Journal of Nonparametric Statistics
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Asymptotic normality of locally modelled regression estimator for functional data Zhiyong Zhou & Zhengyan Lin To cite this article: Zhiyong Zhou & Zhengyan Lin (2016) Asymptotic normality of locally modelled regression estimator for functional data, Journal of Nonparametric Statistics, 28:1, 116-131, DOI: 10.1080/10485252.2015.1114112 To link to this article: http://dx.doi.org/10.1080/10485252.2015.1114112
Published online: 02 Dec 2015.
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Date: 23 February 2016, At: 18:12
Journal of Nonparametric Statistics, 2016 Vol. 28, No. 1, 116–131, http://dx.doi.org/10.1080/10485252.2015.1114112
Asymptotic normality of locally modelled regression estimator for functional data
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Zhiyong Zhou ∗ and Zhengyan Lin Department of Mathematics, Zhejiang University, Hangzhou 310027, People’s Republic of China (Received 17 July 2014; accepted 20 October 2015) We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk’s phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results. Keywords: functional data; locally modelled regression; asymptotic normality; empirical likelihood
1.
Introduction
With the development of modern measuring instruments, functional data have become usual. Functional data analysis has been involved in many fields of applied sciences including chemometrics, biometrics, econometrics and geophysics. Various kinds of statistical methods in this special framework have been proposed (see the monographs such as Bosq 2000; Ramsay and Silverman 2005; Horváth and Kokoszka 2012; Bongiorno, Salinelli, Goia, and Vieu 2014, and references therein). Recently, nonparametric methods have been used to deal with functional data (see Ferraty and Vieu 2006, and references therein). This functional nonparametric regression method is essentially based on an extension of the well-known Nadaraya-Watson kernel estimator to the functional case. The nonparametric kernel estimator for functional data has been considered by many authors. Ferraty, Mas, and Vieu (2007) derived the mean-squared convergence and asymptotic distribution for the estimator, while Ferraty, Laksaci, Tadj, and Vieu (2010) established the uniform almost complete convergence. Meanwhile, the nonparametric k-Nearest-Neighbours (kNN) estimator for functional data has also been investigated (see the recent work of Burba, Ferraty, and Vieu 2009; Kudraszow and Vieu 2013). For the case when both response and predictor are functions, the interested reader may refer Ferraty, Laksaci, Tadj, and Vieu (2011) and Ferraty, Van Keilegom, and Vieu (2012). As for the dependent *Corresponding author. Email:
[email protected] © American Statistical Association and Taylor & Francis 2015
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functional data case, Ferraty and Vieu (2004), Masry (2005) investigated the nonparametric kernel estimator for α-mixing functional data, while Laib and Louani (2010), and Ling, Liang, and Vieu (2015) obtained the asymptotic properties of the nonparametric kernel estimator for functional stationary ergodic data, Benhenni, Hedli-Griche, Rachdi, and Vieu (2008) for the longrange-dependent case. There are also some literatures concerning the semiparametric functional models, like Chen, Hall, and Müller (2011), and Goia and Vieu (2014) for single index functional regression model, Aneiros-Prez and Vieu (2006) for the functional partial linear model. Many other recent related references about the nonparametric functional data analysis include Amiri, Crambes, and Thiam (2014), Ezzahrioui and Ould-Said (2008), Rachdi and Vieu (2007), and so on. In the finite dimensional data case, local linear smoothing technique has various advantages over the kernel method, such as bias reduction and adaptation of edge effects (cf. Fan 1992; Fan and Yao 2003 for an extension discussion). Consequently, the nonparametric estimation based on the local linear approach for functional data have also been investigated recently (see Baíllo and Grané 2009; Boj, Delicado, and Fortiana 2010; Berlinet, Elamine, and Mas 2011 for more details). As a fast local linear regression method, a local modelling approach named locally modelled regression estimator was proposed by Barrientos-Marin, Ferraty, and Vieu (2010). Very recently, Demongeot, Laksaci, Madani, and Rachdi (2013) and Rachdi, Laksaci, Demongeot, Abdali, and Madani (2014) use this method to estimate the conditional density of a scalar response given a functional explanatory variable. Demongeot, Laksaci, Rachdi, and Rahmani (2014) apply it for the conditional cumulative distribution function and Messaci, Nemouchi, Ouassou, and Rachdi (2015) for the conditional quantile. The asymptotic consistency result and the almost complete convergence of the locally modelled regression estimation for regression function have already been provided in BarrientosMarin et al. (2010). The mean-squared convergences of the locally modelled regression estimation for conditional density function and conditional cumulative distribution function have also been established in Rachdi et al. (2014) and Demongeot et al. (2014), respectively. Nevertheless, there is no existing literature which investigates the asymptotic normality of this functional local estimation. This paper attempts to fill this gap. More specifically, we obtain the mean-squared convergence and asymptotic normality of the locally modelled regression estimation for the regression operator, which is significant, from both theoretical and practical points of view. Though our mean-squared convergence result follows from the similar arguments in Rachdi et al. (2014) and Demongeot et al. (2014), it is still new for the regression operator. The convergence result shows the superiority of this method over the kernel method, namely in the bias term (cf. Section 4.1). More importantly, we strictly obtain the asymptotic normality result for the local linear estimator by some complex deduction, which complements the literatures concerning the local linear modelling for functional data. Moreover, we adapt the empirical likelihood method (cf. Owen 1988, 1990; Qin and Lawless 1994 for basic introduction, and Lian 2012; Xiong and Lin 2013 for the extensions to the functional data framework) to construct point-wise confidence intervals for the local linear estimation of the regression function and derive the Wilk’s phenomenon for the empirical likelihood inference. The paper is organised as follows. Section 2 introduces the nonparametric functional regression model and the locally modelled regression method. Some assumptions and notations are given in Section 3. The main asymptotic results and Wilk’s phenomenon for the empirical likelihood inference are established in Section 4. A simulation study is presented in Section 5. Section 6 is devoted to the conclusion and some prospects. Finally, the technical lemmas and the detailed proofs of our main results are postponed to the appendix. The proofs of the technical lemmas can be found in the supplemental material of this paper.
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Model
This paper deals with the nonparametric functional regression model
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Y = m(X ) + ε,
with E(ε|X ) = 0, E(ε2 |X ) = σ 2 (X ),
(1)
where the explanatory variable X is valued in some infinite-dimensional space F and Y is a scalar response. Assume that {(Xi , Yi ), i = 1, 2, . . . , n} is a sample of n random variables which are independent and identically distributed and have the same distribution as (X , Y ), then the Nadaraya-Watson estimator for the regression function m(χ ) = E(Y |X = χ ) is defined as follows: n n Yi K(|δ(Xi , χ )|/h) Ki Yi i=1 , (2) = m ˆ NW (χ ) = i=1 n n K(|δ(X , χ )|/h) i i=1 i=1 Ki where K is the asymmetrical kernel, δ(·, ·) locates one element of F with respect to another one, | · | denotes the absolute value function, h is the bandwidth and Ki = K(|δ(Xi , χ )|/h). The Nadaraya-Watson estimator can be seen as a local constant regression estimator which minimises the objection function: min a
n
(Yi − a)2 K(|δ(Xi , χ )|/h).
i=1
In contrast with the ordinary Nadaraya-Watson estimator, Barrientos-Marin et al. (2010) proposed the functional locally modelled regression estimator for the regression function m ˆ LMR (χ ) = aˆ which is obtained by minimising the following objection function: min
(a,b)∈R2
n
(Yi − a − bβ(Xi , χ ))2 K(|δ(Xi , χ )|/h),
i=1
where β(·, ·) is a predetermined operator defined on F × F such that ∀ ξ ∈ F, β(ξ , ξ ) = 0. Then, by some simple calculation, the functional locally modelled regression estimator can be explicitly written as follows: n n n i=1 j=1 wij Yj j=1 Wj Kj Yj = n , (3) m ˆ LMR (χ ) = n n i=1 j=1 wij j=1 Wj Kj with wij = βi (βi − βj )Ki Kj , Wj = ni=1 (wij /Kj ) = ni=1 βi2 Ki − ( ni=1 βi Ki )βj , where βi = β(Xi , χ ). The locally modelled regression estimation approach is indeed a kind of nonparametric functional local linear regression method, which assumes that a + bβ(·, χ ) is a good approximation of m(·) in a small neighbourhood of χ . The choices of δ(·, ·) and β(·, ·) are crucial to the behaviour of the estimator. Some particular examples for δ(·, ·) and β(·, ·) have been provided in BarrientosMarin et al. (2010). For instance, if the functional data are smooth, we may use β(χ1 , χ2 ) = (q) (q) (q) (q) θ (t)(χ1 (t) − χ2 (t)) dt and the semi-metric |δ(χ1 , χ2 )| = (χ1 (t) − χ2 (t))2 dt, where θ is a given function and χ (q) denotes the qth derivative of χ . Remark 1 Actually, we can see that locally modelled regression estimator is just a weighted Nadaraya-Watson estimator with random weights {Wj , j = 1, 2, . . . , n}. In the case that n i=1 βi Ki ≡ 0, the locally modelled regression goes to the ordinary Nadaraya-Watson estimator. ˆ NW . Thus, m ˆ LMR is more flexible than m
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Assumptions and Notations
In the rest of the paper, for any fixed χ ∈ F, we impose the following assumptions: (H.1) For all r > 0, φχ (r) = P(|δ(X , χ )| ≤ r). Moreover, there exists a function χ (u) such that φχ (uh) = χ (u), ∀u ∈ [0, 1]. lim h→0 φχ (h) (H.2) m and σ 2 is continuous in the neighbourhood of χ , which means that m and σ 2 are both in the set
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f : F → R,
lim
|δ(χ ,χ)|→0
f (χ ) = f (χ ) .
The bi-functional operator β satisfies the following three conditions: ∃0 < C1 < C2 , ∀χ ∈ F, we have C1 |δ(χ , χ )| |β(χ , χ )| C2 |δ(χ , χ )|; supu∈B(χ,r) |β(u, χ ) − δ(u, χ )| = o(r), where B(χ, r) = {z ∈ F : |δ(z, χ )| ≤ r}; h B(χ,h) β(u, χ ) dPX (u) = o( B(χ ,h) β 2 (u, χ ) dPX (u)), with PX (u) is the probability distribution of X . (H.4) The kernel K is supported on [0, 1] and has a continuous derivative K (s) < 0 for s ∈ [0, 1) and K(1) > 0. (H.5) The bandwidth h satisfies limn→∞ h = 0 and limn→∞ nφχ (h) = ∞.
(H.3) (H.3.1) (H.3.2) (H.3.3)
Remark 2 Assumptions (H.1), (H.2), (H.4) and (H.5) are simple adaptations of hypotheses H0 – H2 in Ferraty et al. (2007), when one replaces the semi-metric d(·, ·) by |δ(·, ·)|. Moreover, the assumption (H.3) is the same as the assumption (H3) in Rachdi et al. (2014), which is unrestrictive and is satisfied, for instance if δ(·, ·) = β(·, ·) or if β(x, u) − 1 = 0. lim δ(x,u)→0 δ(x, u) Interested readers would find several examples of δ and β which satisfy this condition (see the comments in Barrientos-Marin et al. 2010). Before giving the main results, we list some notations. In the sequel, we denote 1 j (K j (u)) χ (u) du, where j = 1, 2. Mj = K (1) − 0
N(a, b) = K a (1) −
1
(ub K a (u)) χ (u) du,
0
Moreover, let m ˆ l (χ ) = (1/nE(W1 K1 ))
4. 4.1.
n j=1
forall a > 0 and b = 2, 4.
Wj Kj Yjl , l = 0, 1, then m ˆ LMR (χ ) = m ˆ 1 (χ )/m ˆ 0 (χ ).
Main results Mean-squared convergence
Denote ˆ 1 (χ )] − m(χ ) = Bn (χ ) = E[m =
E(W1 K1 Y1 ) − m(χ ) E(W1 K1 )
E(β12 K1 ) · E[K1 (m(X1 ) − m(χ ))] − E(β1 K1 ) · E[β1 K1 (m(X1 ) − m(χ ))] . E(β12 K1 ) · E(K1 ) − E2 (β1 K1 )
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Theorem 1
When the assumptions (H.1)–(H.5) hold, we have the following asymptotic results:
1 (4) E[m ˆ LMR (χ )] − m(χ ) = Bn (χ ) + O nφχ (h)
and
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Var[m ˆ LMR (χ )] =
1 M2 2 1 σ (χ ) + o . nφχ (h) M12 nφχ (h)
(5)
n Remark 3 Obviously, Bn (χ ) = o(1) because of the continuity of m. And, when i=1 βi Ki ≡ 0, we have E(β1 K1 ) = 0 and the bias term Bn (χ ) goes to (E[K1 (m(X1 ) − m(χ ))])/E(K1 ) = O(h), which is the bias of the Nadaraya-Watson estimator (cf. Ferraty et al. 2007). If we impose the following additional assumption (H.A), then we can obtain the exact order of the bias term Bn (χ ).
(H.A) E[m(X ) − m(χ )|β(X , χ ) = s] = ψ(s) with ψ (0) and ψ (0) exist. Moreover, K = K(|δ(X , χ )|/h) is measurable with respect to β(X , χ ). Under the above assumption, we have E[K1 (m(X1 ) − m(χ ))] = E(E[K1 (m(X1 ) − m(χ ))|β1 ]) = E[K1 ψ(β1 )]
= ψ (0)E(K1 β1 ) + 12 ψ (0)E(β12 K1 ) + o(E(β12 K1 )). Similarly, E[β1 K1 (m(X1 ) − m(χ ))] = E[β1 K1 ψ(β1 )]
= ψ (0)E(β12 K1 ) + 12 ψ (0)E(β13 K1 ) + o(E(β13 K1 )). Then, Bn (χ ) =
1 ψ (0)E2 (β12 K1 ) − 12 ψ (0)E(β1 K1 )E(β13 K1 ) 2 (1 E(β12 K1 ) · E(K1 ) − E2 (β1 K1 )
+ o(1))
ψ (0)N(1, 2) 2 h (1 + o(1)), = 2M1 which follows from Lemma A.1 in the appendix. Therefore, it is quite clear that in comparison to the Nadaraya-Watson method, there is a significant gain on the bias term of the locally modelled regression estimator. Indeed, the bias term for our estimator is O(h2 ) while the Nadaraya-Watson estimator is O(h). The gain on the bias term shows the superiority of the locally modelled regression estimator over the Nadaraya-Watson estimator. What’s more, their asymptotic variance terms are the same, which equal to (1/nφχ (h))(M2 /M12 )σ 2 (χ ). The variance term depends on the function φχ (h) which is closely linked on δ and the latter is related to the topological structure on the functional space F. 4.2.
Asymptotic normality
Theorem 2 When assumptions (H.1)–(H.5) hold, then we have
M2 2 d nφχ (h)(m ˆ LMR (χ ) − m(χ ) − Bn (χ )) −→ N 0, 2 σ (χ ) . M1
(6)
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Remark 4 By the similar proof approach of this theorem and some small modifications, we assert that we can obtain the asymptotic normality of the local linear estimation of the conditional density for functional data, that is
d Y /X nhH φx (hK )(fˆhK ,hH (x, y) − f Y /X (x, y) − BH (x, y)h2H − BK (x, y)h2K ) −→ N(0, VHK (x, y)),
which is of a prospect in Demongeot et al. (2013) and Rachdi et al. (2014). Similarly, the asymptotic normality result for the local linear estimation of the conditional cumulative distribution function for functional data expected in Demongeot et al. (2014) would also be achieved.
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If we estimate the function φχ (h) by its empirical counterpart: φˆ χ (h) =
(i : |δ(Xi , χ )| ≤ h) , n
and impose the additional assumption: (H.6) limn→∞ nφχ (h)Bn (χ ) = 0, then we can cancel the bias term and obtain the following two simpler versions: Corollary 1
When assumptions (H.1)–(H.6) hold, then nφˆ χ (h)(m ˆ LMR (χ ) − m(χ ))
M1 d −→ N(0, 1). √ σ (χ ) M2
(7)
To avoid estimating the constants involved in Corollary 1, one may consider the simple uniform kernel (M1 = M2 = 1) and get the following result: Corollary 2 When assumptions (H.1)–(H.6) hold, K(·) = 1[0,1] (·) and if σˆ 2 (χ ) is a consistent estimator of σ 2 (χ ), then we have
4.3.
nφˆ χ (h) d (m ˆ LMR (χ ) − m(χ )) −→ N(0, 1). σˆ 2 (χ )
(8)
Empirical likelihood
By taking advantage of the asymptotic normality result, we can construct a point-wise confidence interval for the regression function. However, this procedure involves some unknown parameters such as σ 2 (χ ), M1 , M2 and φχ (h), which need to be estimated. Obviously, it itself is a difficult task. Moreover, as illustrated in Lian (2012) which constructed confidence intervals for nonparametric functional operator by using empirical likelihood, the intervals based on the normal approximation may have poor coverage rates for finite sample sizes while empirical likelihood method improves the accuracy of confidence interval. In addition, empirical likelihood method owns the major advantage that it involves no predetermined assumptions on the shapes of the confidence intervals, while the intervals constructed by normal approximation method are always symmetric. In this subsection, we adapt the empirical likelihood method to construct the confidence interval for the local linear estimation in the functional data framework. Next, we provide the main notations and derive the Wilk’s phenomenon for empirical likelihood inference by using the asymptotic normality result obtained in the last subsection.
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Let gni (θ ) = Ki Wi (Yi − θ ), where Ki and Wi are defined in Section 2. Then, the locally modelled regression estimator m ˆ LMR (χ ) satisfies the following estimating equation: n
gni (m ˆ LMR (χ )) = 0.
i=1
For a candidate value θ of the target quantity m(χ ), the empirical likelihood ratio statistic is defined as n n n Rn (θ ) = max npi | pi ≥ 0, pi = 1, gni (θ ) = 0 . (p1 ,p2 ,...,pn )
i=1
i=1
i=1
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Following Owen (1990), the log likelihood ratio is given by ln (θ ) = −2 log Rn (θ ) = 2
n
log(1 + λgni (θ )),
i=1
where λ satisfies
n i=1
gni (θ ) = 0. 1 + λgni (θ )
To obtain the asymptotic distribution of ln (m(χ )), we need the following moment assumption: (H.7) E(Y 4 ) < ∞ and the function W (χ ) = E[|Y − m(χ )|4 |X = χ ] is continuous in a neighbourhood of χ. We remark that this technical condition is also imposed in Xiong and Lin (2013) which considered the empirical likelihood method inference for functional stationary ergodic data. Now, we are ready to provide the main result known as the Wilk’s phenomenon for the empirical likelihood inference. Theorem 3 When assumptions (H.1)–(H.7) hold, then d
ln (m(χ )) −→ χ 2 (1),
(9)
where χ 2 (1) denotes the chi-square distribution with degree of freedom 1. Remark 5 Assume χ12 (α) satisfies P(χ 2 (1) ≤ χ12 (α)) = 1 − α, where 0 < α < 1. Then, by using the above theorem, we can construct a 1 − α confidence interval for the regression function m(χ ) as Iα = {θ, ln (θ ) ≤ χ12 (α)}.
5.
Simulation
In this section, a simple simulation study is presented to illustrate our asymptotic normality result. Similar to the simulation example in Delsol (2009), we generate n = 200 functional data (see Figure 1) by Xi (t) = sin(Wi − π(2t − 1))
for t ∈ [0, 1] and i = 1, 2, . . . , 200,
(10)
where the random variables Wi are i.i.d uniformly distributed in [0, π/4]. The curves are discretised on the same grid which is composed of 100 equidistant values in [0,1]. The real response
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0.0 −1.0
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−0.5
X(t)
0.5
1.0
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0.0
0.2
0.4
0.6
0.8
1.0
t Figure 1.
Simulated curves (n = 200).
Y is generated by Yi = m(Xi ) + εi , where εi are i.i.d normally distributed as N(0, 1) and 3/4 3/4 1 m(Xi ) = (Xi (t))2 dt = 2π cos2 (Wi − π(2t − 1)) dt. 2π 1/2 1/2
(11)
We consider the quadratic kernel K(x) = 32 (1 − x2 )1[0,1) and K(1) > 0. We take β(χ , χ ) = 1 − χ (t)) dt, with θ is the eigenfunction of the empirical covariance opera0 θ (t)(χ (t) n tor (1/n) i=1 (Xi − X¯ )T (Xi − X¯ ) corresponding to the biggest eigenvalues, where X¯ = 1 2 (1/n) ni=1 Xi . Meanwhile, we choose δ(χ , χ ) = 0 (χ (t) − χ (t)) dt. We select the bandwidth h by an adaptation of the cross-validation method. In order to avoid the difficulty in the estimation of the bias term, we may assume assumption (H.6) holds. At the same time, we estimate the parameters M1 , M2 and σ 2 (χ ) by the following the consistent estimators:
n 1 |δ(Xi , χ )| ˆ 1 (χ ) = M K ; h nφˆ χ (h) i=1
n 1 2 |δ(Xi , χ )| ˆ M2 (χ ) = K ; h nφˆ χ (h) i=1 n 2 i=1 K(|δ(Xi , χ )|/h)Yi 2 − (m ˆ NW (χ ))2 . σˆ (χ ) = n i=1 K(|δ(Xi , χ )|/h) Then, by the asymptotic normality result in Section 4, we have ˆ 1 (χ ) M d n = nφˆ χ (h)(m ˆ LMR (χ ) − m(χ )) −→ N(0, 1). ˆ 2 (χ ) σˆ 2 (χ )M
(12)
Since we simulate 200 curves, then we have 200 generations of n . To verify the theoretical result, we list the histogram and Normal Q–Q plot of the 200 generations of n in Figure 2. In the
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Normal Q−Q Plot
1 −1
0
Sample Quantiles
30 20 10 0
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Frequency
40
2
50
Histogram
−2
−1
0
1
2
−3
Γn Figure 2.
−1 0
1
2
3
Theoretical Quantiles
Histogram and Q–Q plot of n .
left plot, the histogram of n is almost symmetric around the zero point and well-shaped similar to the histogram of standard normal distribution. The right plot is the Normal Q–Q plot of n , which is almost a line. Thus, the simulation results indicate that n obeys a standard normal law when n is large, which verifies its limiting distribution. Next, we randomly generated n = 200 or n = 400 curves and 100 testing curves and constructed 95% confidence intervals for them by using both empirical likelihood and normal approximation. The results are based on 100 replications. The confidence intervals based on the normal approximation for any m(χ ) are as follows: ˆ 2 (χ )/nφˆ χ (h)M ˆ 1 (χ ), m ˆ 2 (χ )/nφˆ χ (h)M ˆ 1 (χ )], ˆ LMR (χ ) + z σˆ 2 (χ )M [m ˆ LMR (χ ) − z σˆ 2 (χ )M where z is the 0.975 quantile of the standard normal distribution. The simulation results shown in Table 1 demonstrated the superiority of empirical likelihood-based intervals. For both cases, the empirical likelihood method produces better coverage and shorter intervals compared to the normal approximation. Table 1. Simulation results for the constructed 95% confidence intervals. The numbers shown are the coverage accuracy and the average interval lengths (numbers in the brackets). n
Normal
EL
200
0.876 (2.391) 0.903 (2.140)
0.914 (0.936) 0.940 (0.708)
400
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More discussions about the locally modelled regression for functional data in practical aspects including the choices of the functional operator β, δ, the selection of bandwidth h and the superiority of the locally modelled regression estimaton are referred to Barrientos-Marin et al. (2010), Demongeot et al. (2014) and Rachdi et al. (2014).
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6.
Conclusion and perspectives
In this paper, we obtain the mean-squared convergence and asymptotic normality of the locally modelled regression estimation, which complements the literatures concerning the locally modelled regression estimation for functional data (cf. Barrientos-Marin et al. 2010; Demongeot et al. 2013, 2014; Rachdi et al. 2014; Messaci et al. 2015). We also adapt the empirical likelihood method to construct the point-wise confidence interval for the nonparametric regression function and derive the asymptotic distribution of the empirical likelihood ratio. In addition, a simulation study is presented to verify our theoretical results. However, one issue that has not been addressed is how to extend the independent case considered in this paper to the dependent cases, as considered in Masry (2005), Laib and Louani (2010), and Benhenni et al. (2008). Detailed assumptions and rigorous proofs may involve both new tools and more technicalities and therefore are left for future research. The other issue is possible extensions of the local linear ideas in a purely nonparametric model to the semiparametric models. Further discussions are also left for future research. Acknowledgments The authors would like to thank the Editor, the Associate Editor and the anonymous referees for their helpful comments that greatly improved the paper.
Disclosure statement No potential conflict of interest was reported by the author(s).
Funding This research is supported by the National Natural Science Foundation of China (No. 11171303) and Natural Science Foundation of Zhejiang Province of China (No. LY14A010003).
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Appendix. Technical lemmas and proofs In what follows, we provide the technical lemmas and the proofs of the main results. The proofs of the technical lemmas can be found in the supplemental material of this paper. We use the notations in Section 3 and denote by C some generic constant.
Journal of Nonparametric Statistics Lemma A.1 (a) (b) (c) (d)
Under assumptions (H.1)–(H.5), we have
j
E(K1 ) = Mj φχ (h) + o(φχ (h)), for j = 1, 2; E(K1a β1 ) = o(hφχ (h)), for all a > 0; E(K1a β1b ) = N(a, b)hb φχ (h) + o(hb φχ (h)), for all a > 0 and b = 2, 4; E(W1 K1 ) = (n − 1)E(w12 ) = (n − 1)N(1, 2)M1 h2 φχ2 (h)(1 + o(1)).
Proof
See the supplemental material of this paper.
Lemma A.2
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Under assumptions (H.1)–(H.5), we have
Var[m ˆ 0 (χ )] =
1 nφχ (h)
M2 N(2, 4) 2N(2, 2) + 2 + N 2 (1, 2) N(1, 2)M1 M1
(1 + o(1));
M2 N(2, 4) 2N(2, 2) + (1 + o(1)); + N 2 (1, 2) N(1, 2)M1 M12 1 N(2, 4) M2 M2 2N(2, 2) 2 2 Var[m ˆ 1 (χ )] = + 2 σ (χ ) 2 (1 + o(1)). m (χ ) + nφχ (h) N 2 (1, 2) N(1, 2)M1 M1 M1
Cov(m ˆ 1 (χ ), m ˆ 0 (χ )) =
Proof
m(χ ) nφχ (h)
See the supplemental material of this paper.
Lemma A.3
Under assumptions (H.1)–(H.5), we have p
m ˆ 0 (χ ) −→ E(m ˆ 0 (χ )) = 1.
Proof
See the supplemental material of this paper.
Proof of Theorem 1 For the bias term, by applying the expression (z − 1)2 1 = 1 − (z − 1) + z z to z = m ˆ 0 (χ ), we can obtain that ˆ 1 (χ ) − m(χ )) − (m ˆ 0 (χ ) − 1)m ˆ 1 (χ ) + (m ˆ 0 (χ ) − 1)2 m ˆ LMR (χ ), m ˆ LMR (χ ) − m(χ ) = (m which implies that ˆ 1 (χ )) − m(χ )] − Cov(m ˆ 0 (χ ), m ˆ 1 (χ )) + E[(m ˆ 0 (χ ) − 1)2 m ˆ LMR (χ )]. E[m ˆ LMR (χ )] − m(χ ) = [E(m
(A1)
Concerning the variance term, we have the following decomposition (see the expression (11) in Ferraty et al. 2007):
1 ˆ 1 (χ ), m ˆ 0 (χ ))] Var(m ˆ 1 (χ )) E[m ˆ 1 (χ )Cov(m (E(m ˆ 1 (χ )))2 −4 + 3Var(m ˆ 0 (χ )) +o . 2 3 4 (E(m ˆ 0 (χ ))) (E(m ˆ 0 (χ ))) (E(m ˆ 0 (χ ))) nφχ (h) (A2) ˆ 1 (χ )) = m(χ ) + Bn (χ ) = m(χ )(1 + o(1)), then Theorem 1 follows directly from Lemma A.2 As E(m ˆ 0 (χ )) = 1 and E(m and the last two expressions. Var[m ˆ LMR (χ )] =
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Proof of Theorem 2 Denote Qn (χ ) = (m ˆ 1 (χ ) − E[m ˆ 1 (χ )]) − m(χ )(m ˆ 0 (χ ) − E[m ˆ 0 (χ )]). Since Bn (χ ) =
E[m ˆ 1 (χ )] − m(χ ), E[m ˆ 0 (χ )]
then m ˆ LMR (χ ) − m(χ ) − Bn (χ ) =
ˆ 0 (χ ) − E[m ˆ 0 (χ )]) Qn (χ ) − Bn (χ )(m . m ˆ 0 (χ )
p
Lemma A.3 implies that m ˆ 0 (χ ) → 1. Moreover, Bn (χ ) = o(1) as n → ∞. Then, we can obtain that
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m ˆ LMR (χ ) − m(χ ) − Bn (χ ) = Thus, in order to gain Theorem 2, it suffices to show that
M2 2 nφχ (h)Qn (χ ) −→ N 0, 2 σ (χ ) . M1 d
Write
Qn (χ ) (1 + op (1)). m ˆ 0 (χ )
(A3)
nφχ (h)[m ˆ 1 (χ ) − m(χ )m ˆ 0 (χ )] =
n nφχ (h) Wj Kj (Yj − m(χ )) nE(W1 K1 ) j=1
=
n 1 βi2 Ki · nE(β12 K1 ) i=1
−
1 nE(β1 K1 )
n
n nφχ (h)E(β12 K1 ) Kj (Yj − m(χ )) E(W1 K1 ) j=1
βi Ki ·
i=1
n nφχ (h)E(β1 K1 ) βj Kj (Yj − m(χ )) E(W1 K1 ) j=1
=: An · Bn − Cn · Dn , then
nφχ (h)Qn (χ ) =
nφχ (h)[(m ˆ 1 (χ ) − m(χ )m ˆ 0 (χ )) − E(m ˆ 1 (χ ) − m(χ )m ˆ 0 (χ ))]
= [An Bn − E(An Bn )] − [Cn Dn − E(Cn Dn )]. To show Equation (A3), we only need to show the following two claims: d
Claim 1
An Bn − E(An Bn ) −→ N(0,
Claim 2
Cn Dn − E(Cn Dn ) −→ 0.
M2 2 σ (χ )); M12
p
As for Claim 1, rewrite An Bn − E(An Bn ) = [Bn − E(Bn )] + [(An − 1)Bn − E((An − 1)Bn )]. The Cauchy–Schwarz inequality implies that
E|(An − 1)Bn − E((An − 1)Bn )| ≤ 2E|(An − 1)Bn | ≤ 2 E[(An − 1)2 ] · E(B2n ).
Moreover, by using Lemma A.1, we have E[(An − 1)2 ] = Var(An ) = ≤
1 · n · Var(β12 K1 ) n2 E2 (β12 K1 )
1 · E(β14 K12 ) n(O(h2 φχ (h)))2
1 · O(h4 φχ (h)) nO(h4 φχ2 (h))
1 =O nφχ (h)
=
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129
and E(B2n )
⎤2 ⎡ n nφχ (h)E2 (β12 K1 ) ⎣ = Kj (Yj − m(χ ))⎦ E E2 (W1 K1 ) j=1
=
nφχ (h)O(h4 φχ2 (h)) [nE(K1 (Y1 (n − 1)2 O(h4 φχ4 (h))
=
n [nO(φχ (h)) + n(n − 1)o(φχ2 (h))] (n − 1)2 O(φχ (h))
− m(χ )))2 + n(n − 1)E2 (K1 (Y1 − m(χ )))]
= O(1) + o(nφχ (h)).
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Thus, E|(An − 1)Bn − E((An − 1)Bn )| ≤ 2 E[(An − 1)2 ] · E(B2n )
1 · [O(1) + o(nφχ (h))] = o(1), ≤2 O nφχ (h) which implies that (An − 1)Bn − E((An − 1)Bn ) = op (1). Therefore, to prove Claim 1, we just need to prove that M2 d Bn − E(Bn ) −→ N 0, 2 σ 2 (χ ) . (A4) M1 Denote
n nφχ (h)E(β12 K1 ) [Kj (Yj − m(χ )) − E(Kj (Yj − m(χ )))] E(W1 K1 )
Bn − E(Bn ) =
j=1
=:
n
εnj ,
j=1
where εnj = nφχ (h)E(β12 K1 )/E(W1 K1 )[Kj (Yj − m(χ )) − E(Kj (Yj − m(χ )))] are i.i.d random variables with mean 0. The variance of nj=1 εnj goes to ⎛ E⎝
n
⎞2 εnj ⎠ =
j=1
nφχ (h)E2 (β12 K1 ) · n · Var(K1 (Y1 − m(χ ))). E2 (W1 K1 )
Moreover, we have Var(K1 (Y1 − m(χ ))) = Var(K1 Y1 ) − 2m(χ ) · Cov(K1 Y1 , K1 ) + m2 (χ )Var(K1 ) = [E(K12 Y12 ) − (E(K1 Y1 ))2 ] − 2m(χ )[E(K12 Y1 ) − E(K1 Y1 )E(K1 )] + m2 (χ )[E(K12 ) − E2 (K1 )] = [E((m2 (X1 ) + σ 2 (X1 ))K12 ) − (E(m(X1 )K1 ))2 ] − 2m(χ )[E(m(X1 )K12 ) − E(m(X1 K1 ))E(K1 )] + m2 (χ )[E(K12 ) − E2 (K1 )] = [(m2 (χ ) + σ 2 (χ ))E(K12 )(1 + o(1)) − m2 (χ )E2 (K1 )(1 + o(1))] − 2m(χ )[m(χ )E(K12 )(1 + o(1)) − m(χ )E2 (K1 )(1 + o(1))] + m2 (χ )[E(K12 ) − E2 (K1 )] = σ 2 (χ )E(K12 )(1 + o(1)) − m(χ )E(K12 )o(1) + m2 (χ )E2 (K1 )o(1) = σ 2 (χ )E(K12 )(1 + o(1)) = σ 2 (χ )M2 φχ (h)(1 + o(1)). Then, taking advantage of Lemma A.1, we can obtain that ⎞2 ⎛ n nφχ (h)[N(1, 2)h2 φχ (h)(1 + o(1))]2 ⎝ εnj ⎠ = · nσ 2 (χ )M2 φχ (h)(1 + o(1)) E [(n − 1)N(1, 2)M1 h2 φχ2 (h)(1 + o(1))]2 j=1
=
n2 M2 M2 σ 2 (χ )(1 + o(1)) −→ 2 σ 2 (χ ). (n − 1)2 M12 M1
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By using the central limit theorem, the proof of Equation (A4) is completed if the Lindeberg condition is verified. While, the Lindeberg condition holds since for any η > 0, n
√ 2 2 E[εnj 1(|εnj |>η) ] = nE[εn1 1(|εn1 |>η) ] = E[( nεn1 )2 1(|√nεn1 |>η√n) ] −→ 0,
j=1
as √ M2 2 ) −→ 2 σ 2 (χ ). E[( nεn1 )2 ] = nE(εn1 M1 Next, we present the proof of Claim 2. Rewrite Cn Dn − E(Cn Dn ) = [(Cn − 1)Dn − E((Cn − 1)Dn )] + [Dn − E(Dn )].
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Similar to the proof of Claim 1, we have
E|(Cn − 1)Dn − E((Cn − 1)Dn )| ≤ 2E|(Cn − 1)Dn | ≤ 2 E[(Cn − 1)2 ] · E(D2n )
Since 1 · n · Var(β1 K1 ) n2 E2 (β1 K1 )
E[(Cn − 1)2 ] = Var(Cn ) = ≤
1 · n · E(β12 K12 ) n2 (o(hφχ (h)))2
=
1 · n · O(h2 φχ (h)) n2 o(h2 φχ2 (h))
= o( and
1 ), nφχ (h)
⎤2 ⎡ n 2 (β K ) (h)E nφ χ 1 1 βj Kj (Yj − m(χ ))⎦ E⎣ E(D2n ) = E2 (W1 K1 ) j=1
= =
nφχ (h)o(h2 φχ2 (h)) [nE(β1 K1 (Y1 (n − 1)2 O(h4 φχ4 (h))
− m(χ )))2 + n(n − 1)E2 (β1 K1 (Y1 − m(χ )))]
1 n o [nO(h2 φχ (h)) + n(n − 1)o(h2 φχ2 (h))] (n − 1)2 h2 φχ (h)
= o(1) + o(nφχ (h)), then
E|(Cn − 1)Dn − E((Cn − 1)Dn )| ≤ 2 E[(Cn − 1)2 ] · E(D2n ) = o(1),
which implies that (Cn − 1)Dn − E((Cn − 1)Dn ) = op (1). Therefore, to show Claim 2, it suffices to show Dn − E(Dn ) = op (1). While, E[Dn − E(Dn )]2 = Var(Dn ) =
nφχ (h)E2 (β1 K1 ) · nVar(β1 K1 (Y1 − m(χ ))). E2 (W1 K1 )
By some simple calculation, we can obtain Var(β1 K1 (Y1 − m(χ ))) = σ 2 (χ )E(β12 K12 )(1 + o(1)). Then, Lemma A.1 implies that E[Dn − E(Dn )]2 = =
nφχ (h)E2 (β1 K1 ) · nσ 2 (χ )E(β12 K12 )(1 + o(1)) E2 (W1 K1 ) nφχ (h)o(h2 φχ2 (h)) 2 4 (n − 1) h φχ4 (h)N 2 (1, 2)M12 (1 + o(1))
· nσ 2 (χ )O(h2 φχ (h))
= o(1), which completes the proof of Claim 2. Then, Theorem 2 is proved.
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Finally, we give the proof of Theorem 3 concerning empirical likelihood inference. As a consequence of the above technical lemmas and Theorem 2, we have the following key lemma. j Lemma A.4 Let H = ni=1 Wi Ki and Gj = ni=1 gni (m(χ )) for j = 1, 2. Then, assumptions (H.1)–(H.7) imply the following results: d (a) nφχ (h)G1 /H −→ N(0, (M2 /M12 )σ 2 (χ )) and G1 = Op ( n3 h4 φχ3 (h)); p
(b) G2 = n3 σ 2 (χ )N 2 (1, 2)M2 h4 φχ3 (h)(1 + op (1)) and nφχ (h)G2 /H 2 −→ (M2 /M12 )σ 2 (χ ); (c) max1≤i≤n |gni (m(χ ))| = op ( n3 h4 φχ3 (h)); (d) λ = G1 /G2 + op (1/ n3 h4 φχ3 (h)).
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Proof
See the supplemental material of this paper.
Proof of Theorem 3 By Taylor’s expansion, 2 (m(χ )) + ni (m(χ )), log(1 + λgni (m(χ ))) = λgni (m(χ )) − 12 λ2 gni
where ni (m(χ )) satisfies that for some finite number C > 0, P(|ni (m(χ ))| ≤ C|λgni (m(χ ))|3 ) → 1,
∀1 ≤ i ≤ n.
Lemma A.4 implies that n
|λgni (m(χ ))|3 ≤ |λ|3 · max |gni (m(χ ))| · G2 1≤i≤n
i=1
⎛⎛ ⎞3 ⎞ 1 ⎜⎝ 3 4 3 ⎠ ⎟ = Op ⎝ ⎠ · op ( n3 h4 φχ3 (h)) · Op (n h φχ (h)) = op (1). n3 h4 φχ3 (h)
Therefore, ln (m(χ )) = 2
n
log(1 + λgni (m(χ )))
i=1
= 2λ
n i=1
gni (m(χ )) − λ2
n
2 gni (m(χ )) + op (1)
i=1
⎛ ⎛ ⎞⎤ ⎞⎤2 ⎡ 1 1 G1 G1 ⎝ ⎝ ⎠ ⎦ ⎠⎦ G2 + op (1) ⎣ ⎣ G1 − =2 + op + op G2 G2 n3 h4 φ 3 (h) n3 h4 φ 3 (h) ⎡
χ
χ
G12 + op (1) G2 2 nφχ (h)G1 /H + op (1). = nφχ (h)G2 /H 2 =
Thus, Theorem 3 follows from
p d nφχ (h)G1 /H −→ N(0, (M2 /M12 )σ 2 (χ )) and nφχ (h)G2 /H 2 −→ (M2 /M12 )σ 2 (χ ).
