point Xi , i = 1, . . ., n, and is shown to fail in Kosorok (2008). The reason why our bootstrap procedure works is that our bootstrap version of is ˆn , which is a smooth function. 4. Applications In this section, we consider three specific models: in each model, we detail the estimator n on which the Grenander-type estimator ˆn is based, the construction of X (n)∗ and the assumptions under which the procedure is consistent. Hereafter, p denotes a fixed real in [1, 5/2); we assume that the parametric model under the null hypothesis satisfies the smoothness assumptions (A) and (E) and we consider an estimator ˆn that satisfies (D). Typically, ˆn could be a maximum likelihood or a least-squares estimator under H0 . Note that for every and one then has sup | (t) − ˆn (t)| = OP (n−1/2 ).

(8)

t∈[0, 1]

We begin the section by briefly describing a slight modification of ˆn that is a more natural estimator than ˆn itself in some of the considered models. 4.1. A slight modification of the monotone estimator ˜ n be the continuous version of n , which means that ˜ n (t) = n (t) at every jump point t Let of n and at the boundaries t = 0 and t = 1, and ˜ n is linear in between two consecutive such points. The modification of the monotone estimator we consider is n , the left-hand slope of the least concave majorant of ˜ n , with n (0) = limt↓0 n (t). Assume that for every and , q ˜ E sup | n (t) − n (t)| Cn1−q (9) t∈[0, 1]

for some q > 12 and C > 0. Then, theorem 1 and corollary 1 remain true with ˆn replaced by n . If, furthermore, ∗ E∗ sup | ˜ n (t) − ∗n (t)|q Cn1−q (10) t∈[0, 1]

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∗ with probability tending to one, where ˜ n is computed in the same manner as ˜ n , but with (n) (n)∗ X replaced by X , then theorem 2 and corollary 2 remain true with ˆn replaced by n . It can be shown that (9) and (10) hold in the models we consider next, so the following results remain true with ˆn replaced by n .

4.2. Monotone density Assume X (n) = (X1 , . . ., Xn ) where the Xi s are i.i.d. with a decreasing density on [0, 1], and let n be the corresponding empirical distribution function. Besides (A), (D) and (E), we assume: (Ad) For every , inft (t) > 0. Then, theorem 6 of Durot (2007) shows that (B) and (C) hold with L = (as there is no nuisance parameter here, we omit the subscript in the notation). Setting Lˆ n (1) = 1 and lˆn = ˆn , it follows from (8) and corollary 1 that the test with critical region (5) has asymptotic level . For the bootstrap procedure, let X (n)∗ = (X1∗ , . . ., Xn∗ ), where the Xi∗ s are conditional i.i.d. random variables with density function ˆn . Assume, furthermore, that (D*) holds. It is shown in section I of the online Supporting Information that (B*) and (C*) hold with Ln∗ = ˆn , so ∗ the test with critical region {Spn > qpn } has asymptotic level . 4.3. Monotone regression Assume one observes X (n) = (y1 , . . ., yn ) with yi = (ti ) + i , i = 1, . . ., n. Here, ti = i/n, the i s are i.i.d. random variables with mean zero and finite variance 2 , : [0, 1] → R is an unknown decreasing function, and the nuisance parameter is the common distribution of the i s. Let n be the partial sum process n (t) =

1 yi , n int

t ∈ [0, 1].

Besides (A), (D) and (E), we assume: (Ar) There exists q > 12 such that E|i |q < ∞, and E2i = 2 > 0. The assumption that q > 12 here can probably be weakened at the price of more technicalities. Let ˆ2n be an estimator of 2 such that n1/6 (ˆ2n − 2 ) = oP (1) for all and . One can consider for instance the Rice estimator 1 (yi + 1 − yi )2 ; 2(n − 1) i = 1 n−1

ˆ2n =

(11)

other examples of suitable estimators can be found in Hall & Marron (1990). By theorem 5 of Durot (2007), (B) and (C) hold under our assumptions with L (t) = t2 . Setting Lˆ n (1) = lˆn (t) = ˆ2n , it follows from corollary 1 that the test with critical region (5) has asymptotic level . Now, we turn to the bootstrap calibration. Let ñ be a non-parametric estimator of that satisfies the following assumption. (Ar*) For every and , supt |ñ (t)| = OP (1) and there exists an so > 3/4 such that 2 1 ˜ n (ti ) − (ti ) = OP (n−2so /3 ). n i =1 n



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One can consider for instance a kernel estimator. Alternatively, one can consider ñ := ˆn as it can be shown using theorem 2 of Durot (2007) that n−1/3

n 2 ˆn (ti ) − (ti ) = OP (1). i =1

For every i = 1, . . ., n, let î = yi − ñ (ti )

(12)

and 1 ˆj . n j =1 n

˜i = î −

Conditionally on X (n) , let ∗1 , . . ., ∗n be an n-sample from the distribution ˆn that puts mass 1/n at each point ˜1 , . . ., ñ . Then set X (n)∗ = (y1∗ , . . ., yn∗ ), where for every i = 1, . . ., n, yi∗ = ˆn (ti ) + ∗i . We now have corollary 3. Corollary 3 In the regression setting where (A), (Ar), (D), (E), (D*) and (Ar*) hold, the test with critical ∗ region {Spn > qpn } has asymptotic level . Proof. Assume = for some ∈ . From (Ar) and (Ar*), there exists C > 0 such that q n n 1 1 ∗ ∗ q ˆj C E |i | = î − n = n = i

1

j

1

with probability that tends to one. But there exists C > 0 such that with probability that tends to one, |ˆn (t)| C and |ˆn (t)| C for all t, so 1 ∗ 2C ∗ sup Mn (t) − n int i n t∈[0, 1] with probability that tends to one. Let 2∗ n be the common conditional variance of the variables ∗i . Then, the assumption on ñ ensures that 2 −so /3 ), 2∗ n = + OP (n

and one can prove that (B*) and (C*) hold with Ln∗ (t) = t2∗ n (see section II of the online Supporting Information for details). Therefore, corollary 3 follows from corollary 2. 4.4. Right-censoring model with monotone hazard rate Assume one observes a right-censored sample X (n) = ((X1 , 1 ), . . ., (Xn , n )). Here, Xi = min(Ti , Yi ) and i = 1Ti Yi , where the Ti s are non-negative i.i.d. failure times and the Yi s are i.i.d. censoring times independent of the Ti s. Let be the restriction to [0, 1] of the common hazard rate of the Ti s, that is, = f/(1 − F ) on [0, 1], where f and F are the density and distribution functiona of Ti , respectively. Here, the nuisance parameter is the common distribution function of the Yi s. Let n be the restriction to [0, 1] of the Nelson–Aalen estimator of and ˆn be the Kaplan–Meier estimator of . Besides (A), (D) and (E), we © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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make the following assumption, where F denotes the distribution function of the law with hazard rate . (Ac) has a bounded continuous first derivative on (0, 1) and limt↑1 (t) < 1. Moreover, for every , inft (t) > 0 and F (1) < 1. By theorem 3 of Durot (2007), (B) and (C) hold with t (u) du. L (t) = (1 − F (u))(1 − (u)) 0 But ˆn uniformly converges to in probability at the rate lˆn =

ˆn and Lˆ n (1) = (1 − Fˆn )(1 − ˆn )

1

√

n, so with

lˆn (t) dt,

0

it follows from (8) and corollary 1 that the test with critical region (5) has asymptotic level . We now turn to the bootstrap calibration. Conditionally on X (n) , let T1∗ , . . ., Tn∗ be i.i.d. random variables with hazard rate ˆn and Y1∗ , . . ., Yn∗ be i.i.d. random variables with distribution function ˆn , independent of the Ti∗ s. We define the bootstrap observation by X (n)∗ = ((X1∗ , ∗1 ), . . ., (Xn∗ , ∗n )), where Xi∗ = min(Ti∗ , Yi∗ ) and ∗i = 1Ti∗ Yi∗ . We assume, furthermore, that (D*) holds. Let n be the continuous version of ˆn (we mean, the polygonal function that coincides with ˆn at each jump point of ˆn on [0, 1) and such that ñ (1) = limt↑1 ˆn (t)). Then, sup |ñ (t) − ˆn (t)| = OP (n−1 ),

t∈[0, 1)

so (see section III of the online Supporting Information for more details) similar arguments as before show that (B*) and (C*) hold with t ˆn (u) du. Ln∗ (t) = (1 − F (u))(1 − ñ (u)) 0 ˆn ∗ } has asymptotic level . Therefore, the test with critical region {Spn > qpn

5. Simulation study In this section, we report a simulation study performed in a regression model. Based on observations yi = (ti ) + i ,

i = 1, . . ., n,

(13)

we study several tests for the simple null hypothesis H0s : = 0 ,

(14)

where 0 (t) = −10t, t ∈ [0, 1], and for the composite null hypothesis H0c : ∈ { , ∈ },

(15)

where = R × (−∞, 0) and for every = (a, b), (t) = a + bt, t ∈ [0, 1]. In (13), ti = i/n, the i s are independently drawn from the standard Gaussian distribution, and we vary n, and . The nominal level of the tests we investigate is of 5 per cent. These tests are described in section 5.1. Results about level and power are reported in sections 5.2 and 5.3, respectively. © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.


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5.1. Tests Hereafter, ˆ2n denotes the Rice estimator of 2 given by (11), ˆn denotes the ordinary leastsquares estimator of given by n ¯ ¯ = 1 (ti − t)(yi − y) ¯ bˆ n = i and aˆn = y¯ − bˆt, n ¯2 i = 1 (ti − t) where t¯ and y¯ are the mean values of the ti s and yi s, respectively, and ñ denotes the local linear kernel estimator (see Wand & Jones, 1995) with quartic kernel K (t) =

15 (1 − t2 )2 1|t|1 16

and bandwidth denoted by h. Every bootstrap calibration is performed using Monte Carlo simulations. Moreover, all the tests we investigate are presented next in the case of a composite null hypothesis H0c . The case of a simple null hypothesis H0s is similar, replacing ˆn by 0 in the formulae. n First, we fix p = 1 and consider the two tests described in section 4.3. We denote by S ∗ and Sn the tests based on plug-in and bootstrap calibrations, respectively, where 8k1 and 21 are approximated by 0.17 and 0.82, respectively, and the bootstrap calibration is performed using the local linear kernel estimator ñ with a bandwidth h selected by cross-validation. Note that for time cost necessity, we do not minimize the cross-validation function over the whole interval [0, 1] but over an interval I ⊂ [0, 1] suggested by a preliminary study. Moreover, to assess the qualities of cross-validation in our setting, we also implement the bootstrap calibration where h is fixed to a value h depending on the unknown function , and suggested by a preliminary study of the estimated mean-squared error. We denote by Sn∗o the corresponding test (which, of course, cannot be implemented in practice as it relies on the knowledge of ). Next, we consider three tests that are similar to Sn∗ , but that are calibrated using different bootstrap procedures. Their study attempts to justify the choice of our bootstrap calibration method. The first two modifications consist in replacing the local linear kernel estimator ñ in (12) by either the parametric estimator ˆn or the Grenander-type estimator ˆn , so that the resulting test does not depend on a smoothing parameter. Apart from this replacement, the tests are identical to Sn∗ . The first proposal is denoted by Sn∗p in the sequel, and the second one is denoted by Sn∗m . The third modification we consider consists in studentization: it rejects the null hypothesis when n1/6 √ ˆn 0.17

1 n1/3 S1n − 0.82ˆ2/3 |ˆn (t)/2|1/3 dt n 0

exceeds the -upper percentile point of its bootstrap version, obtained by replacing S1n , ˆn and ˆn by the same quantities based on the bootstrap observation (which is generated as in section 4.3 with ñ as the local linear kernel estimator). This test is denoted by Zn∗ in the sequel. Finally, we consider three competitor tests. Two of them are goodness-of-fit tests that do not take into account the monotonicity constraint. The first one, which is denoted by Tn∗o in the sequel, is similar to ours, but with the Grenander-type estimator ˆn replaced by the local linear kernel estimator ñ : it consists in rejecting the null hypothesis when 0

1

(ñ (t) − ˆn (t))2 dt


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exceeds the -upper percentile point of its bootstrap version, obtained by replacing ñ (t) and ˆn by the same quantities based on the bootstrap observation, generated as in section 4.3. For time cost reasons, we do not select the bandwidth h using cross-validation. Rather, similar to Sn∗o , h is fixed here to a value h depending on the unknown function , suggested by a preliminary study of the estimated mean-squared error. The second competitor is the Kolmogorov–Smirnov test studied by Stute et al. (1998) and is denoted by Dn∗ in the sequel. The last competitor test is the smooth test proposed by Ducharme & Fontez (2004). It is denoted by Rn in the sequel. To our knowledge, it is the only goodness-of-fit test specifically designed for monotone regression functions available in the literature. However, Rn does not allow testing H0s . Indeed, it is designed to test a parametric model for f := /((1) − (0)), where is the derivative of . Thus, it allows testing H0c , which is equivalent to ‘f = 1’, but does not allow testing H0s . Moreover, it requires the regression function to be increasing and positive, so we change yi into −yi to implement the method. The test statistic is defined by (13) in Ducharme & Fontez (2004). The parameter K in this formula is optimized as indicated, taking d = 1 and D = 8, and the critical value is given in table 1 of that paper. 5.2. Levels Hereafter, we investigate the empirical levels of the aforementioned tests. For each pair (n, ), n ∈ {50, 100}, ∈ {0.5, 1}, we generate observations (13) with = 0 . Then for each test, we compute the test statistic and the critical value. In the case of a bootstrap calibration, the critical value is computed using 1000 bootstrap replications. The empirical levels of the tests are then defined as the percentage of rejection of the null hypothesis over 10,000 replications. They are reported in Table 1 for = 0.5. For the sake of briefness, the levels obtained for = 1 are not displayed here, but they lead to similar conclusions. It can be seen in Table 1 that Sˆ n is very anticonservative. This means that for moderate sample sizes, the asymptotic normal approximation is misleading (as already noticed by Durot and Tocquet in the simple null hypothesis case). Similarly, Sn∗m is misleading. The other tests show empirical levels quite close to the nominal level of 5 per cent. However, Sn∗p tends to be conservative when testing H0c , and Sn∗ , which behaves similarly as Sn∗o , is not very accurate for small sample sizes (n = 50). This drawback is overcome by studentization: the empirical level of Zn∗ is very close to 5 per cent in every considered case. Both Dn∗ and Rn perform well for testing H0c , and Dn∗ is slightly anticonservative when testing H0s . The same drawback is observed for Tn∗o , for both simple and composite hypotheses. Table 1. Empirical levels (in per cent) of the tests for H0s and H0c when = 0.5 H0s

H0c

Test

n = 50

n = 100

Sˆ n Sn∗m ∗p Sn Sn∗ Sn∗o Zn∗ Dn∗ Tn∗o Rn

17.7 6.5 4.5 5.8 5.7 5.1 5.9 7.0

15.5 11.5 4.6 5.2 5.2 5.2 5.7 6.2

n = 50 8.3 4.8 3.4 4.3 4.4 4.5 5.0 5.8 4.9


n = 100 8.1 10.8 4.4 4.7 4.7 5.0 5.0 5.6 4.8


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5.3. Powers We now investigate the empirical powers of the tests. As the empirical levels of Sˆ n and Sn∗m are much larger than the nominal level of 5 per cent, we did not study the power of these tests and concentrate on the other ones. We focus on alternative functions of the form (t) = 0 (t) + c(t), where (t) = −1/2

(16)

0.25 −

2 3

t − t0 − 0.5

1t0 tt0 + .

Here, c, √ and t0 are positive constants. The monotonicity constraint on implies imposing c < 3/2 250 5/3. The functions (16) are obtained from the straight line 0 by an additional perturbation which takes the form of a local bump. The constants c and , respectively, adjust the height and width of the bump and the value of t0 is chosen such that the bump is centred. We consider here the cases (, t0 ) = (0.3, 0.35) and c ∈ {10, 20, 30}, (, t0 ) = (0.2, 0.4) and c ∈ {5, 10, 15}. For each pair (n, ), n ∈ {50, 100}, ∈ {0.5, 1} and each considered alternative function, we generate observations (13), with defined by (16). Then for each test, we compute the test statistic and the critical value. The critical value of each test relying on a bootstrap calibration is computed using 800 bootstrap replications. The empirical powers of the tests are then defined as the percentage of rejection over 8000 replications. Performances are reported in Tables 2 and 3 for H0s and H0c , respectively. We only display the results for = 0.5, but the following comments rely on the whole study ( = 0.5 and = 1). It is seen in Tables 2 and 3 that the power of the tests increases with the sample size and the height c of the bump. Over the whole set of results, it appears that Sn∗ and Sn∗o behave similarly. They dominate Sn∗p against all the considered alternatives, and also Zn∗ against most of the considered alternatives: Zn∗ is slightly more powerful against small departures from a composite null hypothesis but is outperformed by Sn∗ and Sn∗o in the other cases. This justifies the choice of our bootstrap method and the use of cross-validation. Globally, Rn is the less powerful test, and Dn∗ is often outperformed by either Tn∗o or Sn∗ (or even by both these tests). Actually, Sn∗ (and Sn∗o ) appears to dominate every other test in the case of a simple Table 2. Empirical powers (in per cent) against alternative functions (16) for H0s when = 0.5 n = 50 (, t0 ) (0.3, 0.35)

(0.2, 0.4)

n = 100

Test

c = 10

c = 20

c = 30

c = 10

c = 20

c = 30

∗p Sn Sn∗ Sn∗o Zn∗ Dn∗ Tno ∗

11.8 14.6 15.6 12.2 10.2 8.4

27.4 52.1 53.0 30.8 20.4 45.3

49.1 81.4 81.2 57.2 36.9 81.2

17.0 19.0 20.4 17.3 13.3 9.1

51.1 70.4 68.6 52.6 37.2 65.6

85.1 95.8 96.5 87.9 66.5 96.2

c=5

c = 10

c = 15

c=5

c = 10

c = 15

10.0 12.3 12.9 10.7 8.3 8.1

15.4 17.2 17.6 17.8 15.4 8.9

12.9 15.7 16.5 14.1 10.7 8.4

25.8 28.7 28.6 28.5 17.0 9.8


6.2 7.9 8.1 6.7 5.9 6.2


7.5 8.8 8.3 8.0 7.0 6.7

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Table 3. Empirical powers (in per cent) against alternative functions (16) for H0c when = 0.5 n = 50 (, t0 ) (0.3, 0.35)

Test ∗p Sn Sn∗ Sn∗o Zn∗ Dn∗ Tn∗o

Rn

(0.2, 0.4)


Rn

n = 100

c = 10

c = 20

c = 30

c = 10

c = 20

c = 30

5.9 7.1 7.3 8.0 9.8 11.2 6.8

14.0 38.9 43.1 19.4 25.2 49.1 14.5

28.2 67.9 69.9 39.7 51.4 77.1 28.7

10.4 11.6 11.4 11.9 15.5 15.5 10.2

33.5 55.3 56.9 37.1 52.8 62.6 30.1

71.7 89.7 90.3 76.5 90.0 93.2 61.8

c=5

c = 10

c = 15

c=5

c = 10

c = 15

4.0 5.0 5.1 5.2 6.0 6.6 5.2

5.7 6.9 6.8 8.1 8.0 9.9 6.9

8.6 9.6 9.9 10.7 12.5 11.7 8.7

16.5 18.6 18.8 20.0 24.2 22.1 17.1

8.0 10.4 10.2 11.3 11.7 11.6 8.2

5.6 6.2 6.3 6.3 6.7 7.2 6.0

null hypothesis, whereas Tn∗o tends to outperform the others in the composite null hypothesis case, except for some cases of small departures, where Dn∗ achieves a slightly better power. 6. Conclusion In this article, we have presented a test of a composite null hypothesis which is specifically designed for monotone functions. The test statistic is based on a non-parametric data-driven estimator of the function under study, so that no extra parameter has to be adjusted to compute the test statistic. To calibrate our test, we have proposed a general bootstrap method covering three statistical frameworks: regression, density and hazard rate functions. We believe that similar constructions could be applied to deal with other statistical models. Acknowledgements The authors are grateful to the editor and the referees for their insightful comments to improve the quality of the article. Supporting Information Additional Supporting Information may be found in the online version of this article: Details on the results in section 4 are available as supporting information for this article. This material may be found in the online version of the article. Please note: Wiley-Blackwell are not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. References Aerts, M., Claeskens, G. & Hart, J. D. (1999). Testing the fit of a parametric function. J. Amer. Statist. Assoc. 94, 869–879. ´ Alcalá, J. T., Cristobal, J. A. & González-Manteiga, W. (1999). Goodness-of-fit test for linear models based on local polynomials. Statist. Probab. Lett. 42, 39–46. © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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Barlow, R. E., Bartholomew, D. J., Bremner, J. M. & Brunk, H. D. (1972). Statistical inference under order restrictions. The theory and application of isotonic regression. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons, London. Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric techniques in statistical inference (ed. M. L. Puri), 177–195. Cambridge University Press, London. Ducharme, G. R. & Fontez, B. (2004). A smooth test of goodness-of-fit for growth curves and monotonic nonlinear regression models. Biometrics 60, 977–986. Durot, C. (2007). On the Lp -error of monotonicity constrained estimators. Ann. Statist. 35, 1080–1104. Durot, C. & Tocquet, A. S. (2001). Goodness-of-fit tests for isotonic regression. ESAIM Probab. Statist. 5, 119–140. Eubank, R. L. & Hart, J. D. (1992). Testing goodness-of-fit in regression via order selection criteria. Ann. Statist. 20, 1412–1425. Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 91, 674–688. Fan, J. & Huang, L. S. (2001). Goodness-of-fit tests for parametric regression models. J. Amer. Statist. Assoc. 96, 640–652. Gray, R. & Pierce, D. (1985). Goodness-of-fit for censored survival data. Ann. Statist. 13, 552–563. Grenander, U. (1956). On the theory of mortality measurement. Part II. Skandinavisk Aktuarietidsskrift 39, 125–153. Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in honor of Jerzy Neyman & Jack Kiefer (eds L. LeCam & R. Olshen), 2, 539–555. Wadsworth Advanced Book & Software, Monterey, California. Hall, P. & Marron, J. S. (1990). On variance estimation in nonparametric regression. Biometrika 77, 415–419. Hardle, W. & Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21, 1926–1947. Hart, J. D. (1997). Nonparametric smoothing and lack-of-fit tests. Springer-Verlag, New York. Huang, J. & Wellner, J. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22, 3–33. Kosorok, M. R. (2008). Bootstrapping the Grenander estimator. In Beyond parametrics in interdisciplinary research: Festschrift in honor of Professor Pranab K. Sen. (eds N. Balakrishnan, E. A. Pena & M. J. Silvapulle), 1, 282–292. Institute of Mathematical Statistics, Beachwood, Ohio. Ledwina, T. (1994). Data-driven version of Neyman’s smooth test of fit. J. Amer. Statist. Assoc. 89, 1000– 1005. Liero, H. & Läuter, H. (2006). Nonparametric estimation and testing in survival models. In Probability, statistics and modelling in public health (eds M. Nikulin, D. Commenges & C. Huber), 319–331. Springer, New York. Liero, H., Läuter, H. & Konakov, V. (1998). Nonparametric versus parametric goodness-of-fit. Statistics 31, 116–149. Neyman, J. (1937). Smooth test for goodness-of-fit. Skandinavisk Aktuarietidsskrift 20, 149–199. ˜ E. (1998). Smooth goodness-of-fit tests for composite hypothesis in hazard based models. Ann. Pena, Statist. 26, 1935–1971. Reboul, L. (2005). Estimation of a function under shape restrictions. Application to reliability. Ann. Statist. 33, 1330–1356. Stute, W. & González-Manteiga, W. (1996). NN goodness-of-fit tests for linear models. J. Statist. Plann. Inference 53, 75–92. Stute, W., González-Manteiga, W. & Presedo Quindimil, M. (1998). Bootstrap approximations in model checks for regression. J. Amer. Statist. Assoc. 93, 141–149. Wand, M. P. & Jones, M. C. (1995). Kernel smoothing. Chapman & Hall, London. Received January 2008, in final form December 2009 Cécile Durot, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France. E-mail: [email protected]

Appendix This appendix is devoted to the proof of theorem 2. The similar proof of theorem 1 is omitted. Note that theorem 2 is a generalization of theorem 2 in Durot (2007) as it provides © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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the asymptotic distribution of the Lp -distance between a monotone estimator and a mono∗ tone function. The generalization is twofold. First, the considered monotone estimator ˆn is computed from an observation X (n)∗ the distribution of which depends on parameters ˆn and ˆn that are allowed to depend on n. As a consequence, the function Ln∗ that appears in the assumption (B*) may also depend on n. Second, we do not consider here the distance between the monotone estimator and the true underlying function ˆn itself, but we consider the distance between the monotone estimator and an estimator of the true underlying function. Despite these differences, the proof of theorem 2 follows the lines of the proof of Durot’s theorem 2. Therefore, details are omitted and we only stress the places where the two proofs differ. Moreover for notational convenience, we sometimes omit subscript n and also subscripts and . Thus, in the sequel, we assume = for some and we write L and l for L and l , respectively. Under assumptions (A), (D) and (E), there exist positive reals c and C that do not depend on n such that the probability that sup |ˆ(t)| C,

t∈[0, 1]

inf |ˆ(t)| > c,

sup |ˆ(t)| C

t∈[0, 1]

(17)

t∈[0, 1]

tend to one as n → ∞. But to prove theorem 2, we can restrict ourselves to an event whose probability tends to one, so we can assume without loss of generality that (17) holds. Likewise, thanks to assumptions (7) and (D), we assume sup |l(t) − l ∗ (t)| Cn−so /3 log n and ˆ − n−1/2 log n. t∈(0, 1)

• Step 1. For every a ∈ R, let ∗ Uˆ (a) = argmax{∗n (u) − au},

(18)

u∈[0, 1]

where argmax denotes the greatest location of maximum. Let gˆ and gˆ ∗ be the inverse functions of ˆ and ˆ ∗ , respectively. Arguing as for Durot’s lemmas 3 and 4 one obtains that there exists K > 0 such that ∗ P∗ [|Uˆ (a) − gˆ(a)| x] K (nx3 )1−q

(19)

for every a ∈ R and x > 0, and P∗ [|Uˆ ∗ (a) − gˆ(a)| x]

K nx(ˆ(gˆ(a)) − a)2

(20)

for every x > 0 and a ∈ ˆ([0, 1]). Arguing as for Durot’s theorem 1 one then obtains that for every p ∈ [1, 2), there exists K > 0 such that E∗ |ˆ∗ (t) − ˆ(t)|p Kn−p /3

for all t ∈ [n−1/3 , 1 − n−1/3 ] and E∗ |ˆ∗ (t) − ˆ(t)|p K [n(t ∧ (1 − t))]−p /2

for all t ∈ (0, n−1/3 ] ∪ [1 − n−1/3 , 1). Now, set x+ = x ∨ 0 for every x ∈ R and let 1 1 ∞ p I1∗ = 1ˆ∗ (t) ˆ∗ (t) + a1/p da dt. (ˆ∗ (t) − ˆ∗ (t))+ dt = 0

0

0

∗

We have ˆ∗ (t) < ˆ∗ (0) for all t > Uˆ (ˆ∗ (0)); so © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.


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I1∗ =

1

(ˆ∗ (0)−ˆ∗ (t))p

where 0 R∗n

1ˆ∗ (t) ˆ∗ (t) + a1/p da dt + Rn ∗ ,

0

0

17

Uˆ ∗ (ˆ∗ (0)) 0

[(ˆ∗ (t) − ˆ∗ (t))+ ]p dt.

But Uˆ ∗ is non-increasing and ˆ∗ (0) = ˆ(0) + OP∗ (n−1/2 ): so, we have

P∗ Uˆ ∗ (ˆ∗ (0)) > n−1/3 log n P∗ Uˆ ∗ (ˆ(0) − n−1/2 log n) > n−1/3 log n + o(1). Similar to step 1 in Durot (2007), (19) then implies n−1/3 log n p ∗ Rn∗ (ˆ (t) − ˆ∗ (t))+ dt + oP∗ (n−p/3−1/6 ) 0

and as ˆ = ˆ∗ + OP∗ (n−1/2 ), Rn∗ = oP∗ (n−p/3−1/6 ). Change of variable b = ˆ∗ (t) + a1/p then yields ˆ∗ (0) Uˆ ∗ (b) I1∗ = p(b − ˆ∗ (t))p−1 dt 1g ∗ (b)Uˆ ∗ (b) db + oP∗ (n−p/3−1/6 ). ˆ∗ (1)

ˆ

gˆ∗ (b)

It can be shown using Taylor’s expansion that there exists K > 0 such that for every b in the range of ˆ∗ ,

[b − ˆ∗ (t)]p−1 − [(g ˆ∗ (b) − t)ˆ ◦ g ˆ(b)]p−1 K |t − g ˆ∗ (b)|p−1 oP∗ (n−1/6 ) + |t − g ˆ∗ (b)|s , where oP∗ is uniform in t and b. Moreover, integrating (19) proves that there exists K > 0 such that for every q < 3(q − 1) and a ∈ R, q ∗ E∗ n1/3 |Uˆ (a) − gˆ(a)| K. (21) Therefore, I1∗ is asymptotically equivalent to ˆ(0) Uˆ ∗ (b) p(t − gˆ∗ (b))p−1 |ˆ ◦ gˆ(b)|p−1 1g ∗ (b)Uˆ ∗ (b) dt db. ˆ(1)

ˆ

gˆ∗ (b)

Hence, I1∗ =

ˆ(0)

ˆ(1)

∗ (Uˆ (b) − gˆ∗ (b))+

p

|gˆ(b)|1−p db + oP∗ (n−p/3−1/6 ).

Repeating the same arguments for the negative part of the integrand yields ∗ = Spn

ˆ(0)

ˆ(1)

p ˆ∗ U (a) − g ˆ∗ (a) |g ˆ(a)|1−p da + oP∗ (n−p/3−1/6 ).

• Step 2. We have sup a∈(ˆ(1), ˆ(0))

|gˆ(a) − gˆ∗ (a)| = OP∗ (n−1/2 ),

∗ so by (22), Spn is asymptotically equivalent to ˆ(0)

p |gˆ(a)|1−p ∗ ˆ∗ da. L (U (a)) − L∗ (gˆ(a)) + gˆ(a) − gˆ∗ (a) l gˆ(a) |l(gˆ(a))|p ˆ(1)


(22)

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We have the representation Bn∗ (t) = W ∗ (t) − ∗ t, where W ∗ is a standard Brownian motion, ∗ ≡ 0 if Bn∗ is a Brownian motion and ∗ is a standard Gaussian variable independent of Bn∗ if Bn∗ is a Brownian bridge. For every a ∈ R, let a = a − n−1/2 ∗ l(gˆ(a)). Similar to Durot’s formula (27), one gets ∗ P∗ (|L∗ (Uˆ (a )) − L∗ (gˆ(a))| > x) K (nx3 )1−q

(23)

for some K > 0 and all x > 0. Integrating this inequality proves that ∗ sup E∗ (n1/3 |L∗ (Uˆ (a )) − L∗ (gˆ(a))|)q K a∈R

for some K > 0 provided q < 3(q − 1). Using again similar arguments as in Durot’s step 2, one ∗ can then perform the change of variable a → a to get that Spn is asymptotically equivalent to

ˆ(0) ˆ(1)

p |gˆ(a)|1−p ∗ ˆ∗ da. L (U (a )) − L∗ gˆ(a) + gˆ(a) − gˆ∗ (a ) l gˆ(a) |l(gˆ(a))|p

Now by (18), ∗

n1/3 (L∗ (Uˆ (a )) − L∗ (gˆ(a))) is the location of the maximum of the process u → D∗ (a, u) + Wg∗ˆ(a) (u) + R∗ (a, u)

(24)

over I ∗ (a) = n1/3 (L∗ (0) − L∗ (gˆ(a))), n1/3 (L∗ (1) − L∗ (gˆ(a))) , where for every a and u,

D∗ (a, u) = n2/3 ˆ ◦ L∗−1 − aL∗−1 (L∗ (gˆ(a)) + n−1/3 u) − n2/3 (ˆ(gˆ(a)) − agˆ(a)),

∗ ∗ ∗ 1/6 W L (gˆ(a)) + n−1/3 u − W ∗ L∗ (gˆ(a)) W(a) ˆ (u) = n and where with probability that tends to one, the remainder term R∗ (a, u) satisfies |R∗ (a, u)| n2/3 sup |∗n (t) − ˆ(t) − n−1/2 Bn∗ ◦ L∗ (t)| t∈[0, 1]

+ Kn−1/6 | ∗ u|(Cn−so /3 log n + n−1/3 |u|). Let ∗ U˜ (a) = argmax {D∗ (a, u) + Wg∗ˆ(a) (u)}. u∈[− log n, log n]

Similar arguments as in step 2 of Durot (2007) then show that n

p/3

∗ = Spn

p |gˆ(a)|1−p

U ˜ ∗ (a) + n1/3 gˆ(a) − g ˆ∗ (a ) l(gˆ(a)) |l(g ˆ(a))|p da

ˆ(0)

ˆ(1)

+ oP∗ (n−1/6 ). Uniformly in a and u ∈ [− log n, log n], D∗ (a, u) can be approximated by © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.


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n2/3 ˆ ◦ L−1 − aL−1 (L(gˆ(a)) + n−1/3 u) − n2/3 (ˆ(gˆ(a)) − agˆ(a)) which, in turn, can be approximated by (gˆ(a)) 2 u , 2(l(gˆ(a)))2 the remainder being of the order O(n−(s∧so )/3 (log n)3 ). Defining ∗ | (t)| 2 ∗ + u (u) W − V˜ (t) = argmax t 2(l(t))2 u∈[− log n, log n] and using similar arguments as in Durot (2007), one then gets ˆ(0) p |g (a)|1−p

˜∗ ∗ ˆ = da np/3 Spn V (gˆ(a)) + n1/3 gˆ(a) − gˆ∗ (a ) l(gˆ(a)) p |l(g ˆ(1) ˆ (a))| + oP∗ (n−1/6 ) 1 ∗

p p V˜ (t) + n1/3 l(t) ˆ∗ (t) − ˆ(t) + n−1/2 ∗ l(t) (t) dt = l(t) (t) 0 + oP∗ (n−1/6 ). • Step 3. Let H = | /l|p and Dn∗ be equal to 1 ∗

p V˜ (t) + n1/3 l(t) ˆ∗ (t) − ˆ(t) + n−1/2 ∗ l(t) − |V˜ ∗ (t)|p H(t) dt. (t) 0 We show here that Dn∗ = oP∗ (n−1/6 ).

(25)

One can approximate Dn∗ by 1 p ∗ −1/2 ∗ V˜ (t) + n1/3 l(t) ˙ ˆ(t)(ˆ∗ − ) − |V˜ ∗ (t)|p H(t) dt ˆ + n l(t) (t) 0 as the distance between Dn∗ and this integral is less than sup

∗

sup |˙ ˆ(t) − ˙ (t)| × OP∗ (n−1/6 ) = oP∗ (n−1/6 ).

ˆ − ˆ ˆ t∈[0, 1] −

We have

∗ ˆ + n−1/2 ∗ l(t) = OP∗ (n−1/6 ) sup n1/3 l(t) ˙ ˆ(t)(ˆ − )

t∈[0, 1] ∗

and V˜ (t) possesses uniformly bounded moments of any order so, similar to Durot (2007), ∗ expanding x → xp around |V˜ (t)| proves that Dn∗ is asymptotically equivalent to p

1 ∗ ∗ ∗ ∗ l(t) V˜ (t) + n1/3 l(t) ˙ ˆ(t)(ˆ∗ − ) ˆ ˜ √ + − | V (t)| |V˜ (t)|p−1 H(t) dt. (t) n 0

For every x 0 and y > 0 we have x − y = (x2 − y2 )/(x + y), so 1 2n1/3 l(t) ˙ ˆ(t)(ˆ∗ − ) ˆ + n−1/2 ∗ l(t) V˜ ∗ (t) ∗ |V˜ (t)|p−1 H(t) dt Dn∗ =p ∗ −1/6 ˜ 2 (t)|V (t)| + OP∗ (n ) 0 + oP∗ (n−1/6 ), © 2010 Board of the Foundation of the Scandinavian Journal of Statistics.

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where OP∗ (n−1/6 ) is uniform in t. Hence, 1 l(t) ˙ ∗ ˆ + n−1/2 ∗ l(t) V˜ ∗ (t)|V˜ ∗ (t)|p−2 H(t) dt Dn∗ = p n1/3 ˆ(t)(ˆ − ) (t) 0 + oP∗ (n−1/6 ). Similar to step 3 in Durot (2007) one obtains that for every non-random function fn such that supn, t | fn (t)| is bounded, 1 ∗ ∗ V˜ (t)|V˜ (t)|p−2 fn (t) dt = oP∗ (1). 0

Therefore, ∗

ˆ P∗ (n1/3 ) + ∗ oP∗ (n−1/6 ), Dn∗ = (ˆ − )o which yields (25). • Step 4. From the preceding steps, p 1 ∗ p/3 ∗ p (t) −1/6 ˜ n Spn = |V (t)| dt + oP∗ (n ). l(t) 0 One then obtains the required result by using the same arguments as in steps 4, 5 and 6 of Durot (2007).
