Abstract
The univariate power-normal distribution is quite useful for modeling many types of real data. On the other hand, multivariate extensions of this univariate distribution are not common in the statistic literature, mainly skewed multivariate extensions that can be bimodal, for example. In this paper, based on the univariate power-normal distribution, we extend the univariate power-normal distribution to the multivariate setup. Structural properties of the new multivariate distributions are established. We consider the maximum likelihood method to estimate the unknown parameters, and the observed and expected Fisher information matrices are also derived. Monte Carlo simulation results indicate that the maximum likelihood approach is quite effective to estimate the model parameters. An empirical application of the proposed multivariate distribution to real data is provided for illustrative purposes.
Highlights
Symmetry 2019, 11, 1509 regarding univariate asymmetric distributions is the work of Durrans [4], who introduced the fractional order statistics distribution with probability density function (PDF) given by φ D (z; α) = α f (z){ F (z)}α−1, z ∈ R, where α > 0 is a shape parameter that controls the amount of asymmetry in the distribution, F is an absolutely continuous cumulative distribution function (CDF), and f = dF is the corresponding PDF
Extensions of the univariate power-normal distribution to the multivariate setup have been little explored in the statistic literature
By employing the frequentist approach, the estimation of the multivariate power-normal distribution parameters is conducted by the maximum likelihood method
Summary
Asymmetric univariate distributions that can be used for explaining real data which are not adequately fitted by the usual normal distribution were studied in Azzalini [1], Fernández and Steel [2], Mudholkar and Hutson [3], Durrans [4], Pewsey et al [5], and Martínez-Flórez et al [6], among others. Symmetry 2019, 11, 1509 regarding univariate asymmetric distributions is the work of Durrans [4], who introduced the fractional order statistics distribution with PDF given by φ D (z; α) = α f (z){ F (z)}α−1 , z ∈ R, where α > 0 is a shape parameter that controls the amount of asymmetry in the distribution, F is an absolutely continuous CDF, and f = dF is the corresponding PDF. A univariate bimodal distribution was introduced in Bolfarine et al [13], whose PDF is given by φ B (z; α, β) = 2αcα f (z) { F (|z|)}α−1 G ( βz), z ∈ R, where α > 0, β ∈ R, F is an absolutely continuous CDF with PDF f = dF symmetric around zero,. We use the notation ABPN( β, α) to refer to this univariate asymmetric bimodal distribution.
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