Abstract
In this paper, we consider rank estimation for partial functional linear regression models based on functional principal component analysis. The proposed rank-based method is robust to outliers in the errors and highly efficient under a wide range of error distributions. The asymptotic properties of the resulting estimators are established under some regularity conditions. A simulation study conducted to investigate the finite sample performance of the proposed estimators shows that the proposed rank method performs well. Furthermore, the proposed methodology is illustrated by means of an analysis of the Berkeley growth data.
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Acknowledgements
Cao’s work is partly supported by the National Natural Science Foundation of China (No. 11701020). Xie’s work is supported by the National Natural Science Foundation of China (No. 11771032, 11571340, 11971045), Beijing Natural Science Foundation under Grant No. 1202001, the Science and Technology Project of Beijing Municipal Education Commission (KM201710005032, KM201910005015), and the International Research Cooperation Seed Fund of Beijing University of Technology (No. 006000514118553).
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Appendix
Appendix
In order to prove the theorem, we make a linear approximation to \(|\epsilon _i-\epsilon _j-t|\) by \(D_{ij}=2I_{\{\epsilon _i>\epsilon _j\}}-1\). One intuitive interpretation of \(D_{ij}\) is that \(D_{ij}\) can be treated as the first derivative of \(|\epsilon _i-\epsilon _j-t|\) at \(t=0\) (Pollard 1991). Let \(R_i=\sum _{j=1}^m\gamma _{0j}{\hat{\xi }}_{ij}-\int _{0}^{1}\beta _0(t)X_i(t)dt=\varvec{U}_i^T\varvec{\gamma }_0-\int _{0}^{1}\beta _0(t)X_i(t)dt\). By Condition C2, one has \(E(D_{ij})=0, \text {Var}(D_{ij})=1\). Define
and
where \(\varvec{u} =(\varvec{u}_1^T,\varvec{u}_2^T)^T\) and \(\varvec{W}_{n}=(\varvec{W}_{n1}^T,\varvec{W}_{n2}^T)^T.\)
Lemma 1
Denote
Under the Conditions C1 and C2, it has
where
where \(\varvec{W}_{n} =( \varvec{W}_{n1}^T,\varvec{W}_{n2}^T)^T,\) \(\tau =\int f^2(t)dt\) and f(t) is the density function of the random error.
Proof of Lemma 1
Let \(M(t)=E(|\epsilon _i-\epsilon _j-t|-|\epsilon _i-\epsilon _j|)\). With Condition C2, it is easy to show that M(t) has a unique minimizer at zero, and its Taylor expansion at origin has the form \(M(t)=\tau t^2+o(t^2)\). Denote \({\mathcal {F}}_n=\{(\varvec{z}_1,\varvec{X}_1),\ldots ,(\varvec{z}_n,\varvec{X}_n)\}.\) Hence, for larger n, we have
where
Via Condition C1 and Cauchy–Schwarz inequality, it has
Note that
For \(\text {A}_1\), by Condition C1 and Hölder inequality, it is obtained
As for \(\text {A}_2\), since
and
one has
Combining above discussions, it follows
Moreover, it has
Taking advantage of the stochastic order of \(B_{n1}\) and \(A_n(\varvec{u})\) and Cauchy–Schwarz inequality, one has
Therefore, it follows
Furthermore, we obtain
Thus,
By elementary calculation, one has
With cross product terms being cancellated, it follows
where
and
By Condition C2 and the weak law of large numbers, one has
and
Therefore, combining (6) and (7) with (8), we can conclude that \(I_1(\varvec{u})=o_p(1)\).
Next, consider the stochastic order of \(I_{2}(\varvec{u})\).
By Lemma 2 of Yu et al. (2016b) and (8), one has
where \(\varvec{\Lambda }= \text {diag}( \lambda _1,\lambda _2,\ldots ,\lambda _m).\) Similar to \(I_{21}(\varvec{u})\), we have \(I_{22}(\varvec{u})=O_p( { \lambda _m^{-1}}).\) Then, invoking Cauchy–Schwarz inequality, one has \(I_{23}(\varvec{u})=O_p( { \lambda _m^{-1}}).\) Thus, \(I_{2}(\varvec{u})=O_p( { \lambda _m^{-1}}).\) Now, we consider \(I_3(\varvec{u})\). Similar to \(B_n(\varvec{u})\), we can show that \(I_3(\varvec{u})=o_{p}\left( n^{\frac{1}{a+2b}}\right) .\)
Considering the stochastic order of \(I_{1}(\varvec{u})\), \(I_{2}(\varvec{u})\) and \(I_{3}(\varvec{u})\), one has
Thus completes the proof of Lemma 1. \(\square\)
Proof of Theorem 1
Write \(\varvec{U}_{i}=({\hat{\xi }}_{i1},\ldots ,{\hat{\xi }}_{im})^T, \varvec{\theta }=(\varvec{\alpha }^T,\varvec{\gamma }^T)^T\). Note that
so \(\hat{\varvec{\theta }}\) minimizes
where \(R_{i}=\sum _{j=1}^{m}{\hat{\xi }}_{ij}\gamma _{0j}-\int _{0}^{1}\beta _0(t)x_i(t)dt.\)
By Lemma 1, with simple calculation, it yields
where
By the definition of \(\hat{\varvec{\theta }}\) and Lemma 1, we can conclude that
And with simple calculation, it yields
and
It can be rewritten as follows
where \(\varvec{Z}=[\varvec{z}_1,\ldots ,\varvec{z}_n]^T\), and \(\tilde{\varvec{Z}}_i\) is the ith column component of \((\varvec{I}-\varvec{S}_m)\varvec{Z},\) here \(\varvec{S}_m=\varvec{U}_m(\varvec{U}_m^T\varvec{U}_m)^{-1}\varvec{U}_m^T.\)
Similar to Lemma 2 of Du et al. (2018), we have
Therefore, according to the law of large numbers, we have
Furthermore, we have \(\Vert \hat{\varvec{\alpha }}-\varvec{\alpha }_0\Vert =O_p(n^{-1/2}).\) Combining this with the definition of \(\varvec{W}_{n2}\) and (10), one has
Using the similar argument as in Theorem 3.2 of Shin (2009) and the condition \(m\sim n^{1/(a+2b)}\), we have
\(\square\)
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Cao, R., Xie, T. & Yu, P. Rank method for partial functional linear regression models. J. Korean Stat. Soc. 50, 354–379 (2021). https://doi.org/10.1007/s42952-020-00075-4
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DOI: https://doi.org/10.1007/s42952-020-00075-4
Keywords
- Rank estimation
- Karhunen–Loève expansion
- Asymptotic normality
- Functional principal component analysis
- Convergence rate