Robust estimation of functional factor models with functional pairwise spatial signs

Abstract

Factor model analysis has emerged as a powerful tool to capture the latent dynamic structure of functional data from a dimension-reduction viewpoint. Conventional methods for estimating the factor model are sensitive to heavy tails and outliers. To address this issue and achieve robustness, we provide an eigenvalue-ratio based method to estimate the number of factors by replacing the covariance operator with the functional pairwise spatial sign operator. Moreover, we propose a two-step robust approach to recover the factor space. The convergence rates of the robust estimators for factor loadings, factor scores, and common components are derived under some mild conditions. Numerical studies and a real data analysis confirm the proposed procedures remain reliable even when the factors and idiosyncratic errors have heavy-tailed distributions.

Appendix

Proof of Lemma 3.1 Based on Assumptions 3.1$-$3.3, we can get $N(C_2+o(1))\le \lambda _i(\Gamma )\le N(C_1+o(1))$ for $i\le k$; $C_2\le \lambda _i(\Gamma )\le C_1$ for $k<i\le N$; $\lambda _i(\Gamma )\le C_1$ for $i>N$. Moreover, by Theorem 2.4 in Zhong et al. (2022b), $\lambda _{i}(K)=\mathbb {E}\left[ \frac{\lambda _{i}(\Gamma ) U_{i}^{2}}{\sum _{i=1}^{\infty } \lambda _{i}(\Gamma ) U_{i}^{2}}\right]$ with $U_{i}=\frac{\langle Y-{\tilde{Y}}, \phi _{i}\rangle }{\sqrt{2 \lambda _{i}(\Gamma )} }$ and $\sum _{i=1}^{\infty } \lambda _{i}(K)=1$. Thus, for $i\le k$,

$$\begin{aligned}\lambda _i(K)\le \frac{\lambda _1(\Gamma )}{\lambda _k(\Gamma )} \mathbb {E}\left[ \frac{U_{i}^{2}}{\sum _{i=1}^{\infty } U_{i}^{2}}\right] \le \frac{C_1}{kC_2}+o(1),\end{aligned}$$

$$\begin{aligned} \begin{aligned} \lambda _i(K)\ge&\mathbb {E}\left[ \frac{\lambda _k(\Gamma )U_{i}^{2}}{\sum _{i=1}^{k} \lambda _i(\Gamma ) U_{i}^{2}+\sum _{i=k+1}^{\infty }C_1 U_{i}^{2}}\right] \\=&\mathbb {E}\left[ \frac{\lambda _k(\Gamma )U_{i}^{2}}{\sum _{i=1}^{k} (\lambda _1(\Gamma )-C_1 )U_{i}^{2}+\sum _{i=1}^{\infty }C_1 U_{i}^{2}}\right] \\=&\frac{\lambda _k(\Gamma )}{\lambda _1(\Gamma )-C_1}\mathbb {E}\left[ \frac{a_1U_{i}^{2}}{\sum _{i=1}^{k} a_1U_{i}^{2}+ 1}\right] , \end{aligned} \end{aligned}$$

where $a_1=\frac{\lambda _1(\Gamma )}{C_1}-1$, and we have

$$\begin{aligned}{} & {} k\mathbb {E}\left[ \frac{a_1U_{i}^{2}}{\sum _{i=1}^{k} a_1U_{i}^{2}+ 1}\right] +\mathbb {E}\left[ \frac{1}{\sum _{i=1}^{k} a_1U_{i}^{2}+ 1}\right] =1.\\{} & {} \mathbb {E}\left[ \frac{1}{\sum _{i=1}^{k} a_1 U_{i}^{2}+ 1}\right] \le \mathbb {E}\left[ \frac{1}{1+a_1 U_{i}^{2}}\right] \le \frac{1}{2}\mathbb {P}(a_1 U_{i}^{2}\le 1)+\frac{1}{2}\mathbb {P}(a_1 U_{i}^{2}> 1)=\frac{1}{2}\mathbb {P}(a_1 U_{i}^{2}\le 1)+\frac{1}{2} \le \frac{1}{2} +o(1).\end{aligned}$$

Similarly,

$$\begin{aligned}\mathbb {E}\left[ \frac{a_1U_{i}^{2}}{\sum _{i=1}^{k} a_1U_{i}^{2}+ 1}\right] \ge \frac{1}{k}(\frac{1}{2}+o(1)),\end{aligned}$$

which implies that $\lambda _i(K)\ge \frac{C_2}{2kC_1}+o(1)$.

For $k<i\le N$, we have

$$\begin{aligned}\lambda _i(K)\le \mathbb {E}\left[ \frac{C_1U_{i}^{2}}{\sum _{i=1+k}^{N} C_2U_{i}^{2}}\right] =\frac{C_1}{(N-k)C_2}=\frac{C_1}{NC_2}(1+o(1))=O\left( \frac{1}{N}\right) .\end{aligned}$$

For $i> N$, we have

$$\begin{aligned}\lambda _i(K)\le \lambda _k(K)=O\left( \frac{1}{N}\right) .\end{aligned}$$

Proof of Theorem 3.1

By Lemma 2.1, Lemma 3.1, and Weyl’s theorem, $\lambda _{r}({\widehat{K}}) \asymp 1, r \le k$ and $\lambda _{r}({\widehat{K}})=O_{p}(N^{-1 / 2}), r>k$. Let $\alpha =N^{-1 / 2}$, then

$$\begin{aligned} \max _{i < k} \frac{\lambda _{i}({\widehat{K}})}{\lambda _{i+1}({\widehat{K}})+c \alpha }=O_{p}(1), \quad \max _{i >k} \frac{\lambda _{i}({\widehat{K}})}{\lambda _{i+1}({\widehat{K}})+c \alpha } \lesssim O_{p}(1), \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{\lambda _{k}({\widehat{K}})}{\lambda _{k+1}({\widehat{K}})+c \alpha } \ge c \alpha ^{-1} \rightarrow \infty , \end{aligned}$$

which concludes the consistency.

Proof of Theorem 3.2 Similar to the proof of Lemma 2.3 in Horváth and Kokoszka (2012), we have

$$\begin{aligned} \max _{1 \le h \le k}\left\| {\widehat{f}}_{h}-sf_{h}\right\| \le \frac{2 \sqrt{2}}{\alpha }\left\| {\widehat{K}}-K\right\| _{\mathcal {S}} \end{aligned}$$

where $\alpha =\min \left\{ \lambda _{1}-\lambda _{2}, \ldots , \lambda _{K-1}-\lambda _{K}, \lambda _{K}\right\}$, and the Hilbert-Schmidt norm of a Hilbert-Schmidt operator S is defined by

$$\begin{aligned} \Vert S\Vert _{\mathcal {S}}^{2}=\sum _{j=1}^{\infty }\left\| S\left( e_{j}\right) \right\| ^{2}, \end{aligned}$$

where $\left\{ e_{1}, e_{2}, \ldots \right\}$ is any orthonormal basis. Then the asymptotic result of the factors follows from Lemma 2.1. In addition, by Assumption 3.3, $\varvec{S}=\textrm{sgn}(\frac{1}{N} \sum _{i=1}^{N}(\widehat{\varvec{l}}_i \varvec{l}_i^{\top })) =\textrm{diag}\{s_1,\dots ,s_k\}$ with entries $\pm 1$. Then,

$$\begin{aligned} \begin{aligned} \frac{1}{N}\Vert \widehat{\varvec{l}}_i - \varvec{Sl}_i \Vert ^2&=\frac{1}{N}\sum _{i=1}^{N}(\widehat{\varvec{l}}_i - \varvec{Sl}_i)^{\top }(\widehat{\varvec{l}}_i - \varvec{Sl}_i) \\&=\frac{1}{N}\sum _{i=1}^{N}\sum _{h=1}^{k}({\widehat{l}}_{ih}-s_h l_{ih})^2. \end{aligned} \end{aligned}$$

Recall that ${\widehat{l}}_{ih}=\langle Y_{i}, {\widehat{f}}_{h}\rangle$, $\frac{1}{N}\sum _{i=1}^{N}\sum _{h=1}^{k}({\widehat{l}}_{ih}-s_h l_{ih})^2=\frac{1}{N}\sum _{i=1}^{N}\sum _{h=1}^{k} \int _\mathcal {I}(Y_i(t)({\widehat{f}}_h(t)-s_h f_h(t))+\epsilon _i(t)s_h f_h(t)+s_h l_{ih}(f_h(t)f_h(t)-1))\textrm{dt}=O_p(N^{-1})$, where the last equality follows by Assumption 3.1 and the orthonormality of the factor curves, which concludes the proof of the theorem.

Proof of Corollary 3.1 By Theorem 3.2 and triangular inequality, we have

$$\begin{aligned} \begin{aligned} \frac{1}{N}\sum _{i=1}^{N}\left\| \sum _{h=1}^{k} {\widehat{l}}_{ih}{\widehat{f}}_h-l_{ih} f_{h}\right\| ^2&=\frac{1}{N}\sum _{i=1}^{N}\left\| \sum _{h=1}^{k} {\widehat{l}}_{ih}{\widehat{f}}_h-s_h{\widehat{l}}_{ih}f_h+s_h{\widehat{l}}_{ih}f_h-l_{ih} f_{h}\right\| ^2 \\ {}&\le \frac{1}{N}\sum _{i=1}^{N}\sum _{h=1}^{k}l_{ih}^2\Vert {\widehat{f}}_{h}-sf_{h}\Vert ^2+\frac{1}{N}\sum _{i=1}^{N}\sum _{h=1}^{k}(s_h {\widehat{l}}_{ih}-l_{ih})^2 \Vert f_h\Vert ^2\\&=O_{P}\left( \frac{1}{N}\right) . \end{aligned} \end{aligned}$$

See Tables 7, 8, 9, 10, 11, 12 and 13.

Table 7 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario III over 500 repetitions, $k=3$

Table 8 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario I over 500 repetitions, $k=2$

Table 9 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario II over 500 repetitions, $k=2$

Table 10 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario III over 500 repetitions, $k=2$

Table 11 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario I over 500 repetitions, $k=4$

Table 12 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario II over 500 repetitions, $k=4$

Table 13 Simulation results for estimating the factor loadings, factor scores, and common components in Scenario III over 500 repetitions, $k=4$