A multivariate powered half-normal distribution

The half-normal distribution has applications in various contexts, particularly in economic analysis (to describe, for example, inefficiency variables), reliability analysis and quality control. However, it seems that, still in the recent literature, interest in forms of multivariate distribution with half-normal marginals is remained at a low level. This problem arises, for example, in the context of economic analysis when it needs to examine inefficiency variables simultaneously. Furthermore, from a robustness perspective, the distributional assumption of half-normality may show itself too strong (or the model too rigorous) and to be inconsistent with the real data, thus motivating the need of a more flexible model. In this paper we consider a powered half-normal distribution as a model more flexible than the half-normal distribution, showing that it can be derived from an half-normal model by means of a power transformation, and propose a multivariate version of this as a model adequate to deal a set of multivariate data when a powered half-normal density, or an ordinary half-normal density as a special case, is the appropriate distribution of the univariate marginal variables. This new multivariate distribution is expressed in closed form, shows symmetry in the component variables and its marginals of any dimension are of powered half-normal form too. We provide explicit expressions of product moments and of the correlation coefficients, and the problem of estimation of the parameters is also discussed. Also the focus is on the univariate marginals. We point out the closeness between the analytical form of a powered half-normal distribution and the Weibull distribution one and, also, a few relationships between this marginal and some well-known distributions are established. Specifically we develop a trivariate powered half-normal distribution and afterwards consider the multivariate case as an immediate extension of this.