Correction: Contreras-Reyes, J.E.; Cortés, D.D. Bounds on Rényi and Shannon Entropies for Finite Mixtures of Multivariate Skew-Normal Distributions: Application to Swordfish (Xiphias gladius Linnaeus). Entropy 2016, 18, 382

The authors were not aware of one error made in the proofreading phase; therefore, we wish to make the following correction to this paper [1]:

On Equation (14), the multinomial (MN) theorem is considered [2] to obtain under the condition with $\mathcal{A}=\left\{k_i\in \mathbb{N}:k_i>0,{\sum }_{i=1}^m{k}_i=\alpha ;i=1,\dots ,m\right\}$, $0<\alpha <\infty$, $\alpha \ne 1$; and an m-component mixture model where ${\pi }_i\ge 0$, ${\sum }_{i=1}^m{\pi }_i=1$, $\tilde{\mathbf{\theta }}=(\tilde{\mathbf{\xi }},\tilde{\mathbf{\Omega }},\tilde{\mathbf{\eta }})$: $\tilde{\mathbf{\xi }}=({\mathbf{\xi }}_1,\dots ,{\mathbf{\xi }}_m)$ a set of m location vector parameters, $\tilde{\mathbf{\Omega }}=({\mathbf{\Omega }}_1,\dots ,{\mathbf{\Omega }}_m)$ a set of m dispersion matrices, $\tilde{\mathbf{\eta }}=({\mathbf{\eta }}_1,\dots ,{\mathbf{\eta }}_m)$ a set of shape vector parameters, and with m mixing weights, $\mathbf{\pi }=({\pi }_1,\dots ,{\pi }_m)$.

Given that integral in (1) is based on the $\alpha$th-order Rényi entropy on variable $\mathbf{y}$: the condition (2) is not accomplished for any $m\in \mathbb{N}$. For example, if $m=2$ and $\alpha =2$, thus $\alpha =k_1+k_2\Rightarrow 2=1+1$ implies that $k_i=1$, $i=1,2$, because $k_i\in \mathbb{N}$, $k_i>0$. In general, for any $m\in \mathbb{N}$, the MN theorem needs $k_i=1$, $i=1,\dots ,m$. Therefore, MN theorem can not be considered for the approximation of the Rényi entropy for any finite mixture, with $\alpha$th-order satisfying $0<\alpha <\infty$ and $\alpha \ne 1$. Only for the limit $\alpha \to 1$, the Shannon entropy is obtained as by applying l’Hôpital’s rule to ${\mathcal{R}}_{\alpha }[\mathbf{Y};\tilde{\mathbf{\theta }},\mathbf{\pi }]$ with respect to $\alpha$. Therefore, Lemma 3 of the paper could be considered in the limit $\alpha \to 1$. However, this is the case of bounds on Shannon entropy addressed in Section 3.1.

In conclusion:

However, these changes do not influence the conclusions and the other results of the paper. The authors would like to apologize for any inconvenience caused.