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Score Tests for Intercept and Slope Parameters of Doubly Multivariate Linear Models with Skew-Normal Errors

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Abstract

We generalize the theory of linear models for doubly multivariate data from matrix-variate normally distributed errors to matrix-variate skew normally distributed errors. In addition, we assume that the covariance matrix \(\varvec{\varOmega }\) defining the location-scale matrix-variate skew normal distribution has block compound symmetry structure. We derive the maximum likelihood estimators of the model’s parameters; the Fisher information matrix for the direct, working, and centered parametrizations; Rao’s score tests and likelihood ratio tests for model building tests of hypotheses; and a hypothesis test for the centered intercept. A profiling argument is used to reduce the dimensionality of the optimization method used to obtain the maximum likelihood estimators and a comprehensive discussion of initial values is provided. Finally, we provide a real-world example to illuminate these derivations and apply a goodness-of-fit test to validate the distributional assumption.

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Acknowledgements

The authors want to thank an anonymous reviewer for the careful reading and valuable suggestions that led to a quite improved version of the manuscript.

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Authors and Affiliations

  1. Department of Management Science and Statistics, The University of Texas at San Antonio, One UTSA Circle, San Antonio, TX, 78249, USA

    Timothy Opheim & Anuradha Roy

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  1. Timothy Opheim
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  2. Anuradha Roy
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Correspondence to Anuradha Roy.

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This article is part of the topical collection “Celebrating the Centenary of Professor C. R. Rao” guest edited by, Ravi Khattree, Sreenivasa Rao Jammalamadaka , and M. B. Rao.

Appendices

A Appendix: Definitions and Results of Special Matrices

Many references exist for the following definitions and propositions. We direct the reader to but one, Magnus and Neudecker [19]. Let \({\varvec{N}}_m\) be the symmetric idempotent \(m^2 \times m^2\) matrix defined as \({\varvec{N}}_m=\frac{1}{2}({\varvec{I}}_{m^2}+ {\varvec{K}}_{m,m})\), where \({\varvec{I}}_{m^2}\) and \({\varvec{K}}_{m,m}\) represent the identity matrix and the commutation matrix, respectively. A unique \(m^2\times m(m+1)/2-\)dimensional matrix \({\varvec{D}}_m\) is called a duplication matrix if

$$\begin{aligned} {\varvec{D}}_m\mathrm{vech}{{\varvec{A}}}=\mathrm{vec}{{\varvec{A}}}. \end{aligned}$$

Using the above definitions, we have the following proposition.

Proposition 1

The following equalities hold:

  1. (i)

    \({\varvec{K}}_{p,m}({\varvec{A}} \otimes {\varvec{y}}) = {\varvec{y}} \otimes {\varvec{A}}\), for any \(m \times n\) matrix \({\varvec{A}}\) and \(p \times 1\) vector \({\varvec{y}}\);

  2. (ii)

    \(\mathrm{vec}({\varvec{A}}\otimes {\varvec{B}})=({\varvec{I}}_l\otimes {\varvec{K}}_{n,k}\otimes {\varvec{I}}_m)(\mathrm{vec}{\varvec{A}}\otimes \mathrm{vec}{\varvec{B}})\) for any \(k\times l\) matrix \({\varvec{A}}\) and \(m\times n\) matrix \({\varvec{B}}\);

  3. (iii)

    \(\mathrm{vec}\left( {\varvec{A}}{\varvec{B}}{\varvec{C}}\right) = \left( {\varvec{C}}^{\prime } \otimes {\varvec{A}}\right) \mathrm{vec}\left( {\varvec{B}}\right) \);

  4. (iv)

    \({\varvec{N}}_m({\varvec{A}}\otimes {\varvec{A}}){\varvec{N}}_m={\varvec{N}}_m({\varvec{A}}\otimes {\varvec{A}})= ({\varvec{A}}\otimes {\varvec{A}}){\varvec{N}}_m\) for any \(m\times m\) symmetric matrix \({\varvec{A}}\);

  5. (v)

    \({\varvec{D}}_m{\varvec{D}}^+_m={\varvec{N}}_m\);

  6. (vi)

    \({\varvec{N}}_m \mathrm{vec}{\varvec{A}} = \mathrm{vec}{\varvec{A}}\) for any \(m\times m\) symmetric matrix \({\varvec{A}}\).

Lemma 1

The following lemma follows from Proposition 1(i):

  1. (i)

    \(({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\mathrm{vec}({\varvec{P}}_n) \otimes {\varvec{I}}_p^2)= \frac{{\varvec{1}}_n}{n} \otimes {\varvec{I}}_p \otimes {\varvec{1}}_n \otimes {\varvec{I}}_p\);

  2. (ii)

    \(({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\mathrm{vec}({\varvec{I}}_n) \otimes {\varvec{I}}_p^2)= \sum _{i=1}^n {\varvec{e}}_i \otimes {\varvec{I}}_p \otimes {\varvec{e}}_i \otimes {\varvec{I}}_p\);

Proof

Starting from the LHS of equation (i), we have

$$\begin{aligned} ({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\mathrm{vec}({\varvec{P}}_n) \otimes {\varvec{I}}_p^2) &= ({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\frac{{\varvec{1}}_n}{n} \otimes {\varvec{1}}_n \otimes {\varvec{I}}_p \otimes {\varvec{I}}_p) \\ &= \frac{{\varvec{1}}_n}{n} \otimes {\varvec{K}}_{p,n}\left( {\varvec{1}}_n\otimes {\varvec{I}}_p\right) \otimes {\varvec{I}}_p \\ &= \frac{{\varvec{1}}_n}{n} \otimes {\varvec{I}}_p \otimes {\varvec{1}}_n \otimes {\varvec{I}}_p. \end{aligned}$$

Likewise, starting from the LHS of equation (ii), we have

$$\begin{aligned} ({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\mathrm{vec}({\varvec{I}}_n) \otimes {\varvec{I}}_p^2) = &\,({\varvec{I}}_n\otimes {\varvec{K}}_{p,n}\otimes {\varvec{I}}_p)(\sum _{i=1}^n {\varvec{e}}_i \otimes {\varvec{e}}_i \otimes {\varvec{I}}_p \otimes {\varvec{I}}_p) \\ \!= & {} \! \sum _{i=1}^n {\varvec{e}}_i \otimes {\varvec{K}}_{p,n} \left( {\varvec{e}}_i \otimes {\varvec{I}}_p \right) \otimes {\varvec{I}}_p \\ \!= & {} \! \sum _{i=1}^n {\varvec{e}}_i \otimes {\varvec{I}}_p \otimes {\varvec{e}}_i \otimes {\varvec{I}}_p. \end{aligned}$$

\(\square \)

B Appendix: Proofs of BCS Related Results

1.1 B.1 Intermediary Expectations

Using the results and notation in Sect. 3 with \(\varvec{\eta }=\varvec{\eta }^{\star },\) note that

$$\begin{aligned} {\varvec{V}} &= {\varvec{P}}_n \otimes \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1}+ n \varvec{\eta }\varvec{\eta }^{\prime }\right) ^{-1} + {\varvec{Q}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{Z}}_2}} \\ {\varvec{W}} &= {\varvec{P}}_n \otimes \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1}+ 2n \varvec{\eta }\varvec{\eta }^{\prime }\right) ^{-1} + {\varvec{Q}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{Z}}_2}} \\ w & \sim N(0, n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }/\left( 1+2n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\varvec{\eta }\right) ) \\ c_k &= (1 + kn\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta })^{-1/2}2(2\pi )^{-k/2} \\ {\varvec{V}}\varvec{\eta }^{\star } &= {\varvec{1}}_n \otimes \frac{\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}{1+n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }} \quad \text{ and } \\ {\varvec{W}}\varvec{\eta }^{\star } &= {\varvec{1}}_n \otimes \frac{\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}{1+2n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}. \end{aligned}$$

Since \({\varvec{z}}_1, \ldots , {\varvec{z}}_N\) are independent, the proof of the expectations in Sect. 5.1 is

$$\begin{aligned} \mathrm{E}\left[ {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star })\right]&= {\varvec{I}}_N \otimes \mathrm{E}\left[ g_2({\varvec{z}}_1^{\prime }\varvec{\eta }^{\star })\right] = - c_2 w_0 {\varvec{I}}_N \\ \mathrm{E}\left[ {\varvec{Z}}\right]&= {\varvec{1}}_N^{\prime } \otimes \sqrt{\frac{2}{\pi }} \frac{\varvec{\varOmega } \varvec{\eta }^{\star }}{\sqrt{1 + \left( \varvec{\eta }^{\star }\right) ^{\prime } \varvec{\varOmega } \varvec{\eta }^{\star }}} = {\varvec{1}}_N^{\prime } \otimes c_1 {\varvec{1}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \\ \mathrm{E}\left[ g_1^{\prime }({\varvec{Z}}^{\prime }\varvec{\eta }^{\star })\right]&= {\varvec{1}}_N^{\prime } \otimes \mathrm{E}\left[ g_1({\varvec{z}}_1^{\prime }\varvec{\eta }^{\star })\right] = c_1 {\varvec{1}}_N^{\prime } \\ \mathrm{E}\left[ {\varvec{Z}}\left( {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star }) \otimes \left( \varvec{\eta }^{\star }\right) ^{\prime }\right) \right]&= {\varvec{1}}_N^{\prime } \otimes \mathrm{E}\left[ {\varvec{z}}_1 g_2({\varvec{z}}_1^{\prime }\varvec{\eta }^{\star })\right] \left( \varvec{\eta }^{\star }\right) ^{\prime } \\&= -{\varvec{1}}_N^{\prime } \otimes \left( \frac{c_2w_1}{\left( \varvec{\eta }^{\star }\right) ^{\prime }{\varvec{W}}\varvec{\eta }^{\star }} {\varvec{W}}\varvec{\eta }^{\star } + c_1 {\varvec{V}}\varvec{\eta }^{\star } \right) \left( \varvec{\eta }^{\star }\right) ^{\prime } \\&= -\left( \frac{c_2 w_1}{\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }} + \frac{n c_1}{1+n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}\right) {\varvec{1}}_N^{\prime } \otimes {\varvec{P}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }\varvec{\eta }^{\prime } \\ \mathrm{E}\left[ {\varvec{Z}}{\varvec{Z}}^{\prime }\right]&= N \mathrm{E}\left[ {\varvec{z}}_1{\varvec{z}}_1^{\prime }\right] = N \varvec{\varOmega } \quad \text{ and } \\ \mathrm{E}\left[ {\varvec{Z}}\left( {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star })\right) {\varvec{Z}}^{\prime }\right]&= N \mathrm{E}\left[ {\varvec{z}}_1 {\varvec{z}}_1^{\prime } g_2 \left( {\varvec{z}}_1^{\prime } \varvec{\eta }^{\star }\right) \right] \\&= N c_2\left[ w_0 \left( {\varvec{W}}-\frac{{\varvec{W}}\varvec{\eta }^{\star }\left( \varvec{\eta }^{\star }\right) ^{\prime }{\varvec{W}}}{\left( \varvec{\eta }^{\star }\right) ^{\prime }{\varvec{W}}\varvec{\eta }^{\star }}\right) + w_2 \frac{{\varvec{W}}\varvec{\eta }^{\star }\left( \varvec{\eta }^{\star }\right) ^{\prime }{\varvec{W}}}{\left( \left( \varvec{\eta }^{\star }\right) ^{\prime } {\varvec{W}}\varvec{\eta }^{\star }\right) ^2}\right] \\&= N c_2 \left[ {\varvec{P}}_n \otimes \left[ w_0 \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}} - \frac{2n \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }\varvec{\eta }^{\prime } \varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{1+2n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}\right. \right. \right. \\&\quad\left. -\,\frac{\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{(\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }) (1+2n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta })} \right) \\&\quad\left. \left. +\, \frac{w_2 \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{n(\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta })^2}\right] + {\varvec{Q}}_n \otimes w_0 \varvec{\varSigma }_{\small {{\varvec{Z}}_2}} \right] . \end{aligned}$$

1.2 B.2 Fisher Information

Using the expectations developed in Appendix B.1 and Lemma 1(i) and (ii) in Appendix A, the blocks of the information matrix reported in Sect. 5.1 are proven as

$$\begin{aligned} I_{11}^{SP}&= - \mathrm{E}\left[ \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left[ -{\varvec{I}}_N \otimes \varvec{\varOmega }^{-1} + {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star }) \otimes \varvec{\eta }^{\star }\left( \varvec{\eta }^{\star }\right) ^{\prime } \right] \left( {\varvec{X}} \otimes {\varvec{I}}_p\right) \right] \\ &= \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left[ {\varvec{I}}_N \otimes \left( \varvec{\varOmega }^{-1} + c_2w_0 \varvec{\eta }^{\star }\left( \varvec{\eta }^{\star }\right) ^{\prime }\right) \right] \left( {\varvec{X}} \otimes {\varvec{I}}_{p}\right) \\ I_{12}^{SP}&^= -\mathrm{E}\left[ -\left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left( {\varvec{Z}}^{\prime } \varvec{\varOmega }^{-1} \otimes \varvec{\varOmega }^{-1}\right) \left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{pn} \otimes {\varvec{I}}_{p}\right] \left[ \mathrm{vec}({\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] {\varvec{D}}_p \right] \\ &= \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left( \left( {\varvec{1}}_N \otimes c_1 {\varvec{1}}_n^{\prime } \otimes \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\right) \varvec{\varOmega }^{-1} \otimes \varvec{\varOmega }^{-1}\right) \left[ \frac{1}{n} {\varvec{1}}_n \otimes \left( {\varvec{I}}_p \otimes {\varvec{1}}_n\right) \otimes {\varvec{I}}_p\right] {\varvec{D}}_p \\ &=c_1 \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left[ {\varvec{1}}_N \otimes \varvec{\eta }^{\prime } \otimes {\varvec{1}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right] {\varvec{D}}_p \\ I_{13}^{SP} &= -\mathrm{E}\left[ -\left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left( {\varvec{Z}}^{\prime } \varvec{\varOmega }^{-1} \otimes \varvec{\varOmega }^{-1}\right) \left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{pn} \otimes {\varvec{I}}_{p}\right] \right. \\ &\quad\left. \left[ \left( \mathrm{vec}({\varvec{I}}_n)-\mathrm{vec}({\varvec{P}}_n)\right) \otimes {\varvec{I}}_{p^2}\right] {\varvec{D}}_p \right] \\ &=c_1 \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \sum _{i=1}^n \left[ {\varvec{1}}_N \otimes \varvec{\eta }^{\prime } \otimes \left( \frac{1}{n} {\varvec{1}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} + \left( {\varvec{e}}_i - \frac{1}{n} {\varvec{1}}_n\right) \otimes \varvec{\varSigma }_{\small {{\varvec{Z}}_2}}^{-1}\right) \right] \\ &{\varvec{D}}_p - I_{12}^{SP} \\ &= \underset{rp \times \frac{1}{2} p(p+1)}{{\varvec{O}}} \\ I_{14}^{SP} &= -\mathrm{E}\left[ \left( {\varvec{X}}^{\prime } \otimes {\varvec{I}}_p\right) \left[ \left( g_1({\varvec{Z}}^{\prime }\varvec{\eta }^{\star }) \otimes {\varvec{I}}_{np}\right) + \left( {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star }) \otimes \varvec{\eta }^{\star }\right) {\varvec{Z}}^{\prime } \right] \left( {\varvec{1}}_n \otimes {\varvec{I}}_p\right) \right] \\ &= - {\varvec{X}}^{\prime }\left( {\varvec{1}}_N \otimes {\varvec{1}}_n\right) \otimes \left( c_1 {\varvec{I}}_p - \left( \frac{c_2 w_1}{\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }} + \frac{n c_1}{1+n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }}\right) \varvec{\eta }\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\right) \\ I_{22}^{SP} &= -\mathrm{E}\bigl [{\varvec{D}}_p^{\prime } \left[ \mathrm{vec}^{\prime }({\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] \\ &\quad\left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{np} \otimes {\varvec{I}}_{p}\right] \left[ \left( \frac{N}{2}\varvec{\varOmega }^{-1} - \varvec{\varOmega }^{-1}{\varvec{Z}}{\varvec{Z}}^{\prime }\varvec{\varOmega }^{-1}\right) \otimes \varvec{\varOmega }^{-1}\right] \\ &\quad\times \left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{pn} \otimes {\varvec{I}}_{p}\right] \left[ \mathrm{vec}({\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] {\varvec{D}}_p\bigr ] \\ &= \frac{N}{2}{\varvec{D}}_p^{\prime }\left[ \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right] {\varvec{D}}_p \\ I_{23}^{SP} &= -\mathrm{E}\bigl [{\varvec{D}}_p^{\prime } \left[ \mathrm{vec}^{\prime }({\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] \\ &\quad\left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{np} \otimes {\varvec{I}}_{p}\right] \left[ \left( \frac{N}{2}\varvec{\varOmega }^{-1} - \varvec{\varOmega }^{-1}{\varvec{Z}}{\varvec{Z}}^{\prime }\varvec{\varOmega }^{-1}\right) \otimes \varvec{\varOmega }^{-1}\right] \\ &\quad \times \left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{pn} \otimes {\varvec{I}}_{p}\right] \left[ \mathrm{vec}({\varvec{I}}_n-{\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] {\varvec{D}}_p\bigr ] \\ &= \frac{N}{2n}\sum _{i=1}^n {\varvec{D}}_p^{\prime }\left[ \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right] {\varvec{D}}_p - I_{22}^{SP} = \underset{ \frac{1}{2} p(p+1) \times \frac{1}{2} p(p+1)}{{\varvec{O}}} \\ I_{24}^{SP} &= -E \left( \frac{\partial ^2 l}{\partial \mathrm{vech}\left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}}\right) \, \partial \varvec{\eta }^{\prime }}\right) = \underset{ \frac{1}{2} p(p+1) \times p}{{\varvec{O}}} \\ I_{33}^{SP} &= - \mathrm{E}\bigl [{\varvec{D}}_p^{\prime } \left[ \mathrm{vec}^{\prime }({\varvec{I}}_n-{\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] \\ &\quad\left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{np} \otimes {\varvec{I}}_{p}\right] \left[ \left( \frac{N}{2}\varvec{\varOmega }^{-1} - \varvec{\varOmega }^{-1}{\varvec{Z}}{\varvec{Z}}^{\prime }\varvec{\varOmega }^{-1}\right) \otimes \varvec{\varOmega }^{-1}\right] \\ &\quad \times \left[ {\varvec{I}}_{n} \otimes {\varvec{K}}_{pn} \otimes {\varvec{I}}_{p}\right] \left[ \mathrm{vec}({\varvec{I}}_n-{\varvec{P}}_n) \otimes {\varvec{I}}_{p^2}\right] {\varvec{D}}_p\bigr ] \\ &= \frac{N(n-1)}{2} {\varvec{D}}_p^{\prime }\left[ \varvec{\varSigma }_{\small {{\varvec{Z}}_2}}^{-1} \otimes \varvec{\varSigma }_{\small {{\varvec{Z}}_2}}^{-1} \right] {\varvec{D}}_p \\ I_{34}^{SP} &= -E \left( \frac{\partial ^2 l}{\partial \mathrm{vech}\left( \varvec{\varSigma }_{\small {{\varvec{Z}}_2}}\right) \, \partial \varvec{\eta }^{\prime }}\right) = \underset{\frac{1}{2} p(p+1) \times p}{{\varvec{O}}} \quad \text{ and } \\ I_{44}^{SP} &= - \mathrm{E}\left[ \left( {\varvec{1}}_n^{\prime } \otimes {\varvec{I}}_p\right) {\varvec{Z}} \left( {\varvec{G}}_2({\varvec{Z}}^{\prime }\varvec{\eta }^{\star })\right) {\varvec{Z}}^{\prime } \left( {\varvec{1}}_n \otimes {\varvec{I}}_p\right) \right] \\ &= -Nn c_2 \left[ w_0 \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}} - \frac{2n \varvec{\varSigma }_{ {\varvec{z}}_1}\varvec{\eta }\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{1+2n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }} \right. \right. \\ &\quad\left. \left. - \frac{\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{(\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }) (1+2n\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\varvec{\eta })} \right) + \frac{w_2 \varvec{\varSigma }_{\small {{\varvec{z}}_1}}\varvec{\eta } \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}}}{n(\varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta })^2}\right] \!. \end{aligned}$$

1.3 B.3 MLEs under \(H_{0,int}: \varvec{\alpha }^{CP}={\varvec{0}}\)

The MLEs under \(H_{0,int}: \varvec{\alpha }^{CP} = {\varvec{0}}\) may be obtained using a similar profiling argument as in Sect. 5.2. First recall that \(\varvec{\alpha } = \varvec{\alpha }^{CP} - \sqrt{2/\pi } \left( 1 + n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) ^{-1/2} \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }\). Let \({\varvec{z}}_{i,\alpha }\) denote \({\varvec{z}}_i\) evaluated at \(\varvec{\alpha }^{CP} = {\varvec{0}}\). From the previous transformation, we have

$$\begin{aligned} {\varvec{z}}_{i,\alpha } = y_i - \left( {\varvec{X}}^{*}_i \otimes {\varvec{I}}_p \right) \mathrm{vec}(\varvec{\gamma }^{\prime }) + \sqrt{2/\pi } \left( 1 + n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) ^{-1/2} \left( {\varvec{1}}_n \otimes {\varvec{I}}_p\right) \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta }. \end{aligned}$$

In addition let \({\varvec{Z}}_\alpha = \left( {\varvec{z}}_{1,\alpha }, \ldots , {\varvec{z}}_{N,\alpha }\right) \) and define

$$\begin{aligned} {\varvec{V}}_1&= \sqrt{2/\pi } \left( 1 + n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) ^{-3/2}\\&\quad\left[ \left( 1+n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) \left( {\varvec{1}}_n \varvec{\eta }^{\prime } \otimes {\varvec{I}}_p \right) - \frac{1}{2} n \left( {\varvec{1}}_n \otimes {\varvec{I}}_p \right) \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \mathrm{vec}(\varvec{\eta }\varvec{\eta }^{\prime }) \right] \!, \\ {\varvec{V}}_2 &= \sqrt{2/\pi } \left( 1 + n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) ^{-3/2}\\&\quad\left[ \left( 1+n \varvec{\eta }^{\prime }\varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \right) \left( {\varvec{1}}_n \otimes {\varvec{I}}_p \right) \varvec{\varSigma }_{\small {{\varvec{z}}_1}} - n \left( {\varvec{1}}_n \otimes {\varvec{I}}_p \right) \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \varvec{\eta } \varvec{\eta }^{\prime } \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \right] . \end{aligned}$$

Since the transformation from \(\varvec{\alpha }\) to \(\varvec{\alpha }^{CP}\) does not introduce any terms including \(\varvec{\varSigma }_{\small {{\varvec{Z}}_2}}\), the MLE of \(\varvec{\varSigma }_{\small {{\varvec{Z}}_2}}\), given \(\varvec{\theta }^{TP} = \left( \mathrm{vec}(\varvec{\gamma }^{\prime }), \mathrm{vech}(\varvec{\varSigma }_{\small {{\varvec{z}}_1}}), \varvec{\eta } \right) \), is

$$\begin{aligned} \widehat{\varvec{\varSigma }}_{{\varvec{Z}}_2, \varvec{\theta }^{TP}} = \frac{1}{N(n-1)} \sum _{i=1}^n \left( {\varvec{e}}_i^{\prime }{\varvec{Q}}_n \otimes {\varvec{I}}_p \right) {\varvec{Z}}_{\alpha }{\varvec{Z}}_{\alpha }^{\prime } \left( {\varvec{Q}}_n {\varvec{e}}_i \otimes {\varvec{I}}_p \right) . \end{aligned}$$

However, since this transformation introduced new terms involving \(\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\), it can be shown that no closed-form expression exists for the MLE of \(\varvec{\varSigma }_{\small {{\varvec{z}}_1}}\) given the other parameters are known.

Plugging in \(\widehat{\varvec{\varSigma }}_{\small {{\varvec{Z}}_2}, \varvec{\theta }^{TP}}\) into the likelihood given in Eq. (8), keeping the transformation of \({\varvec{Z}}\) to \({\varvec{Z}}_\alpha \) in mind, results in the profiled log likelihood (up to an additive constant)

$$\begin{aligned} l \left( \varvec{\theta }^{TP} | {\varvec{Y}}\right)&= - \frac{N}{2} \mathrm{ln}\left| \varvec{\varSigma }_{{{\varvec{z}}_1}} \right| \\&\quad- \frac{N(n-1)}{2} \mathrm{ln}\left| \widehat{\varvec{\varSigma }}{_{{\varvec{Z}}_2}, \varvec{\theta }^{TP}} \right| - \frac{1}{2} \mathrm{vec}^{\prime } ({\varvec{Z}}_{\alpha }) \left( {\varvec{I}}_N \otimes {\varvec{P}}_n \otimes \varvec{\varSigma }_{{{\varvec{z}}_1}} ^{-1}\right) \mathrm{vec}({\varvec{Z}}_{\alpha }) \\&\quad+ \sum _{i=1}^N g_0 \left( {\varvec{z}}_{i,\alpha }^{\prime } \varvec{\eta }^{*}\right), \end{aligned}$$

and its first differential is

$$\begin{aligned} \mathrm{d}l \left( \varvec{\theta }^{TP} | {\varvec{Y}}\right)&= \Biggl [ \left( \sum _{i=1}^n \mathrm{vec}^{\prime } \left[ \left( {\varvec{Q}}_n {\varvec{e}}_i \otimes {\varvec{I}}_p \right) \widehat{\varvec{\varSigma }}_{{{\varvec{Z}}_2}, \varvec{\theta }^{TP}}^{-1} \left( {\varvec{e}}_i^{\prime }{\varvec{Q}}_n \otimes {\varvec{I}}_p \right) {\varvec{Z}}_{\alpha } \right] + \mathrm{vec}^{\prime } \left( {\varvec{Z}}_{\alpha } \right) \right. \\&\quad\left. \left( {\varvec{I}}_N \otimes {\varvec{P}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right) \right) \\&\quad\times \left( {\varvec{X}}^{*} \otimes {\varvec{I}}_p \right) - \sum _{i=1}^n g_1 \left( {\varvec{z}}_{i,\alpha }^{\prime } \varvec{\eta }^{*}\right) \left( \varvec{\eta }^{*}\right) ^{\prime } \left( {\varvec{X}}_i^{*} \otimes {\varvec{I}}_p \right) \Biggr ] \mathrm{d}\mathrm{vec}\left( \varvec{\gamma }^{\prime } \right) \\&\quad- \Biggl [ \frac{N}{2} \mathrm{vec}^{\prime } \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \right) + \sum _{i=1}^n \mathrm{vec}^{\prime } \left[ \left( {\varvec{Q}}_n {\varvec{e}}_i \otimes {\varvec{I}}_p \right) \widehat{\varvec{\varSigma }}_{\small {{\varvec{Z}}_2}, \varvec{\theta }^{TP}}^{-1} \left( {\varvec{e}}_i^{\prime }{\varvec{Q}}_n \otimes {\varvec{I}}_p \right) {\varvec{Z}}_{\alpha } \right] \left( {\varvec{1}}_N \otimes {\varvec{V}}_1 \right) \\&\quad- \frac{1}{2} \left( \mathrm{vec}^{\prime } \left( {\varvec{Z}}_\alpha \right) \otimes \mathrm{vec}^{\prime } \left( {\varvec{Z}}_\alpha \right) \right) \\&\quad\left( {\varvec{I}}_{nN} \otimes {\varvec{K}}_{p,nN} \otimes {\varvec{I}}_p \right) \left[ \mathrm{vec}\left( {\varvec{I}}_N \otimes {\varvec{P}}_n \right) \otimes {\varvec{I}}_{p^2}\right] \left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1}\right) \\&\quad+ \mathrm{vec}^{\prime } \left( {\varvec{Z}}_{\alpha } \right) \left( {\varvec{I}}_N \otimes {\varvec{P}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right) \left( {\varvec{1}}_N \otimes {\varvec{V}}_1 \right) \\&\quad- \sum _{i=1}^N g_1 \left( {\varvec{z}}_{i,\alpha }^{\prime } \varvec{\eta }^{*}\right) \left( \varvec{\eta }^{*}\right) ^{\prime } {\varvec{V}}_1 \Biggr ] {\varvec{D}}_p \mathrm{d}\mathrm{vech}\left( \varvec{\varSigma }_{\small {{\varvec{z}}_1}} \right) \\&\quad- \Biggl [ \sum _{i=1}^n \mathrm{vec}^{\prime } \left[ \left( {\varvec{Q}}_n {\varvec{e}}_i \otimes {\varvec{I}}_p \right) \widehat{\varvec{\varSigma }}_{\small {{\varvec{Z}}_2}, \varvec{\theta }^{TP}}^{-1} \left( {\varvec{e}}_i^{\prime }{\varvec{Q}}_n \otimes {\varvec{I}}_p \right) {\varvec{Z}}_{\alpha } \right] \left( {\varvec{1}}_N \otimes {\varvec{V}}_2 \right) \\&\quad+\mathrm{vec}^{\prime } \left( {\varvec{Z}}_{\alpha } \right) \left( {\varvec{I}}_N \otimes {\varvec{P}}_n \otimes \varvec{\varSigma }_{\small {{\varvec{z}}_1}}^{-1} \right) \left( {\varvec{1}}_N \otimes {\varvec{V}}_2 \right) \\&\quad- \sum _{i=1}^n g_1 \left( {\varvec{z}}_{i,\alpha }^{\prime } \varvec{\eta }^{*}\right) \left( \varvec{\eta }^{*}\right) ^{\prime }{\varvec{V}}_2 \Biggr ] \mathrm{d}\varvec{\eta }. \end{aligned}$$

Using a quasi-Newton method, the profile likelihood can be maximized, and the process may be quickened by the inclusion of the gradient, formed by the above differential. Initial values for the optimization procedure follow from a similar argument discussed in Sect. 5.2.

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Opheim, T., Roy, A. Score Tests for Intercept and Slope Parameters of Doubly Multivariate Linear Models with Skew-Normal Errors. J Stat Theory Pract 15, 30 (2021). https://doi.org/10.1007/s42519-020-00159-8

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