Corrigendum to Maximum Likelihood Estimation of the Multivariate Normal Mixture Model

This article refers to:

Maximum Likelihood Estimation of the Multivariate Normal Mixture Model

Otilia Boldea^a and Jan R. Magnus^b,c

^aDepartment of Econometrics & OR, Tilburg University, Tilburg, The Netherlands

^bDepartment of Econometrics & Data Science, Vrije Universiteit, Amsterdam, The Netherlands

^cTinbergen Institute, Amsterdam, The Netherlands

CONTACT Otilia Boldea [email protected] Department of Econometrics & OR, Tilburg University, Warandelaan 2, 5037 AB, Tilburg, The Netherlands.

Dante Amengual, Gabriele Fiorentini and Enrique Sentana kindly alerted us to an error in Theorem 2 in Boldea and Magnus (2009), where we erroneously stated that the degrees of freedom of our proposed information matrix test is $\mathit{\text{gm}}(m+3)/2$. The degrees of freedom are different, and a generalized inverse should be used in the definition of the IM test. Below is the corrected statement of Theorem 2, which is now a proposition because the proof follows from Amengual, Fiorentini, and Sentana (2024a, 2024b).

Proposition A

(Information Matrix Test). Define the variance matrix: $$\begin{array}{c}\mathbf{\Sigma }(\mathit{\theta })=\frac{1}{n}\sum _{t=1}^n{\mathbf{w}}_t{\mathbf{w}}_t^{\prime }-\left(\frac{1}{n}\sum _{t=1}^n{\mathbf{w}}_t{\mathbf{q}}_t^{\prime }\right){\left(\frac{1}{n}\sum _{t=1}^n{\mathbf{q}}_t{\mathbf{q}}_t^{\prime }\right)}^{-1}\\ \hspace{1em}\times \left(\frac{1}{n}\sum _{t=1}^n{\mathbf{q}}_t{\mathbf{w}}_t^{\prime }\right)\end{array}$$

where ${\mathbf{q}}_t$ denotes the tth increment to the score, and $${\mathbf{w}}_t=\text{vec}(\text{vech}{\mathbf{W}}_t^1,\text{vech}{\mathbf{W}}_t^2,\dots ,\text{vech}{\mathbf{W}}_t^g).$$

Then, evaluated at $\widehat{\mathit{\theta }}$, and under the null hypothesis of correct specification, with ${\mathbf{\Sigma }}^{+}$ equal to the Moore-Penrose generalized inverse of $\mathbf{\Sigma }(\widehat{\mathit{\theta }})$, $$\text{IM}=n\left(\frac{1}{n}\sum _{t=1}^n{\mathbf{w}}_t\right)\prime {\mathbf{\Sigma }}^{+}\left(\frac{1}{n}\sum _{t=1}^n{\mathbf{w}}_t\right)$$ asymptotically follows a ${\chi }^2$-distribution with $\mathit{\text{gm}}(m+1)(m+2)(m+7)/24$ degrees of freedom.

Our 2009 Appendix mistakenly suggested that the proof of Theorem 2 “follows from […] the development in Lancaster (1984)”. The results in Lancaster (1984) are insufficient to arrive at the correct degrees of freedom.

Instead, the proof of Proposition A follows directly from Amengual, Fiorentini, and Sentana (2024a, 2024b). Amengual, Fiorentini, and Sentana (2024a) show in their Proposition 1 that the IM test for a multivariate normal distribution (so for g = 1) coincides with a test that the expectations of all distinct third- and fourth-order multivariate Hermite polynomials are zero. Since there are $(\begin{array}{c}m+k-1\\ k\end{array})$ unique kth order polynomials, and the Hermite polynomials are orthogonal, the IM test for g = 1 converges to a ${\chi }^2$-distribution with $(\begin{array}{c}m+2\\ 3\end{array})+(\begin{array}{c}m+3\\ 4\end{array})=m(m+1)(m+2)(m+7)/24$ degrees of freedom. Based on these results, Amengual, Fiorentini, and Sentana (2024b) show in their Proposition 4 that a reparameterized version of our test converges to a ${\chi }^2$-distribution with $\mathit{\text{gm}}(m+1)(m+2)(m+7)/24$ degrees of freedom.

The error in Theorem 2 also implies that we used the wrong critical values in Table 5. We reran the simulations in Table 5 with the correct degrees of freedom, and we report them here. The corrected code can be found at https://github.com/oboldea/MultivariateNormalMixtures. The rest of the article is not affected.

Table 5 Size of IM test, simulation results.

	Critical values
n	17.34	21.60	25.99	28.87	34.81
100	0.9945	0.9939	0.9924	0.9893	0.9778
500	0.9309	0.8503	0.7468	0.6740	0.5280
1000	0.8490	0.7062	0.5554	0.4674	0.2992
$\infty$	0.5000	0.2500	0.1000	0.0500	0.0100