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

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.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Akdemir D, Gupta AK (2010) A matrix variate skew distribution. Eur J Pure Appl Math 3(2):128–140

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Arellano-Valle RB, Azzalini A (2006) On the unification of families of skew-normal distributions. Scand J Stat 33:561–574

    MathSciNet  Article  Google Scholar 

  3. 3.

    Arellano-Valle RB, Azzalini A (2008) The centred parametrization for the multivariate skew-normal distribution. J Multivar Anal 99(7):1362–1382

    MathSciNet  Article  Google Scholar 

  4. 4.

    Arnold SF (1973) Application of the theory of products of problems to certain patterned covariance matrices. Ann Stat 1(4):682–699

    MathSciNet  Article  Google Scholar 

  5. 5.

    Arnold SF (1979) Linear models with exchangeably distributed errors. J Am Stat Assoc 74:194–199

    MathSciNet  Article  Google Scholar 

  6. 6.

    Azzalini A, Dalla Valle A (1996) The multivariate skew-normal distribution. Biometrika 83:715–726

    MathSciNet  Article  Google Scholar 

  7. 7.

    Azzalini A, Capitanio A (2014) The skew-normal and related families. Cambridge University Press, New York

    Google Scholar 

  8. 8.

    Balakrishnan N, Capitanio A, Scarpa B (2014) A test for multivariate skew-normality based on its canonical form. J Multivar Anal 128:19–32

    MathSciNet  Article  Google Scholar 

  9. 9.

    Chen JT, Gupta AK (2005) Matrix variate skew normal distributions. Stat: J Theor Appl Stat 39(3):247–253

    MathSciNet  Article  Google Scholar 

  10. 10.

    Dutilleul P (1999) The MLE algorithm for the matrix normal distribution. J Stat Comput Simul 64:105–123

    Article  Google Scholar 

  11. 11.

    Gupta A, Gonzalez-Farias G, Dominguez-Molina J (2004) A multivariate skew normal distribution. J Multivar Anal 89(1):181–190

    MathSciNet  Article  Google Scholar 

  12. 12.

    Harrar SW, Gupta AK (2008) On matrix variate skew-normal distributions. Stat: J Theor Appl Stat 42(2):179–194

    MathSciNet  Article  Google Scholar 

  13. 13.

    Jana S, Balakrishnan N, Jemila H (2018) Estimation of the parameters of the extended growth curve model under multivariate skew normal distribution. J Multivar Anal 166:111–128

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kheradmandi A, Abdolrahman R (2015) Estimation in skew-normal linear mixed measurement error models. J Multivar Anal 136:1–11

    MathSciNet  Article  Google Scholar 

  15. 15.

    Lachos VH, Bolfarine H, Arellano-Valle RB, Montenegro LC (2007) Likelihood-based inference for multivariate skew-normal regression models. Commun Stat-Theory Methods 36(9):1769–1786

    MathSciNet  Article  Google Scholar 

  16. 16.

    Leiva R (2007) Linear discrimination with equicorrelated training vectors. J Multivar Anal 98(2):384–409

    MathSciNet  Article  Google Scholar 

  17. 17.

    Lu N, Zimmerman DL (2005) The likelihood ratio test for a separable covariance matrix. Stat Probab Lett 73:449–457

    MathSciNet  Article  Google Scholar 

  18. 18.

    Lin TI, Wang WL (2013) Multivariate skew-normal at linear mixed models for multi-outcome longitudinal data. Stat Model 13(3):199–221

    MathSciNet  Article  Google Scholar 

  19. 19.

    Magnus JR, Neudecker H (1986) Symmetry, 0–1 matrices and Jacobians: a review. Econom Theory 2:157–190

    Article  Google Scholar 

  20. 20.

    Magnus JR, Neudecker H (1988) Matrix differential calculus with applications in statistics and econometrics. Wiley, New York

    Google Scholar 

  21. 21.

    Opheim T, Roy A (2019) Revisiting the linear models with exchangeably distributed errors. Proc Am Stat Assoc 2677–2686

  22. 22.

    Opheim T, Roy A (2021) Linear models for multivariate repeated measures data with block exchangeable covariance structure. Comput Stat. https://doi.org/10.1007/s00180-021-01064-9

  23. 23.

    Rao CR (1945) Familial correlations or the multivariate generalizations of the intraclass correlation. Curr Sci 14:66–67

    Google Scholar 

  24. 24.

    Rao CR (1948) Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Math Proc Camb Philos Soc 44:50–57

    MathSciNet  Article  Google Scholar 

  25. 25.

    Rao CR (1953) Discriminant functions for genetic differentiation and selection. Sankhya 12:229–246

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Rao CR (2005) Score test: historical review and recent developments. In: Balakrishnan N, Nagaraja HN, Kannan N (eds) Advances in Ranking and Selection, Multiple Comparisons, and Reliability. Statistics for Industry and Technology, Birkhäuser Boston, pp 3–20. https://doi.org/10.1007/0-8176-4422-9_1

  27. 27.

    Roy A (2007) A note on testing of Kroneckar product covariance structures for doubly multivariate data. In: Proceedings of the American Statistical Association, Seattle, Washington, pp 2157–2162

  28. 28.

    Roy A, Leiva R, Žežula I, Klein D (2015) Testing of equality of mean vectors for paired doubly multivariate observations in blocked compound symmetric covariance matrix setup. J Multivar Anal 137:50–60

    MathSciNet  Article  Google Scholar 

  29. 29.

    Roy A, Zmyślony R, Fonseca M, Leiva R (2016) Optimal estimation for doubly multivariate data in blocked compound symmetric covariance structure. J Multivar Anal 144:81–90

    MathSciNet  Article  Google Scholar 

  30. 30.

    Roy A, Filipiak K, Klein D (2018) Testing a block exchangeable covariance matrix. Stat: A J Theor Appl Stat 52(2):393–408

    MathSciNet  Article  Google Scholar 

  31. 31.

    Roy A, Leiva R (2011) Estimating and testing a structured covariance matrix for three-level multivariate data. Commun Stat Theory Methods 40(10):1945–1963

    MathSciNet  Article  Google Scholar 

  32. 32.

    Sperling MR, Gur RC, Alavi A, Gur RE, Resnick S, O’Connor MJ, Reivich M (1990) Subcortical metabolic alterations in partial epilepsy. Epilepsia 31(2):145–155

    Article  Google Scholar 

  33. 33.

    Szatrowski TH (1982) Testing and estimation in the block compound symmetry problem. J Educ Stat 7(1):3–18

    Article  Google Scholar 

  34. 34.

    Tsukada S (2018) Hypothesis testing for independence under blocked compound symmetric covariance structure. Commun Math Stat 6:163–184

    MathSciNet  Article  Google Scholar 

  35. 35.

    Wang WL (2015) Approximate methods for maximum likelihood estimation of multivariate nonlinear mixed-effects models. Entropy 17:5353–5381

    Article  Google Scholar 

  36. 36.

    Wilks S (1938) The large sample distribution of the likelihood ratio for testing composite hypotheses. Ann Math Stat 9:60–62

    Article  Google Scholar 

  37. 37.

    Žežula I, Klein D, Roy A (2018) Testing of multivariate repeated measures data with block exchangeable covariance structure. Test 27(2):360–378

    MathSciNet  Article  Google Scholar 

Download references

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.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Anuradha Roy.

Ethics declarations

Conflict of Interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

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}$$

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}$$

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Keywords

  • Maximum likelihood estimates
  • Model selection
  • Hypotheses tests
  • Repeated measures data
  • Skew-normal distributions

Mathematics Subject Classification

  • 62J05
  • 62H10
  • 62H12
  • 62H15