Asymptotic power comparison of T²-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.

2000 Mathematics Subject Classification:

• 62H15
• 62F03
• 62F05

Keywords:

• Missing dataset
• asymptotic approximations
• power analysis

Appendix A Proofs and auxiliary results A.1. Proof of Theorem 2.1

We define the following random variables: $$\begin{array}{l}{\mathbf{z}}_1^{\left(1\right)}\sim {\mathcal{N}}_{p_1}(0,{\mathbf{I}}_{p_1}),{\mathbf{z}}_2^{\left(1\right)}\sim {\mathcal{N}}_{p_2}(0,{\mathbf{I}}_{p_2}),{\mathbf{z}}_1^{\left(2\right)}\sim {\mathcal{N}}_{p_1}(0,{\mathbf{I}}_{p_1})\\ {\mathbf{W}}^{\left(1\right)}=\left(\begin{array}{cc}{\mathbf{W}}_{11}^{\left(1\right)}& {\mathbf{W}}_{12}^{\left(1\right)}\\ {\mathbf{W}}_{21}^{\left(1\right)}& {\mathbf{W}}_{22}^{\left(1\right)}\end{array}\right)\sim \mathcal{W}(n_1,{\mathbf{I}}_p),{\mathbf{W}}_{11}^{\left(2\right)}\sim \mathcal{W}(n_2,{\mathbf{I}}_{p_1})\end{array}$$ where ${\mathbf{W}}_{11}^{(1)}$ is a $p_1\times p_1$ matrix, and ${\mathbf{W}}_{22}^{(1)}$ is a $p_2\times p_2$ matrix. Furthermore, ${\mathbf{z}}_1^{(1)},{\mathbf{z}}_2^{(1)},$ ${\mathbf{z}}_1^{(2)},{\mathbf{W}}^{(1)},{\mathbf{W}}_{11}^{(2)}$ are mutually independent. Using these random variables, we obtain the following stochastic representations: $$\begin{array}{l}\sqrt{n}(\widehat{\mathit{\mu }}-{\mathit{\mu }}_0)\stackrel{d}{=}\mathbf{A}\{\sqrt{\frac{n}{N_1}}\left(\begin{array}{cc}\frac{N_1}{N}{\mathbf{I}}_{p_1}& \mathbf{O}\\ -\frac{N_2}{N}{\mathbf{W}}_{21}^{\left(1\right)}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}& {\mathbf{I}}_{p_2}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(1\right)}+\sqrt{N_1}{\mathit{\delta }}_1\\ {\mathbf{z}}_2^{\left(1\right)}+\sqrt{N_1}{\mathit{\delta }}_{2·1}\end{array}\right)\\ +\frac{\sqrt{n{N}_2}}{N}\left(\begin{array}{cc}{\mathbf{I}}_{p_1}& \mathbf{O}\\ {\mathbf{W}}_{21}^{\left(1\right)}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}& \mathbf{O}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(2\right)}+\sqrt{N_2}{\mathit{\delta }}_1\\ 0\end{array}\right)\}\\ n\mathbf{C}\stackrel{d}{=}\frac{n}{N^2}\mathbf{A}\{{\mathbf{W}}^{\left(1\right)}+\left(\begin{array}{cc}{\tilde{\mathbf{W}}}_{11}& {\tilde{\mathbf{W}}}_{11}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}{\mathbf{W}}_{12}^{\left(1\right)}\\ {\mathbf{W}}_{21}^{\left(1\right)}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}{\tilde{\mathbf{W}}}_{11}& {\mathbf{W}}_{21}^{\left(1\right)}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}{\tilde{\mathbf{W}}}_{11}{\mathbf{W}}_{11}^{{\left(1\right)}^{-1}}{\mathbf{W}}_{12}^{\left(1\right)}\end{array}\right)\\ +\left(\begin{array}{cc}\mathbf{O}& \mathbf{O}\\ \mathbf{O}& a{\mathbf{W}}_{22·1}^{\left(1\right)}\end{array}\right)\}{\mathbf{A}}^{\top }\end{array}$$ where $$\begin{array}{l}\mathbf{A}=\left(\begin{array}{cc}{\mathbf{\Sigma }}_{11}^{1/2}& \mathbf{O}\\ {\mathbf{\Sigma }}_{21}{\mathbf{\Sigma }}_{11}^{-1/2}& {\mathbf{\Sigma }}_{22·1}^{1/2}\end{array}\right),a=\frac{N^2}{N_1^2}\left\{1+\frac{N_2}{N(N_1-p_1-2)}\right\}-1\\ {\tilde{\mathbf{W}}}_{11}={\mathbf{W}}_{11}^{\left(2\right)}+\mathbf{w}{\mathbf{w}}^{\top },{\mathbf{W}}_{22·1}^{\left(1\right)}={\mathbf{W}}_{22}^{\left(1\right)}-{\mathbf{W}}_{21}^{\left(1\right)}{\mathbf{W}}_{11}^{\left(1\right)}{}^{-1}{\mathbf{W}}_{12}^{\left(1\right)}\end{array}$$

Here, $\mathbf{w}=\sqrt{1/N}\left(\sqrt{N_2}{\mathbf{z}}_1^{(1)}-\sqrt{N_1}{\mathbf{z}}_1^{(2)}\right)$

First, we obtain the stochastic expansion of $\sqrt{n}(\widehat{\mathit{\mu }}-{\mathit{\mu }}_0).$ Define ${\mathbf{V}}^{(1)}$ and ${\mathbf{V}}_{11}^{(2)}$ respectively as $${\mathbf{V}}^{\left(1\right)}=\left(\begin{array}{cc}{\mathbf{V}}_{11}^{\left(1\right)}& {\mathbf{V}}_{12}^{\left(1\right)}\\ {\mathbf{V}}_{21}^{\left(1\right)}& {\mathbf{V}}_{22}^{\left(1\right)}\end{array}\right),{\mathbf{V}}_{11}^{\left(2\right)}=n_2^{-1/2}({\mathbf{W}}_{11}^{\left(2\right)}-n_2{\mathbf{I}}_{p_1})$$ where $${\mathbf{V}}_{11}^{\left(1\right)}=n_1^{-1/2}({\mathbf{W}}_{11}^{\left(1\right)}-n_1{\mathbf{I}}_{p_1}),{\mathbf{V}}_{12}^{\left(1\right)}={\mathbf{V}}_{21}^{{\left(1\right)}^{\top }}=n_1^{-1/2}{\mathbf{W}}_{12}^{\left(1\right)},{\mathbf{V}}_{22}^{\left(1\right)}=n_1^{-1/2}({\mathbf{W}}_{22}^{\left(1\right)}-n_1{\mathbf{I}}_{p_2})$$

Then the stochastic expansion of $\sqrt{n}(\widehat{\mathit{\mu }}-{\mathit{\mu }}_0)$ is (A.1) $$\sqrt{n}(\widehat{\mathit{\mu }}-{\mathit{\mu }}_0)\stackrel{d}{=}\mathbf{A}\left\{{\mathit{\epsilon }}_0+\frac{1}{\sqrt{n}}{\mathit{\epsilon }}_1+\frac{1}{n}{\mathit{\epsilon }}_2+o_p\left(n^{-1}\right)\right\}$$(A.1) where $$\begin{array}{l}{\mathit{\epsilon }}_0=\left(\begin{array}{cc}\sqrt{{\gamma }_1}{\mathbf{I}}_{p_1}& \mathbf{O}\\ \mathbf{O}& \frac{1}{\sqrt{{\gamma }_1}}{\mathbf{I}}_{p_2}\end{array}\right)\left\{\left(\begin{array}{c}{\mathbf{z}}_1^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_1\\ {\mathbf{z}}_2^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_{2·1}\end{array}\right)+\frac{1}{2n\sqrt{{\gamma }_1}}\left(\begin{array}{c}\sqrt{n}{\mathit{\delta }}_1\\ \sqrt{n}{\mathit{\delta }}_{2·1}\end{array}\right)\right\}\\ +\left(\begin{array}{cc}\sqrt{{\gamma }_2}{\mathbf{I}}_{p_1}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}\end{array}\right)\left\{\left(\begin{array}{c}{\mathbf{z}}_1^{\left(2\right)}+\sqrt{{\gamma }_{2}n}{\mathit{\delta }}_1\\ 0\end{array}\right)+\frac{1}{2n\sqrt{{\gamma }_2}}\left(\begin{array}{c}\sqrt{n}{\mathit{\delta }}_1\\ 0\end{array}\right)\right\}\\ {\mathit{\epsilon }}_1=\left(\begin{array}{cc}\mathbf{O}& \mathbf{O}\\ -\frac{{\gamma }_2}{{\gamma }_1}{\mathbf{V}}_{21}^{\left(1\right)}& \mathbf{O}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_1\\ {\mathbf{z}}_2^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_{2·1}\end{array}\right)+\left(\begin{array}{cc}\mathbf{O}& \mathbf{O}\\ \sqrt{\frac{{\gamma }_2}{{\gamma }_1}}{\mathbf{V}}_{21}^{\left(1\right)}& \mathbf{O}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(2\right)}+\sqrt{{\gamma }_{2}n}{\mathit{\delta }}_1\\ 0\end{array}\right)\\ {\mathit{\epsilon }}_2=\left(\begin{array}{cc}\frac{1-4{\gamma }_1}{2\sqrt{{\gamma }_1}}{\mathbf{I}}_{p_1}& \mathbf{O}\\ \frac{{\gamma }_2}{{\gamma }_1^{3/2}}{\mathbf{V}}_{21}^{\left(1\right)}{\mathbf{V}}_{11}^{\left(1\right)}& -\frac{1}{2{\gamma }_1^{3/2}}{\mathbf{I}}_{p_2}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_1\\ {\mathbf{z}}_2^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_{2·1}\end{array}\right)+\left(\begin{array}{cc}\frac{1-4{\gamma }_1}{2\sqrt{{\gamma }_1}}{I}_{p_1}& \mathbf{O}\\ -\frac{\sqrt{{\gamma }_2}}{{\gamma }_1}{\mathbf{V}}_{21}^{\left(1\right)}{\mathbf{V}}_{11}^{\left(1\right)}& \mathbf{O}\end{array}\right)\left(\begin{array}{c}{\mathbf{z}}_1^{\left(2\right)}+\sqrt{{\gamma }_{2}n}{\mathit{\delta }}_1\\ 0\end{array}\right)\end{array}$$

Next, we derive the stochastic expansion of ${(n\mathbf{C})}^{-1}.$ The result is as follows: (A.2) $${\left(n\mathbf{C}\right)}^{-1}\stackrel{d}{=}{\left({\mathbf{A}}^{\top }\right)}^{-1}\left\{{\mathbf{C}}_0-\frac{1}{\sqrt{n}}{\mathbf{C}}_1+\frac{1}{n}{\mathbf{C}}_2+o_p\left(n^{-1}\right)\right\}{\mathbf{A}}^{-1}$$(A.2) where $${\mathbf{C}}_0=\left(\begin{array}{cc}{\mathbf{I}}_{p_1}& \mathbf{O}\\ \mathbf{O}& {\gamma }_1{\mathbf{I}}_{p_2}\end{array}\right),{\mathbf{C}}_1=\sqrt{{\gamma }_1}{\mathbf{V}}^{\left(1\right)}+\sqrt{{\gamma }_2}\left(\begin{array}{cc}{\mathbf{V}}_{11}^{\left(2\right)}& \mathbf{O}\\ \mathbf{O}& \mathbf{O}\end{array}\right),{\mathbf{C}}_2=\left(\begin{array}{cc}{\mathbf{C}}_{2,11}& {\mathbf{C}}_{2,12}\\ {\mathbf{C}}_{2,21}& {\mathbf{C}}_{2,22}\end{array}\right)$$

Here, $$\begin{array}{l}{\mathbf{C}}_{2,11}={(\sqrt{{\gamma }_1}{\mathbf{V}}_{11}^{\left(1\right)}+\sqrt{{\gamma }_2}{\mathbf{V}}_{11}^{\left(2\right)})}^2+{\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{21}^{\left(1\right)}-(\sqrt{{\gamma }_2}{\mathbf{z}}_1^{\left(1\right)}-\sqrt{{\gamma }_1}{\mathbf{z}}_1^{\left(2\right)}){(\sqrt{{\gamma }_2}{\mathbf{z}}_1^{\left(1\right)}-\sqrt{{\gamma }_1}{\mathbf{z}}_1^{\left(2\right)})}^{\top }+4{\mathbf{I}}_{p_1}\\ {\mathbf{C}}_{2,12}={\mathbf{C}}_{2,21}^{\top }={\mathbf{V}}_{11}^{\left(1\right)}{\mathbf{V}}_{12}^{\left(1\right)}+{\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{22}^{\left(1\right)},{\mathbf{C}}_{2,22}={\mathbf{V}}_{21}^{\left(1\right)}{\mathbf{V}}_{12}^{\left(1\right)}+{\left({\mathbf{V}}_{22}^{\left(1\right)}\right)}^2+(2-{\gamma }_2{p}_1){\mathbf{I}}_{p_2}\end{array}$$

Finally, we derive the stochastic expansion of T². From (A.1) and (A.2), the stochastic expansion of T² is obtained as $$T^2=Q_0+\frac{1}{\sqrt{n}}{Q}_1+\frac{1}{n}{Q}_2+o_p\left(n^{-1}\right)$$ where $Q_0={\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_0{\mathit{\epsilon }}_0,Q_1=2{\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_0{\mathit{\epsilon }}_1-{\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_1{\mathit{\epsilon }}_0,Q_2=2{\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_0{\mathit{\epsilon }}_2+{\mathit{\epsilon }}_1^{\top }{\mathbf{C}}_0{\mathit{\epsilon }}_1-2{\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_1{\mathit{\epsilon }}_1+{\mathit{\epsilon }}_0^{\top }{\mathbf{C}}_2{\mathit{\epsilon }}_0$

We define $$\tilde{\mathbf{z}}={({\mathbf{z}}_1^{{\left(1\right)}^{\top }}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_1^{\top }, {\mathbf{z}}_1^{{\left(2\right)}^{\top }}+\sqrt{{\gamma }_{2}n}{\mathit{\delta }}_1^{\top })}^{\top },{\tilde{\mathbf{z}}}_2^{\left(1\right)}={\mathbf{z}}_2^{\left(1\right)}+\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_{2·1},\mathbf{B}=\mathbf{b}{\mathbf{b}}^{\top }, {\mathbf{B}}_{\perp }={\mathbf{b}}_{\perp }{\mathbf{b}}_{\perp }^{\top }$$ where $\mathbf{b}={(\sqrt{{\gamma }_1},\sqrt{{\gamma }_2})}^{\top }$ and ${\mathbf{b}}_{\perp }={(\sqrt{{\gamma }_2},-\sqrt{{\gamma }_1})}^{\top }.$ Then, we express Q₀, Q₁ and Q₂ as follows: $$\begin{array}{l}{Q}_0={\tilde{\mathbf{z}}}^{\top }(\mathbf{B}\otimes {\mathbf{I}}_{p_1})\tilde{\mathbf{z}}+{\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}{\tilde{\mathbf{z}}}_2^{\left(1\right)}\\ Q_1=-{\tilde{\mathbf{z}}}^{\top }(\sqrt{{\gamma }_1}\mathbf{B}\otimes {\mathbf{V}}_{11}^{\left(1\right)}+\sqrt{{\gamma }_2}\mathbf{B}\otimes {\mathbf{V}}_{11}^{\left(2\right)})\tilde{\mathbf{z}}-\frac{2}{\sqrt{{\gamma }_1}}{\tilde{\mathbf{z}}}^{\top }({\mathbf{e}}_1\otimes {\mathbf{V}}_{12}^{\left(1\right)}){\tilde{\mathbf{z}}}_2^{\left(1\right)}-\frac{1}{\sqrt{{\gamma }_1}}{\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}{\mathbf{V}}_{22}^{\left(1\right)}{\tilde{\mathbf{z}}}_2^{\left(1\right)}\\ Q_2={\tilde{\mathbf{z}}}^{\top }\mathbf{D}\tilde{\mathbf{z}}+2{\tilde{\mathbf{z}}}^{\top }\mathbf{E}{\tilde{\mathbf{z}}}_2^{\left(1\right)}+{\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}\mathbf{F}{\tilde{\mathbf{z}}}_2^{\left(1\right)}+\left\{2{\mathbf{b}}^{\top }\otimes {\left(\sqrt{n}{\mathit{\delta }}_1\right)}^{\top }\right\}\tilde{\mathbf{z}}+\frac{1}{\sqrt{{\gamma }_1}}{\left(\sqrt{n}{\mathit{\delta }}_{2·1}\right)}^{\top }{\tilde{\mathbf{z}}}_2^{\left(1\right)}\end{array}$$ where $$\begin{array}{l}\mathbf{D}=\mathbf{B}\otimes {\mathbf{C}}_{2,11}+\left\{\left(\begin{array}{cc}1& \sqrt{{\gamma }_1/{\gamma }_2}\\ \sqrt{{\gamma }_2/{\gamma }_1}& 1\end{array}\right)-4\mathbf{B}\right\}\otimes {\mathbf{I}}_{p_1}\\ +\left\{\frac{{\gamma }_2}{{\gamma }_1}{\mathbf{B}}_{\perp }+2\left(\begin{array}{cc}{\gamma }_2& -\sqrt{{\gamma }_1{\gamma }_2}\\ {\gamma }_2\sqrt{{\gamma }_2/{\gamma }_1}& -{\gamma }_2\end{array}\right)\right\}\otimes \left({\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{21}^{\left(1\right)}\right)\\ \mathbf{E}=\sqrt{\frac{{\gamma }_2}{{\gamma }_1}}\left(\begin{array}{c}\sqrt{{\gamma }_2/{\gamma }_1}\\ -1\end{array}\right)\otimes ({\mathbf{V}}_{11}^{\left(1\right)}{\mathbf{V}}_{12}^{\left(1\right)}+{\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{22}^{\left(1\right)})+\left(\begin{array}{c}1\\ \sqrt{{\gamma }_2/{\gamma }_1}\end{array}\right)\otimes {\mathbf{C}}_{2,12},\mathbf{F}=\frac{1}{{\gamma }_1}({\mathbf{C}}_{2,22}-{\mathbf{I}}_{p_2})\end{array}$$

The characteristic function of T² can be expressed as $${\psi }_{T^2}\left(t\right)=\mathrm{E}\left[\text{exp}\hspace{0.17em}\left(\mathit{it}{Q}_0\right)\left\{1+\frac{\mathit{it}}{\sqrt{n}}{Q}_1+\frac{\mathit{it}}{n}\left(Q_2+\frac{\mathit{it}}{2}{Q}_1^2\right)+o_p\left(n^{-1}\right)\right\}\right]$$

To evaluate the expectation (A.1), we consider the following transformation: $$\tilde{\mathbf{z}}=\left\{{(I_2-2it\mathbf{B})}^{-1/2}\otimes {\mathbf{I}}_{p_1}\right\}{\mathbf{y}}_1+\frac{1}{1-2it}\mathit{b}\otimes \sqrt{n}{\mathit{\delta }}_1,{\tilde{\mathbf{z}}}_2^{\left(1\right)}={(1-2it)}^{-1/2}{\mathbf{y}}_2+\frac{1}{1-2it}\sqrt{{\gamma }_{1}n}{\mathit{\delta }}_{2·1}$$

The Jacobians are ${(1-2it)}^{-p-1/2}$ and ${(1-2it)}^{-p-1}.$ Thus we obtain $$\text{exp}\hspace{0.17em}\left(\mathit{it}{Q}_0\right){\varphi }_{p_2}\left({\mathbf{z}}_2^{\left(1\right)}\right)\prod _{i=1}^2{\varphi }_{p_1}\left({\mathbf{z}}_1^{\left(i\right)}\right)d{\mathbf{z}}_1^{\left(1\right)}d{\mathbf{z}}_2^{\left(1\right)}d{\mathbf{z}}_1^{\left(2\right)}={\psi }_0\left(t\right){\varphi }_{p_2}\left({\mathbf{y}}_2\right){\varphi }_{2p_1}\left({\mathbf{y}}_1\right)d{\mathbf{y}}_{1}d{\mathbf{y}}_2$$ where ${\varphi }_d(·)$ denotes the probability density function of ${\mathcal{N}}_d(0,{\mathbf{I}}_d),$ and $${\psi }_0\left(t\right)={(1-2it)}^{-p/2}\hspace{0.17em}\text{exp}\hspace{0.17em}\left(\frac{\mathit{it}\Delta }{1-2it}\right)$$

Through this variable transformation, the characteristic function (A.1) is rewritten as follows: $${\psi }_{T^2}\left(t\right)={\psi }_0\left(t\right){\mathrm{E}}_{\ast }\left\{1+\frac{\mathit{it}}{\sqrt{n}}{Q}_1+\frac{\mathit{it}}{n}\left(Q_2+\frac{\mathit{it}}{2}{Q}_1^2\right)+o_p\left(n^{-1}\right)\right\}$$

This notation ${\mathrm{E}}_{\ast }$ expresses the expectation of ${\mathbf{y}}_1,{\mathbf{y}}_2,{\mathbf{V}}^{(1)},$ and ${\mathbf{V}}_{11}^{(2)}.$

We now evaluate the expectation of higher-order terms in the characteristic function (A.1). Using the moments stated in subsection A.2, the expectations of each term are obtained as follows: $$\begin{array}{l}{\mathrm{E}}_{\ast }\left(Q_1\right)=0\\ {\mathrm{E}}_{\ast }\left(Q_2\right)=\frac{{\gamma }_2{p}_1{p}_2}{{\gamma }_1}+\left(\frac{1}{1-2it}\right)\left\{p_1(p+p_2+2)+\frac{p_2(p_2+2)}{{\gamma }_1}+2{\Delta }_1+{\Delta }_{2·1}\right\}\\ +{\left(\frac{1}{1-2it}\right)}^2\left\{(p+2){\Delta }_1+({\gamma }_1{p}_1+p_2+2){\Delta }_{2·1}\right\}\\ {\mathrm{E}}_{\ast }\left(Q_1^2\right)=\frac{1}{1-2it}\frac{4{\gamma }_2{p}_1{p}_2}{{\gamma }_1}+2{\left(\frac{1}{1-2it}\right)}^2\left\{p_1(p+p_2+2)+\frac{p_2(p_2+2)}{{\gamma }_1}+2{\gamma }_2{p}_1{\Delta }_{2·1}\right\}\\ +4{\left(\frac{1}{1-2it}\right)}^3\left\{(p+2){\Delta }_1+({\gamma }_1{p}_1+p_2+2){\Delta }_{2·1}\right\}\\ +2{\left(\frac{1}{1-2it}\right)}^4({\Delta }_1^2+{\gamma }_1{\Delta }_{2·1}^2+2{\gamma }_1{\Delta }_{2·1}{\Delta }_1)\end{array}$$

Organizing by the power of ${(1-2it)}^{-1},$ we obtain (A.3) $${\psi }_{T^2}\left(t\right)={\psi }_0\left(t\right)\left\{1+\frac{1}{8n}\sum _{j=0}^1{c}_j\left(\mathit{\theta }\right){(1-2it)}^{-j}\right\}+o\left(n^{-1}\right)$$(A.3)

Performing an inverse Fourier transform on each term in (A.3), we finally obtain (i) in Theorem 2.1.

Next, we will show (ii). The stochastic expansion of Λ is obtained as $$\Lambda =Q_0+Q_1+{\tilde{Q}}_2+o_p\left(n^{-1}\right)$$ where $$\begin{array}{l}{\tilde{Q}}_2={\tilde{\mathbf{z}}}^{\top }\tilde{\mathbf{D}}\tilde{\mathbf{z}}+{\tilde{\mathbf{z}}}^{\top }\tilde{\mathbf{E}}{\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}+{\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}\tilde{\mathbf{F}}{\tilde{\mathbf{z}}}_2^{\left(1\right)}+\left\{2{\mathbf{b}}^{\top }\otimes {\left(\sqrt{n}{\mathit{\delta }}_1\right)}^{\top }\right\}\tilde{\mathbf{z}}+\frac{1}{\sqrt{{\gamma }_1}}{\left(\sqrt{n}{\mathit{\delta }}_{2·1}\right)}^{\top }{\tilde{\mathbf{z}}}_2^{\left(1\right)}\\ \hspace{1em}\hspace{1em}-\frac{1}{2}{\left\{{\tilde{\mathbf{z}}}^{\top }\right(\mathbf{B}\otimes {\mathbf{I}}_{p_1}\left)\tilde{\mathbf{z}}\right\}}^2-\frac{1}{{\gamma }_1}\left[{\tilde{\mathbf{z}}}^{\top }\right\{\left({\mathbf{e}}_1{\mathbf{e}}_1^{\top }\right)\otimes {\mathbf{I}}_{p_1}\left\}\tilde{\mathbf{z}}\right]\left({\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}{\tilde{\mathbf{z}}}_2^{\left(1\right)}\right)-\frac{1}{2{\gamma }_1}{\left({\tilde{\mathbf{z}}}_2^{{\left(1\right)}^{\top }}{\tilde{\mathbf{z}}}_2^{\left(1\right)}\right)}^2\\ \hspace{1em}\hspace{1em}-{\left[{\tilde{\mathbf{z}}}^{\top }\right\{\left(\mathbf{b}{\mathbf{b}}_{\perp }^{\top }\right)\otimes {\mathbf{I}}_{p_1}\left\}\tilde{\mathbf{z}}\right]}^2\end{array}$$

Here, $$\begin{array}{l}\tilde{\mathbf{D}}=\mathbf{B}\otimes \{{\gamma }_1{\mathbf{V}}_{11}^{{\left(1\right)}^2}+{\gamma }_2{\mathbf{V}}_{11}^{{\left(2\right)}^2}+\sqrt{{\gamma }_1{\gamma }_2}({\mathbf{V}}_{11}^{\left(1\right)}{\mathbf{V}}_{11}^{\left(2\right)}+{\mathbf{V}}_{11}^{\left(2\right)}{\mathbf{V}}_{11}^{\left(1\right)}\left)\right\}+\frac{1}{{\gamma }_1}\left\{\right({\mathbf{e}}_1{\mathbf{e}}_1^{\top })\otimes ({\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{21}^{\left(1\right)}\left)\right\}\\ \hspace{1em}\hspace{1em}+\frac{{\gamma }_1-{\gamma }_2}{2\sqrt{{\gamma }_1{\gamma }_2}}({\mathbf{b}}_{\perp }{\mathbf{b}}^{\top }+\mathbf{b}{\mathbf{b}}_{\perp }^{\top })\otimes {\mathbf{I}}_{p_1}\\ \tilde{\mathbf{E}}=\frac{2}{{\gamma }_1}\{{\mathbf{e}}_1\otimes ({\mathbf{V}}_{11}^{\left(1\right)}{\mathbf{V}}_{12}^{\left(1\right)}+{\mathbf{V}}_{12}^{\left(1\right)}{\mathbf{V}}_{22}^{\left(1\right)}\left)\right\},\tilde{\mathbf{F}}=\frac{1}{{\gamma }_1}({\mathbf{V}}_{22}^{{\left(1\right)}^2}+{\mathbf{V}}_{21}^{\left(1\right)}{\mathbf{V}}_{12}^{\left(1\right)})\end{array}$$

The expectations of ${\tilde{Q}}_2$ is obtained as follows: $$\begin{array}{l}{\mathrm{E}}_{\ast }\left({\tilde{Q}}_2\right)=\frac{{\gamma }_2{p}_1{p}_2}{{\gamma }_1}+\frac{1}{1-2it}\left\{p_1(p+1)+\frac{p_2(p+1)}{{\gamma }_1}+2{\Delta }_1+{\Delta }_{2·1}-\frac{{\gamma }_2{p}_1{p}_2}{{\gamma }_1}\right\}\\ +{\left(\frac{1}{1-2it}\right)}^2\left\{(p+1)({\Delta }_1+{\Delta }_{2·1})-\frac{p_1(p_1+2)}{2}-\frac{p_2(p_2+2)}{2{\gamma }_1}-p_1(p_2+{\gamma }_2{\Delta }_{2·1})\right\}\\ -{\left(\frac{1}{1-2it}\right)}^3\left\{\right(p_1+2){\Delta }_1+(p_2{\Delta }_1+p_1{\gamma }_1{\Delta }_{2·1})+(p_2+2\left){\Delta }_{2·1}\right\}\\ -{\left(\frac{1}{1-2it}\right)}^4\left(\frac{{\Delta }_1^2}{2}+{\gamma }_1{\Delta }_1{\Delta }_{2·1}+\frac{{\gamma }_1{\Delta }_{2·1}^2}{2}\right)\end{array}$$

Let ${\psi }_{\Lambda }(t)$ be the characteristic function of Λ. Organizing by the power of ${(1-2it)}^{-1},$ we obtain (A.4) $${\psi }_{\Lambda }\left(t\right)={\psi }_0\left(t\right)+\frac{{\psi }_0\left(t\right)}{8n}\sum _{j=0}^4\left\{c_j\right(\mathit{\theta })+d_j(\mathit{\theta }\left)\right\}{(1-2it)}^{-j}+o\left(n^{-1}\right)$$(A.4)

Performing an inverse Fourier transform on each term in (A.4), we finally obtain (ii) in Theorem 2.1.

A.2 Proof of Theorem 2.2

Let ${\psi }_{L(M)}(t)$ be the characteristic function of $L(M)=\Lambda /(1+M/n).$ Organizing by the power of ${(1-2it)}^{-1},$ we obtain $${\psi }_{L\left(M\right)}\left(t\right)={\psi }_0\left(t\right)+\frac{{\psi }_0\left(t\right)}{8n}\sum _{j=0}^1{\tilde{c}}_j{(1-2it)}^{-j}+o\left(n^{-1}\right)$$ where $${\tilde{c}}_j={(-1)}^{j}2\left(2Mp-\frac{p^2-p_1^2{\gamma }_2}{{\gamma }_1}\right)$$

Hence if $M=(p^2-p_1^2{\gamma }_2)/(2p{\gamma }_1),{\psi }_{L(M)}(t)={\psi }_0(t)+o(n^{-1}).$ Performing an inverse Fourier transform, we finally obtain Theorem 2.2.

A.3 Some moments

Assume that $\mathbf{W}\sim \mathcal{W}(n,{\mathbf{I}}_p),\mathbf{V}=n^{-1/2}(\mathbf{W}-n{\mathbf{I}}_p),$ and $\mathbf{A}$ and $\mathbf{B}$ are p × p symmetric matrices. We divide $\mathbf{V}$ into p₁ and p₂ as follows: $$\mathbf{V}=\left(\begin{array}{cc}{\mathbf{V}}_{11}& {\mathbf{V}}_{12}\\ {\mathbf{V}}_{21}& {\mathbf{V}}_{22}\end{array}\right)$$

Then it holds that $$\begin{array}{l}\left(\mathrm{i}\right)\mathrm{E}\left({\mathbf{V}}_{\mathit{ij}}\right)=\mathbf{O},\left(\text{ii}\right)\mathrm{E}\left({\mathbf{V}}_{12}{\mathbf{V}}_{21}\right)=p_2{\mathbf{I}}_{p_1}, \mathrm{E}\left({\mathbf{V}}_{21}{\mathbf{V}}_{12}\right)=p_1{\mathbf{I}}_{p_2}\\ \left(\text{iii}\right)\mathrm{E}\left\{\text{tr}\right(\mathbf{AVBV}\left)\right\}=\text{tr}\left(\mathbf{AB}\right)+\text{tr}\left(\mathbf{A}\right)\text{tr}\left(\mathbf{B}\right),\left(\text{iv}\right)\mathrm{E}\left\{\text{tr}\right(\mathbf{AV}\left)\text{tr}\right(\mathbf{BV}\left)\right\}=2\text{tr}\left(\mathbf{AB}\right)\end{array}$$

These results can be directly derived from some results of the moments of the Wishart matrix. For some results of the moment about the Wishart matrix, see Gupta and Nagar (2000).

Additional information

Funding

The first and second authors’ research was supported in part by a Grant-in-Aid for Young Scientists (B) (17K14238, 16K17642) from the Japan Society for the Promotion of Science.