Abstract
In two sample problems it is of interest to examine the difference between the two regression curves or to detect whether certain functions are adequate to describe the overall trend of the difference. In this paper, we propose a simultaneous confidence band (SCB) as a global inference method with asymptotically correct coverage probabilities for the difference curve based on the weighted local linear kernel regression estimates in each sample. Our procedure allows for random designs, different sample sizes, heteroscedastic errors, and especially missing covariates. Simulation studies are conducted to investigate the finite sample properties of the new SCB which support our asymptotic theory. The proposed SCB is used to analyze two data sets, one of which is concerned with human event-related potentials data which are fully observed and the other is concerned with the Canada 2010/2011 youth student survey data with partially missing covariates, leading to a number of discoveries.
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Acknowledgements
We would like to thank the Editors and two referees for their constructive and helpful comments that substantially improved an earlier version of this paper.This research was supported in part by the National Natural Science Foundation of China Award NSFC #11901521, First Class Discipline of Zhejiang–A (Zhejiang Gongshang University–Statistics), Zhejiang Province Statistical Research Program #20TJQN04, and the Simons Foundation Mathematics and Physical Sciences Program Award #499650.
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Appendix
Appendix
For any functions \(\varphi _{n}\left( x\right) \) and \(\phi _{n}\left( x\right) ,x\in {\mathcal {D}}\), we use \(\varphi _{n}\left( x\right) =\) \( U_{p}\left( \phi _{n}\left( x\right) \right) \) and \(\varphi _{n}\left( x\right) =u_{p}\left( \phi _{n}\left( x\right) \right) \) to mean “\(\varphi _{n}\left( x\right) /\phi _{n}\left( x\right) \) is bounded and \(\varphi _{n}\left( x\right) /\phi _{n}\left( x\right) \) tends to 0 as \(n\rightarrow \infty \) for \(x\in {\mathcal {D}}\) uniformly in probability.”
1.1 A.1 Preliminaries
In this subsection, we give some lemmas that are needed in our theoretical development. The following lemma is Theorem 1 in Cai et al. (2020).
Lemma A.1
Under Assumptions (A1)–(A5), as \(n\rightarrow \infty \),
where \(\psi _{k}\left( x\right) =n_{k}^{-1}f_{k}^{-1}\left( x\right) \sum \limits _{i=1}^{n_{k}}\frac{\delta _{ik}}{\pi _{ik}}K_{h}\left( X_{ik}-x\right) \varepsilon _{ik}\) for \(k=1,2.\)
By (A.4) in Cai et al. (2020), one has that conditional on \(\Delta _{nk}=n_{0k}\), \(\psi _{k}\left( x\right) \) is a stochastic process with mean zero and variance \( n_{k}^{-2}n_{0k}h^{-1}d_{k}\left( x\right) \tau _0 \left\{ 1 \right. \) \(\left. +u_{p}\left( 1\right) \right\} \). Here \(\{ n_{0k}\}\) is a sequence of numbers related to \(n_k\) with \(1\le n_{0k}\le n_k\). By (7) it is clear that there exists a constant \(r_k>0\) such that \(r_k \le \Delta _{nk}/n_{k}\le 1\) in probability as \(n_k \rightarrow \infty \). Therefore, we only need to consider \(n_{0k} \ge r_k\times n_k\). That is, \(n_{0k}\) and \(n_k\) have the same order as \(n_k \rightarrow \infty \).
Due to the i.i.d. assumption of the data, conditional on \(\Delta _{nk}=n_{0k}\) is equivalent to conditional on \({\varvec{\delta }}_k=(\delta _{1k},\ldots ,\delta _{n_{k}k})^T\) in which there are \(n_{0k}\) elements equal to 1 and the other \((n_k-n_{0k})\) elements equal to 0. Without loss of generality, let \(\delta _{ik}=1\) for \(i=1,2,\ldots ,n_{0k}\) and \( \delta _{ik}=0\) for \(i=n_{0k}+1,\ldots ,n_{k}, k=1,2\). Then conditional on \(\Delta _{nk}=n_{0k}\), one can write
Furthermore, conditional on \(\Delta _{nk}=n_{0k}\) define a rescaled stochastic process of \(\psi _{k}\left( x\right) \):
Also define a Gaussian stochastic process \(\xi _{n_{0k}}^{*}\left( x\right) \):
where \(d_{nk}\left( x\right) =f_{k}^{-2}\left( x\right) f_{X_{1k}|\delta _{1k}=1}\left( x\right) \mathop {\mathrm{E}}\left\{ \varepsilon _{1k}^{2}\pi _{k}^{-2}\left( m_{k}\left( x\right) +\varepsilon _{1k}\right) I\left( \left| \varepsilon _{1k}\right| \le \kappa _{n}\right) \big | \right. \) \(X_{1k}=x,\left. \delta _{1k}=1\right\} \) with \(\kappa _{n}=n^{\gamma },2/\left( 3\eta \right) \le \gamma \le 6^{-1}\), in which \(\eta \) is given in Assumption (A2), and \(W_{n_{0k}k}\left( T\left( u,\varepsilon _{k}\right) \right) \) is a sequence of Wiener processes. Here \(T\left( \cdot ,\cdot \right) \) is the Rosenblatt quantile transformation of \((X_k,\varepsilon _k)\) in Rosenblatt (1952) which produces mutually independent uniform random variables on \(\left[ 0,1\right] ^{2}\). By Lemmas A.6–A.8 in Cai et al. (2020), one obtains the following result.
Lemma A.2
Under Assumptions (A1)–(A5), as \(n_{0k}\rightarrow \infty \),
and \(\xi _{n_{0k}}^{*}\left( x\right) \) is a Gaussian process with mean zero and covariance function uniformly approximated by \(\tau _h(x-x^{\prime })\).
1.2 A.2 Proofs of the main results in Section 2
Proof of Proposition 1
Let \(G_{nk}^{*}\left( x\right) =(n\Delta _{nk})^{1/2} n_k^{-1}d_{k}^{1/2}\left( x\right) \xi _{n_{0k}}^{*}\left( x\right) \). Lemma A.2 implies that conditional on \(\Delta _{nk}=n_{0k} \), \(G_{nk}^{*}\left( x\right) \) is a Gaussian process with mean zero and covariance function uniformly approximated by
Next, notice that conditional on \(\Delta _{nk}=n_{0k}\), \( \sqrt{nh} \psi _{k}\left( x\right) =(n \Delta _{nk})^{1/2} n_k^{-1} \) \(d_{k}^{1/2} \left( x\right) \xi _{n_{0k}}\left( x\right) . \) This together with (A.3) concludes that conditional on \(\Delta _{nk}=n_{0k}, \)
As is discussed below Lemma A.1, \(n_{0k}\) and \(n_k,n\) have the same order as \(n \rightarrow \infty \). Since \(h\ll n^{-1/5}\log ^{-1/5}n\) given in Assumption (A5), by ( A.1) and the definition of \(G_{nk}(x)\) one has that, conditional on \(\Delta _{nk}=n_{0k}\),
as \(n_{0k} \rightarrow \infty \). That is, for every \(\varepsilon >0\), there exist \(M_{\varepsilon }\) and \(N_{0k}\) such that
for all \(n_{0k}\ge N_{0k}\). On the other hand, (7) implies that there exists \(N_{k}>N_{0k}\) such that \(P\left( \Delta _{nk}\ge N_{0k}\right) >1-\varepsilon /2\) for \(n_{k}\ge N_{k}\). Therefore, for \(n_{k} \ge N_{k}\),
By definition, this means that unconditionally \(\sup _{x\in \left[ a_{0}b_{0} \right] }\left| G_{nk}^{*}\left( x\right) -G_{nk}(x)\right| =o_{p}\left( \log ^{-1/2}n\right) \), completing the proof. \(\square \)
Proof of Theorem 1
It is readily seen that s(x) is bounded away from zero for all \( x\in [a_{0},b_{0}]\). By Equation (4), it is clear that
Using the triangle inequality repeatedly, one can easily obtain that, for any real numbers a and b,
Thus for any given \(\epsilon > 0\), it is readily shown that for any \(t \in {\mathbb {R}}\)
Therefore,
which with (A.4) concludes that
Similarly, by symmetry,
Note that in light of Lemma A.2\( D_{n}^{*}\left( x\right) \) can be viewed asymptotically as a Gaussian process with mean zero and covariance function \(\Sigma _n(x,x^{\prime })\) which converges to \(\Sigma (x,x^{\prime })\) (that equals 1 when \(x = x^{\prime }\), and equals to 0 otherwise). Then it is seen that \(S_n^* =\sup _{x\in \left[ a_{0},b_{0}\right] }| D_{n}^{*}\left( x\right) | \) has a smooth limiting distribution, say, \(\Psi (t)\). Therefore, one can easily obtain that
and that \(\lim _{\epsilon \rightarrow 0}\sup _{t\in {\mathbb {R}}} |\Psi (t+\epsilon ) -\Psi (t) |= 0\). These two equations imply that for any small \(\epsilon > 0\)
Moreover, by (A.5) and (A.7) one has that
Since \(\epsilon > 0\) can be arbitrary small and the left hand side does not depend on \(\epsilon \),
Likewise, by (A.6) and (A.7) one can obtain that
Therefore,
Similarly, one can also obtain
Define \(|y|_+ = y\) if \(y\ge 0\) and \(|y|_+ = 0\) if \(y < 0\), and \(|y|_- = -y\) if \(y\le 0\) and \(|y|_- = 0\) if \(y > 0\). Then \(|k(t)| = |k(t)|_+ +|k(t)|_-\) for any real function k(t) and \(t \in {\mathbb {R}}\). Thus,
Letting \(k(t) = P\left( \sup _{x\in \left[ a_{0},b_{0}\right] } | D_{n}\left( x\right) | \le t\right) - P\left( \sup _{x\in \left[ a_{0},b_{0}\right] } | D_{n}^{*}\left( x\right) | \le t \right) \) in (A.10) and using (A.8) and (A.9) one has that
completing the proof of Theorem 1.
1.3 A.3 Proofs of some statements in the Introduction
This subsection contains the main steps to obtain that under certain strong conditions the extreme value distribution of the standardized version of the estimation error \({\tilde{m}}_{1}(x)-{\tilde{m}}_{2}(x)-\left( m_{1}(x)-m_{2}(x)\right) \) is a standard Gumbel distribution, as stated in Theorem A.1 below. The proofs are similar to those in Cai et al. (2020) for the single population problem. Theorem A.1 reveals that (i) when the data are fully observed, if \(\sigma _{1}^{2}(x)\) is proportional to \(\sigma _{2}^{2}(x)\) and \(f_{1}\left( x\right) \) and \( f_{2}\left( x\right) \) are equal, the extreme value distribution of the estimation error is a Gumbel distribution; (ii) when the covariates are MAR, the result is also true under the strong condition that \(d_1(x)\) is proportional to \(d_2(x)\) which is difficult to check. The following lemmas are needed in the proof of Theorem A.1.
Lemma A.3
If the Gaussian process \(\zeta \left( s\right) ,0\le s\le T\) is stationary with mean zero and variance one, and covariance function satisfying
for some constants \(C>0\) and \(0< a \le 2\), then as \(T\rightarrow \infty \),
in which \(\rho _{T}=\left( 2\log T\right) ^{1/2}\) and
where \(H_a\) is a certain strictly positive constant (\(H_{1}=1,H_{2}=\pi ^{-1/2}\)).
This result is a direct conclusion of Theorems 11.1.5 and 12.3.5 of Leadbetter et al. (1983).
By (A.4) in Cai et al. (2020), one can easily obtain that conditional on \(\Delta _{nk}=n_{0k}\), \(\psi _{1}\left( x\right) -\psi _{2}\left( x\right) \) is a stochastic process with mean 0 and variance
where \(v_{k}(x) = \tau _0d_k(x)\), for \( k=1,2\). Define the standardized version of \(\psi _{1}\left( x\right) -\psi _{2}\left( x\right) \) as follows:
for \(x\in \left[ a_0,b_{0}\right] \), where \(r_{n}\left( x\right) =\frac{n_{1}^{2}\Delta _{n2}d_{2}(x)}{n_{2}^{2}\Delta _{n1}d_{1}(x)}\).
Lemma A.4
Under Assumptions (A1)–(A5), if there exists a constant \(c>0\) such that \(d_2(x)=cd_1(x)\), then one has that as \(n_{01},n_{02}\rightarrow \infty \), for any \(t\in {\mathbb {R}}\),
where
Proof of Lemma A.4
By Lemmas A.6–A.8 in Cai et al. (2020), it is readily seen that \(\xi _{n}\left( x\right) \) has the same absolute maximum asymptotic distribution as
where \(W_{1}\left( u\right) \) and \(W_{2}\left( u\right) \) are two independent two-sided Wiener processes on \(\left( -\infty ,+\infty \right) \) . The absolute maximum of \(\xi _{n}^{*}\left( x\right) \) has the following probability law:
Let \(\eta _{n}\left( t\right) =\frac{\int K\left( s-t\right) dW_{1}\left( s\right) }{\tau _0^{1/2}\sqrt{ 1+r_{n}\left( ht\right) }}-\frac{\int K\left( s-t\right) dW_{2}\left( s\right) }{\tau _0^{1/2}\sqrt{1+r_{n}^{-1}\left( ht\right) }}\). Then \(\eta _{n}\left( t\right) \) is a Gaussian process with mean 0, variance 1 and covariance function
which holds since \(d_2(\cdot )=cd_1(\cdot )\) implying that \(r_n(ht_1)=r_n(ht_2)\).
Define \(\eta ^{*}_{n}\left( t\right) =\eta _{n}\left( t+a_0/h\right) \) for \(0 \le t\le b_0/h-a_0/h\). Then the covariance function of \(\eta ^{*}_{n}\left( t\right) \) satisfies
fulfilling the conditions in Lemma A.3 with \(T=b_0/h-a_0/h\) and \(a=2\). Therefore, one has that
as \(n_{01},n_{02}\rightarrow \infty \). Hence, \( P\{ \rho _{h}( \sup _{t\in \left[ a_0/h,b_0/h\right] }| \eta _{n}\left( t \right) | -\gamma _{h} ) \le t \vert \Delta _{n1}=n_{01}\), \(\Delta _{n2}=n_{02}\} \) and \( P\{ \rho _{h}\left( \sup _{t\in \left[ a_0,b_{0}\right] }| \xi _{n}^{*}\left( t\right) | -\gamma _{h} \right) \le t \vert \Delta _{n1}=n_{01},\Delta _{n2}=n_{02}\} \) have the same limiting distribution as well, completing the proof. \(\square \)
Theorem A.1
Under Assumptions (A1)–(A5), if there exists a constant \(c>0\) such that \(d_2(x)=cd_1(x)\), then one has that for any \(t\in {\mathbb {R}}\)
as \(n\rightarrow \infty \), where \(\rho _{h},\gamma _{h},v_{k}(x),k=1,2,\) are given in Lemma A.4.
Proof of Theorem A.1
By Lemma A.4 and using the total probability formula similar to the proof of Theorem 2 in Cai et al. (2020), one can immediately obtain that
as \(n\rightarrow \infty \). Furthermore, according to Lemma A.1, one has that
This together with \(h^2\sqrt{nh\log n}\rightarrow 0\) obtained by \(h\ll n^{-1/5}\log ^{-1/5} n \) given in Assumption (A5) and the Slutsky Theorem concludes that
\(\square \)
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Cai, L., Wang, S. Global statistical inference for the difference between two regression mean curves with covariates possibly partially missing. Stat Papers 62, 2573–2602 (2021). https://doi.org/10.1007/s00362-020-01208-x
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DOI: https://doi.org/10.1007/s00362-020-01208-x