Scandinavian Journal of Statistics
Volume 49, Issue 3 pp. 1418-1419
CORRIGENDUM
Open Access

Corrigendum to “On the unification of families of skew-normal distributions” published in Scand. J. Stat. vol. 33, pp. 561–574

Reinaldo B. Arellano-Valle, 

Reinaldo B. Arellano-Valle

Departamento de Estadística, Pontificia Universidad Católica de Chile, Santiago, Chile

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Adelchi Azzalini, 

Corresponding Author

Adelchi Azzalini

Dipartimento di Scienze Statistiche, Università di Padova, Padova, Italia

Correspondence Adelchi Azzalini, Dipartimento di Scienze Statistiche, Università di Padova, Padova, Italia.

Email: [email protected]

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First published: 24 April 2022
https://doi.org/10.1111/sjos.12594

Although the expression indicated at the end of Appendix A provides the correct density of the required conditional distribution, it is formally incorrect to denote it as a SUN distribution with the stated parameters, since they do not meet the condition introduced in Section 2.1 that the Γ $$ \Gamma $$ component of the parameter set is a correlation matrix. The expression of Γ 1 . 2 $$ {\Gamma}_{1.2} $$ given in Appendix A has instead diagonal elements less than 1.

There is a fix to the problem, based on the simple fact the probability Φ m ( γ 1 . 2 ; Γ 1 . 2 ) $$ {\Phi}_m\left({\gamma}_{1.2};{\Gamma}_{1.2}\right) $$ appearing in the denominator of f Y 1 | Y 2 = y 2 $$ {f}_{Y_1\mid {Y}_2={y}_2} $$ can be rewritten as Φ m ( D γ 1 . 2 ; D Γ 1 . 2 D ) $$ {\Phi}_m\left(D\kern0.3em {\gamma}_{1.2};D\kern0.3em {\Gamma}_{1.2}\kern0.3em D\right) $$ for any positive definite diagonal matrix D $$ D $$ , and a similar equality holds for the probability in the numerator. Denote by Diag ( Γ 1 . 2 ) $$ \mathrm{Diag}\left({\Gamma}_{1.2}\right) $$ the diagonal matrix formed by the diagonal elements of Γ 1 . 2 $$ {\Gamma}_{1.2} $$ and define
Δ ‾ 1 . 2 = Δ 1 . 2 Diag ( Γ 1 . 2 ) − 1 / 2 Γ ‾ 1 . 2 = Diag ( Γ 1 . 2 ) − 1 / 2 Γ 1 . 2 Diag ( Γ 1 . 2 ) − 1 / 2 , γ ‾ 1 . 2 = Diag ( Γ 1 . 2 ) − 1 / 2 γ 1 . 2 , Ω ‾ 1 . 2 ∗ = Γ ‾ 1 . 2 Δ ‾ 1 . 2 ⊤ Δ ‾ 1 . 2 Ω ‾ 11 . 2 $$ {\displaystyle \begin{array}{ll}{\overline{\Delta}}_{1.2}& ={\Delta}_{1.2}\kern0.3em \mathrm{Diag}{\left({\Gamma}_{1.2}\right)}^{-1/2}\kern0em \\ {}{\overline{\Gamma}}_{1.2}& =\mathrm{Diag}{\left({\Gamma}_{1.2}\right)}^{-1/2}{\Gamma}_{1.2}\mathrm{Diag}{\left({\Gamma}_{1.2}\right)}^{-1/2},\\ {}{\overline{\gamma}}_{1.2}& =\mathrm{Diag}{\left({\Gamma}_{1.2}\right)}^{-1/2}\kern0.3em {\gamma}_{1.2},\\ {}{\overline{\Omega}}_{1.2}^{\ast }& =\left(\begin{array}{cc}{\overline{\Gamma}}_{1.2}& {\overline{\Delta}}_{1.2}^{\top}\\ {}{\overline{\Delta}}_{1.2}& {\overline{\Omega}}_{11.2}\end{array}\right)\kern0em \end{array}} $$
so that we can write Y 1 | Y 2 = y 2 ∼ SUN d 1 , m ( ξ 1 . 2 , γ ‾ 1 . 2 , ω ‾ 1 , Ω ‾ 1 . 2 ∗ ) $$ {Y}_1\mid {Y}_2={y}_2\sim {\mathrm{SUN}}_{d_1,m}\left({\xi}_{1.2},{\overline{\gamma}}_{1.2},{\overline{\omega}}_1,{\overline{\Omega}}_{1.2}^{\ast}\right) $$ , whose parameters identify the same probability distribution as the one indicated in the paper, but complying with the condition that the Γ $$ \Gamma $$ component is a correlation matrix.

It is certainly unfortunate that an inappropriate statement was made in the paper, but we underline that anyone using the distribution with the parameters originally indicated did not incur in an error because of the equivalence between that density and the one indicated here. A second consideration is that the present adjustment is based on the existence of equivalent densities, if the condition on the scale factor of the Γ $$ \Gamma $$ matrix is not imposed. This reinforces the recurrent indication in the paper of the appropriateness of this condition to avoid overparameterization.

ACKNOWLEDGEMENTS

Open Access Funding provided by Universita degli Studi di Padova within the CRUI-CARE Agreement.

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