Generalized Skew-Normal Negentropy and Its Application to Fish Condition Factor Time Series
The problem of measuring the disparity of a particular probability density function from a normal one has been addressed in several recent studies. The most used technique to deal with the problem has been exact expressions using information measures over particular distributions. In this paper, we consider a class of asymmetric distributions with a normal kernel, called Generalized Skew-Normal (GSN) distributions. We measure the degrees of disparity of these distributions from the normal
... om the normal distribution by using exact expressions for the GSN negentropy in terms of cumulants. Specifically, we focus on skew-normal and modified skew-normal distributions. Then, we establish the Kullback-Leibler divergences between each GSN distribution and the normal one in terms of their negentropies to develop hypothesis testing for normality. Finally, we apply this result to condition factor time series of anchovies off northern Chile. Entropy 2017, 19, 528 2 of 18 priors. The paper provides additional insights, to those provided in Vidal et al.  , on the interpretation of this distance and also discusses the usage of the KL divergence among several other distances. Some recent applications of measuring the disparity of a particular pdf from the normal one using negentropy include those by Gao and Zhang  and Wang et al.  , where the negentropy method has been successfully applied to seismic wavelet estimation. Pires and Ribeiro  considered the negentropy to measure the distance of non-Gaussian information from the normal one in independent components, with application to Northern Hemispheric winter monthly variability of a high-dimensional quasi-geostrophic atmospheric model. Furthermore, Pires and Hannachi  used a tensorial invariant approximation of the multivariate negentropy in terms of a linear combination of squared coskewness and cokurtosis. Then, the method was applied to global sea surface temperature anomalies, after data anomalies were tested through a non-Gaussian distribution. In this paper, we develop a procedure, based on KL divergences, to test the significance of the skewness parameter in the Generalized Skew-Normal (GSN) distributions, a flexible class of distributions that includes the SN and normal ones as particular cases. We consider asymptotic expansions of moments and cumulants for the negentropy of two particular cases: the SN and Modified Skew-Normal (MSN) distributions. Given that SN distributions do not accomplish the regularity condition of Fisher Information Matrix (FIM) at η = 0, normality is tested based on the MSN distribution  . This allows one to implement an asymptotic normality test for testing the significance of the skewness parameter. Numerical results are studied by: (a) comparing numerical integration methods with proposed asymptotic expansions; (b) comparing the asymptotic test with the likelihood ratio test and the asymptotic normality test given by Arrué et al.  ; and (c) applying the proposed test to condition factor time series of anchovy (Engraulis ringens). This paper is organized as follows: information theoretic measures are described in Section 2. In Section 3, we provide an asymptotic expansion in terms of the corresponding cumulants for the GSN, SN and MSN negentropies. We also express the KL and J divergences among each GSN distribution and the normal one in terms of negentropies (as cumulants' expansion series) to develop the hypothesis test about the significance of the skewness parameter together with a simulation study (Section 4). A simulation study is given in Section 5. In Section 6, the real data of the condition factor time series of anchovies off northern Chile illustrate the usefulness of the developed methodology. The discussion concludes the paper.