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Linear hypothesis testing in high-dimensional one-way MANOVA: a new normal reference approach

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Abstract

For the general linear hypothesis testing problem for high-dimensional data, several interesting tests have been proposed in the literature. Most of them have imposed strong assumptions on the underlying covariance matrix so that their test statistics under the null hypothesis are asymptotically normally distributed. In practice, however, these strong assumptions may not be satisfied or hardly be checked so that these tests are often applied blindly in real data analysis. Their empirical sizes may then be much larger or smaller than the nominal size. For these tests, this is a size control problem which cannot be overcome via purely increasing the sample size to infinity. To overcome this difficulty, in this paper, a new normal-reference test using the centralized \(L^2\)-norm based test statistic with three cumulant matched chi-square approximation is proposed and studied. Some theoretical discussion and two simulation studies demonstrate that in terms of size control, the new normal-reference test performs very well regardless of if the high-dimensional data are nearly uncorrelated, moderately correlated, or highly correlated and it outperforms two existing competitors substantially. Two real high-dimensional data examples motivate and illustrate the new normal-reference test.

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Acknowledgements

The work was supported by the National University of Singapore academic research grant R-155-000-187-114. The authors thank the Editor, the Associate Editor and an anonymous reviewer for their insightful comments and suggestions.

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Correspondence to Tianming Zhu.

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Appendix: Technical proofs

Appendix: Technical proofs

Proof of Theorem 1

We first prove the first expression in (a). Note that as \(n\rightarrow \infty \), we have \(\text{ tr }({\hat{{{{\varvec{\varSigma }}}}}})/\text{tr }({{{\varvec{\varSigma }}}})\rightarrow 1\) in probability for all p. We can write \(T_{n,p,0}=S_{n,p,0}[1+o_p(1)]\) and \({\tilde{T}}_{n,p,0}={\tilde{S}}_{n,p,0}[1+o_p(1)]\) where

$$\begin{aligned} S_{n,p,0}=\Vert \mathbf{C}({\hat{{{\varvec{\mu }}}}}-{{\varvec{\mu }}})\Vert ^2-q\text{ tr }({{{\varvec{\varSigma }}}}), \; \text{ and } \; {\tilde{S}}_{n,p,0}=\frac{\Vert \mathbf{C}({\hat{{{\varvec{\mu }}}}}-{{\varvec{\mu }}})\Vert ^2-q\text{ tr }({{{\varvec{\varSigma }}}})}{\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^2)}}. \end{aligned}$$

Therefore, it is sufficient to show

$$\begin{aligned} {\tilde{S}}_{n,p,0}{\mathop {\longrightarrow }\limits ^{L}}\zeta , \end{aligned}$$
(A.1)

where \(\zeta \) is defined in Theorem 1(a). For this end, set \(\mathbf{w}_{i}=\sqrt{n_{i}}({\bar{{\mathbf{y}}}}_{i}-{{\varvec{\mu }}}_{i}),\ i=1,\dots ,k\) and let \(\mathbf{w}=(\mathbf{w}_{1}^\top ,\ldots ,\mathbf{w}_{k}^\top )^\top \). We have

$$\begin{aligned} \text{ E }(\mathbf{w})=\mathbf{0}_{kp},\ \ \text{ and } \ \ \text{ Cov }(\mathbf{w})=\mathbf{I}_k\otimes {{{\varvec{\varSigma }}}}. \end{aligned}$$
(A.2)

Then we can write

$$\begin{aligned} S_{n,p,0}=\mathbf{w}^\top [(\mathbf{B}^\top \mathbf{B})\otimes \mathbf{I}_{p}]\mathbf{w}-q\text{ tr }({{{\varvec{\varSigma }}}}), \end{aligned}$$

where

$$\begin{aligned} \mathbf{B}=({\tilde{\mathbf{G}}}\mathbf{D}{\tilde{\mathbf{G}}}^\top )^{-1/2}{\tilde{\mathbf{G}}}\mathbf{D}^{1/2}. \end{aligned}$$

It is easy to see that we have \(\mathbf{B}\mathbf{B}^\top =\mathbf{I}_{q}\) and \(\mathbf{B}^\top \mathbf{B}\) is idempotent with \(\text{ tr }(\mathbf{B}^\top \mathbf{B})=q\).

To prove (A.1), set \(\mathbf{w}_{n,p}=(\mathbf{B}\otimes \mathbf{I}_{p})\mathbf{w}\). We have \(\text{ E }(\mathbf{w}_{n,p})=\mathbf{0}\) and \(\text{ Cov }(\mathbf{w}_{n,p})=\mathbf{I}_q\otimes {{{\varvec{\varSigma }}}}\). Let \(\mathbf{b}_l^\top , l=1,\ldots , q\) be the rows of \(\mathbf{B}\) so that \(\mathbf{B}=[\mathbf{b}_1,\ldots ,\mathbf{b}_q]^\top \). Set \(\mathbf{w}_{n,p,l}=(\mathbf{b}_l^\top \otimes \mathbf{I}_p)\mathbf{w}, l=1,\ldots , q\). Then \(\mathbf{w}_{n,p,l}, l=1,\ldots , q\) are uncorrelated with \(\text{ E }(\mathbf{w}_{n,p,l})=0\) and \(\text{ Cov }(\mathbf{w}_{n,p,l})={{{\varvec{\varSigma }}}}\). It follows that \(S_{n,p,0}=\sum _{l=1}^q\Vert \mathbf{w}_{n,p,l}\Vert ^2-q\text{ tr }({{{\varvec{\varSigma }}}})\).

Let \(\mathbf{u}_{p,r},\ r=1,\dots ,p\) denote the eigenvectors associated with the decreasing-ordered eigenvalues \(\lambda _{p,r},\ r=1,\dots ,p\) of \({{{\varvec{\varSigma }}}}\). We have \(\mathbf{w}_{n,p,l}=\sum _{r=1}^{p}\xi _{l,r}^{(n,p)}\mathbf{u}_{p,r}\) where \(\xi _{l,r}^{(n,p)}=\mathbf{w}_{n,p,l}^\top \mathbf{u}_{p,r}.\) It is known that \(\xi _{l,r}^{(n,p)},\ r=1,\ldots ,p\) are uncorrelated and \(\text{ E }(\xi _{l,r}^{(n,p)})=0\) and \(\text{ Var }(\xi _{l,r}^{(n,p)})=\lambda _{p,r},\ r=1,\ldots ,p\). Set \(\mathbf{h}_l=(\mathbf{b}_l\otimes \mathbf{I}_{p})\mathbf{u}_{p,r}=(\mathbf{h}_{l,1}^\top ,\ldots ,\mathbf{h}_{l,k}^\top )^\top \) where \(\mathbf{h}_{l,i},\ i=1,\dots ,k\) are \(p\times 1\) vectors. Then we have \(\xi _{l,r}^{(n,p)}=\mathbf{w}^\top (\mathbf{b}_l\otimes \mathbf{I}_{p})\mathbf{u}_{p,r}=\sum _{i=1}^{k}\mathbf{h}_{l,i}^\top \mathbf{w}_{i}\). It follows that

$$\begin{aligned} (\xi _{l,r}^{(n,p)})^{2}=\left( \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i}\right) ^{2} =\sum _{i=1}^{k}(\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})^{2}+2\sum _{1\le i<j\le k}(\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})(\mathbf{h}_{l,j}^{\top }\mathbf{w}_{j}). \end{aligned}$$

Notice that \(\text{ E }(\mathbf{w}_{i})=\mathbf{0}\), \(\text{ Cov }(\mathbf{w}_{i})={{{\varvec{\varSigma }}}},\;i=1,\dots ,k\) and \(\mathbf{w}_{i},\ i=1,\dots ,k\) are independent, we have

$$\begin{aligned} \text{ Var }\left[ (\xi _{l,r}^{(n,p)})^{2}\right] =\sum _{i=1}^{k}\text{ Var }\left[ (\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})^{2}\right] +4\sum _{1\le i<j\le k}\text{ Var }\left[ (\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})(\mathbf{h}_{l,j}^{\top }\mathbf{w}_{j})\right] . \end{aligned}$$

By some algebra, we have

$$\begin{aligned} \text{ Var }\left[ (\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})^{2}\right]= & {} 2(\mathbf{h}_{l,i}^{\top } {{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2}\\&+\frac{\text{ E }\left\{ [\mathbf{h}_{l,i}^{\top }({\mathbf{y}}_{i1} -{{\varvec{\mu }}}_i)]^{4}\right\} -3(\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2}}{n_{i}},\;i=1,\ldots ,k, \end{aligned}$$

and

$$\begin{aligned} \text{ Var }\left[ (\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i})(\mathbf{h}_{l,j}^{\top }\mathbf{w}_{j})\right] =\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\mathbf{h}_{l,j}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,j},\ 1\le i<j\le k. \end{aligned}$$

In addition, under Conditions C1 and C2, we have

$$\begin{aligned} \text{ E }\left\{ [\mathbf{h}_{l,i}^{\top }({\mathbf{y}}_{i1}-{{\varvec{\mu }}}_i)]^{4}\right\} \le (3+\varDelta )(\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2},\ i=1,\ldots ,k. \end{aligned}$$

It follows that we have

$$\begin{aligned} \text{ Var }\left[ (\xi _{l,r}^{(n,p)})^{2}\right]= & {} \sum _{i=1}^{k}\text{ Var }\left[ (\mathbf{h}_{l,i}^{\top } \mathbf{w}_{i})^{2}\right] +4\sum _{1\le i<j\le k}\text{ Var }\left[ (\mathbf{h}_{l,i}^{\top }\mathbf{w}_{i}) (\mathbf{h}_{l,j}^{\top }\mathbf{w}_{j})\right] \nonumber \\\le & {} \sum _{i=1}^{k}\left[ 2+\frac{\varDelta }{n_{i}}\right] (\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2}+4\sum _{1\le i<j\le k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\mathbf{h}_{l,j}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,j}\nonumber \\= & {} 2\sum _{i=1}^{k}\sum _{j=1}^{k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\mathbf{h}_{l,j}^{\top } {{{\varvec{\varSigma }}}}\mathbf{h}_{l,j}+\frac{\varDelta }{n_{i}}\sum _{i=1}^{k}(\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2}\nonumber \\\le & {} 2\left( \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\right) ^{2} +\frac{\varDelta }{n_{\min }}\sum _{i=1}^{k}(\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i})^{2}\nonumber \\\le & {} 2\left( \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\right) ^{2} +\frac{\varDelta }{n_{\min }}\left( \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\right) ^{2}\nonumber \\= & {} (2+\varDelta /n_{\min })\left( \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top } {{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}\right) ^{2}\nonumber \\= & {} (2+\varDelta /n_{\min })\lambda _{p,r}^{2},\;r=1,\dots ,k, \end{aligned}$$
(A.3)

where \(n_{\min }=\min _{i=1}^{n}n_{i}\) and we use the fact that

$$\begin{aligned} \sum _{i=1}^{k}\mathbf{h}_{l,i}^{\top }{{{\varvec{\varSigma }}}}\mathbf{h}_{l,i}=\mathbf{h}_l^{\top }(\mathbf{I}_k\otimes {{{\varvec{\varSigma }}}}) \mathbf{h}_l=\mathbf{u}_{p,r}^{\top }(\mathbf{b}_l^\top \otimes \mathbf{I}_{p})(\mathbf{I}_k\otimes {{{\varvec{\varSigma }}}})(\mathbf{b}_l\otimes \mathbf{I}_{p})\mathbf{u}_{p,r} =\lambda _{p,r}. \end{aligned}$$

Write \({\tilde{S}}_{n,p,0}=\sum _{r=1}^p \left[ \sum _{l=1}^{q}(\xi _{l,r}^{(n,p)})^{2}-q\lambda _{p,r}\right] /\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^{2})}\) and set \({\tilde{S}}_{n,p,0}^{(m)}=\sum _{r=1}^{m}[\sum _{l=1}^q (\xi _{l,r}^{(n,p)})^{2} -q\lambda _{p,r}]/\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^{2})}\) for some \(m<p\). We have \(|\psi _{{\tilde{S}}_{n,p,0}}(t)-\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)|\le |t|\big [\text{ E }({\tilde{S}}_{n,p,0}-{\tilde{S}}_{n,p,0}^{(m)})^{2}\big ]^{1/2}\), where \(\psi _{{\tilde{S}}_{n,p,0}}(t)\) and \(\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)\) are the characteristic functions of \({\tilde{S}}_{n,p,0}\) and \({\tilde{S}}_{n,p,0}^{(m)}\), respectively. Note that

$$\begin{aligned} \begin{aligned} \text{ E }\left( {\tilde{S}}_{n,p,0}-{\tilde{S}}_{n,p,0}^{(m)}\right) ^{2}&= \text{ E }\left\{ \sum _{r=m+1}^{p}\left[ \sum _{l=1}^q (\xi _{l,r}^{(n,p)} )^{2}-q\lambda _{p,r}\right] /\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^{2})}\right\} ^{2}\\&= \text{ Var }\left\{ \sum _{r=m+1}^{p}\sum _{l=1}^q (\xi _{l,r}^{(n,p)})^{2} \right\} /\left[ 2q\text{ tr }({{{\varvec{\varSigma }}}}^{2})\right] \\&\le \left\{ \sum _{r=m+1}^{p}\sum _{l=1}^q \sqrt{\text{ Var }\left[ (\xi _{l,r}^{(n,p)})^{2} \right] }\right\} ^{2}/\left[ 2q\text{ tr }({{{\varvec{\varSigma }}}}^2)\right] . \end{aligned} \end{aligned}$$

By (A.3), we have

$$\begin{aligned} \left\{ \sum _{r=m+1}^{p}\sum _{l=1}^q \sqrt{\text{ Var }\left[ (\xi _{l,r}^{(n,p)})^{2}\right] }\right\} ^{2}\le q^2\left( 2+\frac{\varDelta }{n_{\min }}\right) \left( \sum _{r=m+1}^{p}\lambda _{p,r}\right) ^{2}. \end{aligned}$$

It follows that

$$\begin{aligned} \begin{aligned} |\psi _{{\tilde{S}}_{n,p,0}}(t)-\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)|&\le |t|\left[ q\left( 1+\frac{\varDelta }{2n_{\min }}\right) \right] ^{1/2} \left( \sum _{r=m+1}^{p}\lambda _{p,r}\right) /\left[ \text{ tr }({{{\varvec{\varSigma }}}}^{2})\right] ^{1/2}\\&= |t|\left[ q\left( 1+\frac{\varDelta }{2n_{\min }}\right) \right] ^{1/2}\sum _{r=m+1}^{p}\rho _{p,r}. \end{aligned} \end{aligned}$$

Let t be fixed. By Condition C4, for any fixed q, as \(p\rightarrow \infty \), we have \(\sum _{r=1}^{\infty } \rho _{r}<\infty \) and

$$\begin{aligned} \sum _{r=m+1}^{p}\rho _{p,r}=\sum _{r=1}^{p}\rho _{p,r}-\sum _{r=1}^{m}\rho _{p,r}\rightarrow \sum _{r=1}^{\infty }\rho _{r}-\sum _{r=1}^{m}\rho _{r}\rightarrow \sum _{r=m+1}^{\infty }\rho _{r}. \end{aligned}$$

By letting \(m\rightarrow \infty \), we further have \( \sum _{r=m+1}^{\infty }\rho _{r}\rightarrow 0.\) Thus, for any given \(\epsilon >0\), there exist \(P_1\), \(M_1\) and \(N_{1}\), depending on t and \(\epsilon \), such that for any \(p\ge P_1\), \(m\ge M_1\) and \(n\ge N_{1}\), we have

$$\begin{aligned} |\psi _{{\tilde{S}}_{n,p,0}}(t)-\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)|\le \epsilon . \end{aligned}$$
(A.4)

For any fixed \(p\ge P_1, m\ge M_1\), by the central limit theorem, it is easy to show that as \(n\rightarrow \infty \), we have \({\tilde{S}}_{n,p,0}^{(m)}{\mathop {\longrightarrow }\limits ^{L}}{\tilde{S}}_{p,0}^{(m)}\) where \({\tilde{S}}_{p,0}^{(m)}= \sum _{l=1}^q\sum _{r=1}^{m} \rho _{p,r} (A_r-q)/\sqrt{2q}\) since as \(n\rightarrow \infty \), \(\xi _{l,r}^{(n,p)}{\mathop {\longrightarrow }\limits ^{L}}N(0,\lambda _{p,r})\) and \(\xi _{l,r}^{(n,p)}\)’s are asymptotically independent for \(r=1,\ldots , m;\; l=1,\ldots , q\). That is, under Condition C3, there exists \(N_{2}\), depending on pmt and \(\epsilon \), such that for any \(n\ge N_{2}\) we have

$$\begin{aligned} \big |\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)-\psi _{{\tilde{S}}_{p,0}^{(m)}}(t)\big |\le \epsilon . \end{aligned}$$
(A.5)

Recall that \(\zeta =\sum _{r=1}^{\infty }\rho _{r}(A_{r}-q)/\sqrt{2q}\). Set \(\zeta ^{(m)}=\sum _{r=1}^{m}\rho _{r}(A_{r}-q)/\sqrt{2q}\). Then, under Condition C4, for any fixed m, as \(p \rightarrow \infty \), we have \({\tilde{S}}_{p,0}^{(m)}{\mathop {\longrightarrow }\limits ^{L}}\zeta ^{(m)}\). That is, there exists a \(P_2\), depending on mt and \(\epsilon \), such that for any \(p\ge P_2\) we have

$$\begin{aligned} \big |\psi _{{\tilde{S}}_{p,0}^{(m)}}(t)-\psi _{\zeta ^{(m)}}(t)\big |\le \epsilon . \end{aligned}$$
(A.6)

Furthermore, we have

$$\begin{aligned} \big |\psi _{\zeta ^{(m)}}(t)-\psi _{\zeta }(t)\big |\le & {} |t| \bigg \{\text{ E }\bigg [\sum _{r=m+1}^{\infty }\rho _{r}(A_{r}-q)/\sqrt{2q}\bigg ]^2\bigg \}^{1/2}\\\le & {} |t| \bigg \{\text{ Var }\bigg [\sum _{r=m+1}^{\infty }\rho _{r} (A_{r}-q)/\sqrt{2q}\bigg ] \bigg \}^{1/2}\\= & {} |t|\bigg (\sum _{r=m+1}^{\infty }\rho _{r}^{2} \bigg )^{1/2} \le |t| \bigg (\sum _{r=m+1}^{\infty } \rho _{r}\bigg ), \end{aligned}$$

which, under Condition C4, tends to 0 as \(m\rightarrow \infty \). Thus, there exists \(M_2\), depending on t and \(\epsilon \), such that for any \(m\ge M_2\) we have

$$\begin{aligned} \big |\psi _{\zeta ^{(m)}}(t)-\psi _{\zeta }(t)\big |\le \epsilon . \end{aligned}$$
(A.7)

It follows from (A.4)–(A.7) that for any \(n\ge \max (N_{1},N_{2})\), \(p\ge \max (P_1,P_2)\) and \(m\ge \max (M_1,M_2)\) we have

$$\begin{aligned} \big |\psi _{{\tilde{S}}_{n,p,0}}(t)-\psi _{\zeta }(t)\big |\le & {} \big |\psi _{{\tilde{S}}_{n,p,0}}(t)-\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)\big | + \big |\psi _{{\tilde{S}}_{n,p,0}^{(m)}}(t)-\psi _{{\tilde{S}}_{p,0}^{(m)}}(t)\big |\\&+\big |\psi _{{\tilde{S}}_{p,0}^{(m)}}(t)-\psi _{\zeta ^{(m)}}(t)\big | +\big |\psi _{\zeta ^{(m)}}(t)-\psi _{\zeta }(t)\big | \le 4\epsilon . \end{aligned}$$

The convergence in distribution of \({\tilde{S}}_{n,p,0}\) to \(\zeta \) given in (A.1) follows as we can let \(\epsilon \rightarrow 0\). The first expression of Theorem 1(a) is then proved.

Notice that when the k samples (1) are normally distributed, Conditions C1 and C2 are automatically satisfied so that under Conditions C3 and C4, the second expression of (15) follows immediately since under the Gaussian assumption, we have \(T_{n,p,0}{\mathop {=}\limits ^{d}}T_{n,p,0}^{*}\).

We now prove (b). Under Conditions C1, C2, C3, C5, and C6, the first expression of (16) can be proved using the central limit theorem of martingale difference as in Bai and Saranadasa (1996), and the second expression of (16) can be directly proved by the Lindeberg–Feller central limit theorem.

Finally, (17) can also be shown by Pólya’s theorem (see, e.g., Lemma 2.11 of Vaart 1998) using the same proof for Eq. (7) in Zhang et al. (2020a). \(\square \)

Proof of Theorem 2

Under the local alternative (23), we have \(T_{n,p}=\left[ T_{n,p,0}+\text{ tr }(\mathbf{M}^\top \mathbf{H}\mathbf{M})\right] [1+o_p(1)]\). In addition, under the given conditions, we have \({\hat{\beta }}_0/\beta _0{\mathop {\longrightarrow }\limits ^{P}}1, {\hat{\beta }}_1/\beta _1{\mathop {\longrightarrow }\limits ^{P}}1\) and \({\hat{d}}/d{\mathop {\longrightarrow }\limits ^{P}}1\) as \(n, p\rightarrow \infty \). We first prove (a). Under Conditions C1, C2, C3, and C4, Theorem 1(a) indicates that as \(n,p\rightarrow \infty \), we have \({\tilde{T}}_{n,p,0}=T_{n,p,0}/\sqrt{\frac{2q(v_k+q)}{v_k}\text{ tr }({{{\varvec{\varSigma }}}}^2)}{\mathop {\longrightarrow }\limits ^{L}}\zeta \) where \(\zeta \) is defined in Theorem 1(a). It follows that as \(n, p\rightarrow \infty \), we have

$$\begin{aligned}&\Pr \left[ T_{n,p}\ge {\hat{\beta }}_0+{\hat{\beta }}_1\chi _{{\hat{d}}}^2(\alpha )\right] \nonumber \\&\quad = \Pr \left[ {\tilde{T}}_{n,p,0} \ge \frac{\beta _0+ \beta _1 \chi ^2_{d}(\alpha )}{\sqrt{\frac{2q(v_k+q)}{v_k}\text{ tr }({{{\varvec{\varSigma }}}}^2)}}-\frac{\text{ tr }(\mathbf{M}^\top \mathbf{H}\mathbf{M})}{\sqrt{\frac{2q(v_k+q)}{v_k}\text{ tr }({{{\varvec{\varSigma }}}}^2)}} \right] [1+o(1)] \nonumber \\&\quad = \Pr \left[ \zeta \ge \frac{\chi _{d}^2(\alpha )-d}{\sqrt{2d}} -\frac{n\text{ tr }(\mathbf{M}^\top \mathbf{H}^*\mathbf{M})}{\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^2)}} \right] [1+o(1)], \end{aligned}$$
(A.8)

where \(\mathbf{H}^*\) is defined in (24).

We now prove (b). Under the given conditions, Theorem 1(b) indicates that as \(n\rightarrow \infty \), we have \({\tilde{T}}_{n,p,0} {\mathop {\longrightarrow }\limits ^{L}}{\mathcal {N}}(0,1)\) and \({\tilde{T}}_{n,p,0}^* {\mathop {\longrightarrow }\limits ^{L}}{\mathcal {N}}(0,1)\). Therefore, the skewness of \(T_{n,p,0}^*\) will also tend to 0. This, together with (19), shows that as \(n,p\rightarrow \infty \), we have \(d\rightarrow \infty \) and \((\chi _d^2(\alpha )-d)/\sqrt{2d}\rightarrow z_{\alpha }\) where \(z_{\alpha }\) denotes the upper \(100\alpha \)-percentile of \({\mathcal {N}}(0,1)\). Then by (A.8), as \(n,p\rightarrow \infty \), we have

$$\begin{aligned} \begin{aligned}&\Pr \left[ T_{n,p}\ge {\hat{\beta }}_0+{\hat{\beta }}_1\chi _{{\hat{d}}}^2(\alpha )\right] \\&\quad = \Pr \left[ {\tilde{T}}_{n,p,0} \ge \frac{\beta _0+ \beta _1 \chi ^2_{d} (\alpha )}{\sqrt{\frac{2q(v_k+q)}{v_k}\text{ tr }({{{\varvec{\varSigma }}}}^2)}}-\frac{\text{ tr }(\mathbf{M}^\top \mathbf{H}\mathbf{M})}{\sqrt{\frac{2q(v_k+q)}{v_k}\text{ tr }({{{\varvec{\varSigma }}}}^2)}} \right] [1+o(1)] \\&\quad = \varPhi \left[ -z_{\alpha }+\frac{n\text{ tr }(\mathbf{M}^\top \mathbf{H}^*\mathbf{M})}{\sqrt{2q\text{ tr }({{{\varvec{\varSigma }}}}^2)}} \right] [1+o(1)], \end{aligned} \end{aligned}$$

where \(\varPhi (\cdot )\) denotes the cumulative distribution of \({\mathcal {N}}(0,1)\). The proof is complete. \(\square \)

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Zhu, T., Zhang, JT. Linear hypothesis testing in high-dimensional one-way MANOVA: a new normal reference approach. Comput Stat 37, 1–27 (2022). https://doi.org/10.1007/s00180-021-01110-6

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