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A High-Dimensional Test for Multivariate Analysis of Variance Under a Low-Dimensional Factor Structure

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Abstract

In this paper, the problem of high-dimensional multivariate analysis of variance is investigated under a low-dimensional factor structure which violates some vital assumptions on covariance matrix in some existing literature. We propose a new test and derive that the asymptotic distribution of the test statistic is a weighted distribution of chi-squares of 1 degree of freedom under the null hypothesis and mild conditions. We provide numerical studies on both sizes and powers to illustrate performance of the proposed test.

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Acknowledgements

The authors thank the Editor-in-Chief, Professor Niansheng Tang and two anonymous referees for their constructive comments, suggestions and detailed advice that vastly improved this article. Cao’s research is supported by the National Statistical Science Research Program (No. 2020LY002)the National Natural Science Foundation of China (Nos. 11601008, 11526070) and Doctor Startup Foundation of Anhui Normal University (No. 2016XJJ101). He’s research is supported by Anhui Provincial Natural Science Foundation (No. 2008085MA08). Huang’s research is supported by Anhui Provincial Natural Science Foundation (No. 1908085MA20).

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Appendices

Appendix A: Some Lemmas

The following lemma is obtained from Theorem 1 in [9].

Lemma A.1

Under the assumptions (A1) and (A2), it holds that

  1. (1)

    \(\lambda _{l}(\varvec{\Sigma })/p=\lambda _{l}(\varvec{\Sigma }_{{\mathbf{A}}})+o(1)\) for \(1\le l\le q\);

  2. (2)

    \(\lambda _{l}(\varvec{\Sigma })\le c\) for \(l>q\) where c is a constant.

Lemma A.2

Let m-dimensionally random vector \({\mathbf{Y}}=(y_{1},\ldots ,y_{m})^{^{{\mathrm{T}}}}\) satisfy that \(\{y_{i}\}_{i=1}^{m}\) are mutually independent, and \(\text{ E }({\mathbf{Y}})={\mathbf{0}}\), \(\text{ Cov }({\mathbf{Y}})={\mathbf{I}}_{m}\) and \(\text{ E }(y_{i}^{4})=3+\gamma \), where \(\gamma \ge -2\) is a known constant. Then for any \(m\times m\) symmetric matrix \({\mathbf{G}}\), there has \(\text{ Var }({\mathbf{Y}}^{^{{\mathrm{T}}}}{\mathbf{G}}{\mathbf{Y}})=2\text{ tr }({\mathbf{G}}^{2})+\gamma \text{ tr }({\mathbf{G}}\circ {\mathbf{G}})\), where \(\circ \) denotes the Hadamard product of matrices.

Proof

The proof is straight. We can also see Proposition A.1 in [6]. \(\square \)

Lemma A.3 can be obtained by Lemma A.11 of [1], where the proof is omitted.

Lemma A.3

Under the factor model (1.1), as \(j\in \{1,\ldots ,q\}\) and \(p, n\rightarrow \infty \),

$$\begin{aligned} \lambda _{j}({\mathbf{S}})/p=\lambda _{j}(\varvec{\Sigma })/p+o_{p}(1). \end{aligned}$$

Appendix B: Proofs of Theorems

Proof of Theorem 2.1

First, under the null hypothesis \(H_{0}\),

$$\begin{aligned} \Vert \overline{{\mathbf{X}}}_{i}-\overline{{\mathbf{X}}}_{j}\Vert ^{2}&=\Vert {\mathbf{A}}(\overline{{\mathbf{Z}}}_{i} -\overline{{\mathbf{Z}}}_{j})+(\overline{{\mathbf{e}}}_{i}-\overline{{\mathbf{e}}}_{j})\Vert ^{2}\\&=\Vert {\mathbf{A}}(\overline{{\mathbf{Z}}}_{i}-\overline{{\mathbf{Z}}}_{j})\Vert ^2+2(\overline{{\mathbf{e}}}_{i} -\overline{{\mathbf{e}}}_{j})^{^{{\mathrm{T}}}}{\mathbf{A}}(\overline{{\mathbf{Z}}}_{i}-\overline{{\mathbf{Z}}}_{j}) +\Vert \overline{{\mathbf{e}}}_{i}-\overline{{\mathbf{e}}}_{j}\Vert ^2\\&=:I_{ij}+2V_{ij}+U_{ij}, \end{aligned}$$

where \(\overline{{\mathbf{Z}}}_{g}=\frac{1}{n_{g}}\sum _{i=1}^{n_{g}}{\mathbf{Z}}_{gi}\) and \(\overline{{\mathbf{e}}}_{g}=\frac{1}{n_{g}}\sum _{i=1}^{n_{g}}{\mathbf{e}}_{gi}\). Then we have

$$\begin{aligned} \sum _{i<j}^{k}p^{-1}n_{ij}\Vert \overline{{\mathbf{X}}}_{i}-\overline{{\mathbf{X}}}_{j}\Vert ^{2} =\sum _{i<j}^{k}p^{-1}n_{ij}I_{ij}+2\sum _{i<j}^{k}p^{-1}n_{ij}V_{ij} +\sum _{i<j}^{k}p^{-1}n_{ij}U_{ij}. \end{aligned}$$

We firstly consider the term \(\sum _{i<j}^{k}p^{-1}n_{ij}I_{ij}\). It straightly shows that \(\sqrt{n_{g}}\overline{{\mathbf{Z}}}_{g}\) \({\mathop {\longrightarrow }\limits ^{d}}N(0,{\mathbf{I}}_{q})\) as \(n\rightarrow \infty \) by the central limit theorem for \(g\in \{1,\ldots ,k\}\). Denote \(\overline{{\mathbf{Z}}}=(\sqrt{n_{1}}\overline{{\mathbf{Z}}}_{1}^{^{{\mathrm{T}}}},\ldots ,\sqrt{n_{k}}\overline{{\mathbf{Z}}}_{k}^{^{{\mathrm{T}}}})^{^{{\mathrm{T}}}}\), then we have \(\overline{{\mathbf{Z}}}{\mathop {\longrightarrow }\limits ^{d}}N({\mathbf{0}},{\mathbf{I}}_{kq})\) and \(\overline{{\mathbf{Z}}}_{i}-\overline{{\mathbf{Z}}}_{j}=({\mathbf{c}}_{ij}^{^{{\mathrm{T}}}}\otimes {\mathbf{I}}_{q})\overline{{\mathbf{Z}}}\). Therefore,

$$\begin{aligned} \sum _{i<j}^{k}p^{-1}n_{ij}I_{ij}&=\sum _{i<j}^{k}n_{ij} \overline{{\mathbf{Z}}}^{^{{\mathrm{T}}}}({\mathbf{c}}_{ij}\otimes {\mathbf{I}}_{q})\varvec{\Sigma }_{{\mathbf{A}},p}({\mathbf{c}}_{ij}^{^{{\mathrm{T}}}} \otimes {\mathbf{I}}_{q})\overline{{\mathbf{Z}}}\nonumber \\&=\overline{{\mathbf{Z}}}^{^{{\mathrm{T}}}}\left( {\mathbf{C}}\otimes \varvec{\Sigma }_{{\mathbf{A}},p}\right) \overline{{\mathbf{Z}}}\nonumber \\&=\overline{{\mathbf{Z}}}^{^{{\mathrm{T}}}}\left( {\mathbf{C}}\otimes \varvec{\Sigma }_{{\mathbf{A}}}\right) \overline{{\mathbf{Z}}}+o_{p}(1), \end{aligned}$$
(B.1)

where the last equality holds via the assumption (A2).

Let the spectral decompositions of matrices \(\varvec{\Sigma }_{{\mathbf{A}}}\) and \({\mathbf{C}}\) be \({\mathbf{Q}}\varvec{\Lambda }{\mathbf{Q}}^{^{{\mathrm{T}}}}\) and \({\mathbf{Q}}_{1}\varvec{\Delta }{\mathbf{Q}}_{1}^{^{{\mathrm{T}}}}\), respectively, then

$$\begin{aligned} ({\mathbf{Q}}_{1}^{^{{\mathrm{T}}}}\otimes {\mathbf{Q}}^{^{{\mathrm{T}}}})\overline{{\mathbf{Z}}}{\mathop {\longrightarrow }\limits ^{d}}N({\mathbf{0}},{\mathbf{I}}_{kq}). \end{aligned}$$
(B.2)

It is noted that the rank of \({\mathbf{C}}\) is \(k-1\), which combining with (B.2) shows that

$$\begin{aligned} \overline{{\mathbf{Z}}}^{^{{\mathrm{T}}}}\left( {\mathbf{C}}\otimes \varvec{\Sigma }_{{\mathbf{A}}}\right) \overline{{\mathbf{Z}}} {\mathop {\longrightarrow }\limits ^{d}}\sum _{i=1}^{k-1}\sum _{j=1}^{q}\psi _{i} \lambda _{j}(\varvec{\Sigma }_{{\mathbf{A}}})z_{ij}. \end{aligned}$$
(B.3)

Next we deal with the term \(\sum _{i<j}^{k}p^{-1}n_{ij}V_{ij}\). It is easy to show that \(\text{ E }(V_{ij})=0\) and \(\text{ Var }(V_{ij})=n_{ij}^{-2}\text{ tr }({\mathbf{A}}{\mathbf{A}}^{^{{\mathrm{T}}}}{\mathbf{H}})\). Thus, by assumptions (A1) and (A2), we have

$$\begin{aligned} \text{ Var }(p^{-1}n_{ij}V_{ij})&=p^{-1}\text{ tr }(p^{-1}{\mathbf{A}}{\mathbf{A}}^{^{{\mathrm{T}}}}{\mathbf{H}}) \le {p}^{-1}\text{ tr }(\varvec{\Sigma }_{{\mathbf{A}},p})\lambda _{1}({\mathbf{H}})=o(1), \end{aligned}$$

which shows that

$$\begin{aligned} \sum \limits _{i<j}^{k}p^{-1}n_{ij}V_{ij}=o_{p}(1). \end{aligned}$$
(B.4)

For the three term \(\sum _{i<j}^{k}p^{-1}n_{ij}U_{ij}\), we have

$$\begin{aligned} \text{ E }\left( \sum \limits _{i<j}^{k}p^{-1}n_{ij}U_{ij} \right) =\frac{k(k-1)}{2p}\text{ tr }({\mathbf{H}}). \end{aligned}$$

It follows from Lemma A.2 and \(U_{ij}=\left( n_{ij}^{\frac{1}{2}}{\mathbf{H}}^{-\frac{1}{2}}(\overline{{\mathbf{e}}}_{i}-\overline{{\mathbf{e}}}_{j})\right) ^{^{{\mathrm{T}}}}(n_{ij}^{-1}{\mathbf{H}})\left( n_{ij}^{\frac{1}{2}}{\mathbf{H}}^{-\frac{1}{2}}(\overline{{\mathbf{e}}}_{i}-\overline{{\mathbf{e}}}_{j})\right) \) that \(\text{ Var }(U_{ij})=o(p^{2}n_{ij}^{-2})\). Hence, \(\text{ Var }\left( \sum \nolimits _{i<j}^{k}p^{-1}n_{ij}U_{ij}\right) =o(1)\) which results in

$$\begin{aligned} \sum \limits _{i<j}^{k}p^{-1}n_{ij}U_{ij}-\frac{k(k-1)}{2p}\text{ tr }({\mathbf{H}})=o_{p}(1). \end{aligned}$$
(B.5)

Lastly, this proof is completed via (B.1) and (B.3)–(B.5). \(\square \)

Proof of Theorem 2.3

In order to prove this theorem, we only need to show \(p^{-1}\text{ tr }({\mathbf{S}})-p^{-1}\text{ tr }(\varvec{\Sigma }){\mathop {\longrightarrow }\limits ^{P}}0\) and \(p^{-1}\sum _{l=1}^{q}\lambda _{l}({\mathbf{S}})-\text{ tr }(\varvec{\Sigma }_{{\mathbf{A}}}){\mathop {\longrightarrow }\limits ^{P}}0\). Note that

$$\begin{aligned} {\mathbf{S}}&=\frac{1}{n-k}\sum _{g=1}^{k}\sum _{i=1}^{n_{g}}({\mathbf{X}}_{gi} -\overline{{\mathbf{X}}}_{g})({\mathbf{X}}_{gi}-\overline{{\mathbf{X}}}_{g})^{^{{\mathrm{T}}}}\nonumber \\&=\frac{1}{n-k}\sum _{g=1}^{k}\sum _{i=1}^{n_{g}}\left\{ {\mathbf{A}}({\mathbf{Z}}_{gi} -\overline{{\mathbf{Z}}}_{g})+({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})\right\} \left\{ {\mathbf{A}}({\mathbf{Z}}_{gi} -\overline{{\mathbf{Z}}}_{g})+({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})\right\} ^{^{{\mathrm{T}}}}\nonumber \\&=\frac{1}{n-k}\sum _{g=1}^{k}\sum _{i=1}^{n_{g}}\Big \{{\mathbf{A}}({\mathbf{Z}}_{gi} -\overline{{\mathbf{Z}}}_{g})({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})^{^{{\mathrm{T}}}}{\mathbf{A}}^{^{{\mathrm{T}}}} +{\mathbf{A}}({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})^{^{{\mathrm{T}}}}\nonumber \\&\quad +({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})^{^{{\mathrm{T}}}}{\mathbf{A}}^{^{{\mathrm{T}}}} +({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})^{^{{\mathrm{T}}}}\Big \}\nonumber \\&=:{\mathbf{R}}_{1}+{\mathbf{R}}_{3}+{\mathbf{R}}_{3}^{^{{\mathrm{T}}}}+{\mathbf{R}}_{2}. \end{aligned}$$
(B.6)

Firstly, it follows from \(\frac{1}{n_{g}}\sum _{i=1}^{n_{g}}({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})^{^{{\mathrm{T}}}}{\mathop {\longrightarrow }\limits ^{P}}{\mathbf{I}}_{q}\) that \({\mathbf{R}}_{1}={\mathbf{A}}{\mathbf{A}}^{^{{\mathrm{T}}}}\{1+o_{p}(1)\}\) as \(n\rightarrow \infty \). Consequently,

$$\begin{aligned} p^{-1}\text{ tr }({\mathbf{R}}_{1})&=\text{ tr }(p^{-1}{\mathbf{A}}^{^{{\mathrm{T}}}}{\mathbf{A}})\{1+o_{p}(1)\}=\text{ tr }(\varvec{\Sigma }_{{\mathbf{A}}})+o_{p}(1). \end{aligned}$$
(B.7)

For the term \({\mathbf{R}}_{2}\), it is easy to get \(\text{ E }({\mathbf{R}}_{2})={\mathbf{H}}\) and

$$\begin{aligned} \text{ Var }\left\{ \sum _{i=1}^{n_{g}}({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})^{^{{\mathrm{T}}}} ({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})\right\} =c_{1}(n_{g}-1)\text{ tr }({\mathbf{H}}^2), \end{aligned}$$

where \(c_{1}\) is a known and finite constant. Thus we have

$$\begin{aligned} \text{ Var }\{\text{ tr }({\mathbf{R}}_{2})\}&=(n-k)^{-2}\sum _{g=1}^{k}\text{ Var }\left\{ \sum _{i=1}^{n_{g}} ({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})^{^{{\mathrm{T}}}}({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})\right\} \\&=\frac{c_{1}}{n-k}\text{ tr }({\mathbf{H}}^2)=O(pn^{-1}), \end{aligned}$$

which suggests that as \(n,p\rightarrow \infty \),

$$\begin{aligned} p^{-1}\text{ tr }({\mathbf{R}}_{2})-p^{-1}\text{ tr }({\mathbf{H}}){\mathop {\longrightarrow }\limits ^{P}}0. \end{aligned}$$
(B.8)

Next for the term \({\mathbf{R}}_{3}\). It straightly shows that \(\text{ E }({\mathbf{R}}_{3})=0\) and

$$\begin{aligned} \text{ Var }\left\{ \sum _{i=1}^{n_{g}}({\mathbf{e}}_{gi}-\overline{{\mathbf{e}}}_{g})^{^{{\mathrm{T}}}} {\mathbf{A}}({\mathbf{Z}}_{gi}-\overline{{\mathbf{Z}}}_{g})\right\} =(n_{g}-1)\text{ tr }({\mathbf{A}}{\mathbf{A}}^{^{{\mathrm{T}}}}{\mathbf{H}}). \end{aligned}$$

Thus,

$$\begin{aligned} \text{ Var }\{\text{ tr }({\mathbf{R}}_{3})\}=(n-k)^{-1}\text{ tr }({\mathbf{A}}{\mathbf{A}}^{^{{\mathrm{T}}}}{\mathbf{H}})\le (n-k)^{-1}p \lambda _{1}({{\mathbf{H}}})\text{ tr }(\varvec{\Sigma }_{{\mathbf{A}},p})=O((n-k)^{-1}p), \end{aligned}$$

which results in

$$\begin{aligned} p^{-1}\text{ tr }({\mathbf{R}}_{3}){\mathop {\longrightarrow }\limits ^{P}}0. \end{aligned}$$
(B.9)

Lastly, \(p^{-1}\text{ tr }({\mathbf{S}})-p^{-1}\text{ tr }(\varvec{\Sigma }){\mathop {\longrightarrow }\limits ^{P}}0\) holds from (B.6)–(B.9).

For the statement \(p^{-1}\sum _{l=1}^{q}\lambda _{l}({\mathbf{S}})-\text{ tr }(\varvec{\Sigma }_{{\mathbf{A}}}){\mathop {\longrightarrow }\limits ^{P}}0\), which evidently holds according to Lemmas A.1 and A.3. Therefore, the proof of Theorem 2.3 is completed. \(\square \)

Proof of Theorem 2.5

It is easy to get \(\hat{\xi }_{\alpha }{\mathop {\longrightarrow }\limits ^{P}}\xi _{\alpha }\). Thus the theorem follows if we can verify that \(\widehat{T}{\mathop {\longrightarrow }\limits ^{P}}\infty \). Denote \({\mathbf{Y}}_{gi}:={\mathbf{X}}_{gi}-\varvec{\mu }_{g}\), then

$$\begin{aligned} \widehat{T}&=p^{-1}\sum _{i<j}^{k}n_{ij}\Vert \overline{{\mathbf{Y}}}_{i}-\overline{{\mathbf{Y}}}_{j} \Vert ^{2}-\frac{k(k-1)}{2p}\left\{ \text{ tr }({\mathbf{S}})-\sum _{l=1}^{\hat{q}}\lambda _{l}({\mathbf{S}})\right\} \nonumber \\&\quad +2p^{-1}\sum _{i<j}^{k}n_{ij}(\varvec{\mu }_{i}-\varvec{\mu }_{j})^{^{{\mathrm{T}}}}(\overline{{\mathbf{Y}}}_{i} -\overline{{\mathbf{Y}}}_{j})+\delta \nonumber \\&=:\omega +2p^{-1}\sum _{i<j}^{k}n_{ij}Q_{ij}+\delta , \end{aligned}$$
(B.10)

where \(\omega =p^{-1}\sum _{i<j}^{k}n_{ij}\Vert \overline{{\mathbf{Y}}}_{i}-\overline{{\mathbf{Y}}}_{j}\Vert ^{2} -\frac{k(k-1)}{2p}\left\{ \text{ tr }({\mathbf{S}})-\sum _{l=1}^{\hat{q}}\lambda _{l}({\mathbf{S}})\right\} \), \(Q_{ij}=(\varvec{\mu }_{i}-\varvec{\mu }_{j})^{^{{\mathrm{T}}}}(\overline{{\mathbf{Y}}}_{i}-\overline{{\mathbf{Y}}}_{j})\) and \(\overline{{\mathbf{Y}}}_{g}=n_{g}^{-1}\sum _{i=1}^{n_{g}}{\mathbf{Y}}_{gi}\).

It follows from Theorems 2.1 and 2.3 that

$$\begin{aligned} \omega {\mathop {\longrightarrow }\limits ^{P}}\sum _{i=1}^{k-1}\sum _{j=1}^{q}\psi _{i} \lambda _{j}(\varvec{\Sigma }_{{\mathbf{A}}})z_{ij}. \end{aligned}$$
(B.11)

For the term \(Q_{ij}\), it is easy to get \(\text{ E }(Q_{ij})=0\) and \(\text{ Var }(Q_{ij})=n_{ij}^{-1}(\varvec{\mu }_{i}-\varvec{\mu }_{j})^{^{{\mathrm{T}}}}\varvec{\Sigma }(\varvec{\mu }_{i}-\varvec{\mu }_{j}){\le }n_{ij}^{-1}\lambda _{1}(\varvec{\Sigma })\Vert \varvec{\mu }_{i}-\varvec{\mu }_{j}\Vert ^{2}\). Thus, \(\text{ Var }(p^{-1}n_{ij}Q_{ij})\le {p}^{-1}\lambda _{1}(\varvec{\Sigma })\delta _{ij}=O(\delta _{ij})\), which shows that \(p^{-1}n_{ij}Q_{ij}=O_{P}(\delta _{ij}^{1/2})\). Then we have

$$\begin{aligned} p^{-1}\sum _{i<j}^{k}n_{ij}Q_{ij}=\sum _{i<j}^{k}\delta _{ij}^{1/2}O_{P}(1). \end{aligned}$$
(B.12)

Lastly, the conclusion holds immediately from (B.10)–(B.12) and \(\sqrt{\delta }\le \sum _{i<j}^{k}\delta _{ij}^{1/2}\). \(\square \)

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Cao, M., Zhao, Y., Xu, K. et al. A High-Dimensional Test for Multivariate Analysis of Variance Under a Low-Dimensional Factor Structure. Commun. Math. Stat. 10, 581–597 (2022). https://doi.org/10.1007/s40304-020-00236-1

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