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The normal distribution and its perturbation have left an immense mark on the statistical literature. Several generalized forms exist to model different skewness, kurtosis, and body shapes. Although they provide better fitting capabilities, these generalizations do not have parameters and formulae with a clear meaning to the practitioner on how the distribution is being modeled. We propose a neat integration approach generalization which intuitively gives direct control of the body and tail shape, the body-tail generalized normal (BTGN). The BTGN provides the basis for a flexible distribution, emphasizing parameter interpretation, estimation properties, and tractability. Basic statistical measures are derived, such as the density function, cumulative density function, moments, moment generating function. Regarding estimation, the equations for maximum likelihood estimation and maximum product spacing estimation are provided. Finally, real-life situations data, such as log-returns, time series, and finite mixture modeling, are modeled using the BTGN. Our results show that it is possible to have more desirable traits in a flexible distribution while still providing a superior fit to industry-standard distributions, such as the generalized hyperbolic, generalized normal, tail-inflated normal, and t distributions. [ABSTRACT FROM AUTHOR]

South Africa has great potential for considering wind energy as an alternative resource. The climatology allows for significant wind energy production. An accurate joint description of the wind speed (linear) and wind direction (circular) characteristics is important for wind farm development. In this paper, a bivariate class of flexible joint probability density functions of wind speed and wind direction for the use in wind energy analysis is presented. This joint model accounts for bimodality, skewness, and a dependency structure between the wind speed and wind direction. For the joint probabilistic description of the wind speed and wind direction, special cases of this bivariate class are evaluated, namely the semi-parametric Möbius model on the disc, the Möbius distribution on the disc, and the Beta type III Möbius distribution on the disc. These three special cases are applied to wind speed and wind direction data recorded every ten minutes at two locations in South Africa. Evaluation of the models is based on three different information criteria and normalized deviation. Overall, the semi-parametric model is superior to the parametric models based on the performance measures. The wind energy potential at the two locations is evaluated using the semi-parametric model. [ABSTRACT FROM AUTHOR]

The joint modeling of angular and linear observations is crucial as data of this nature are prevalent in multiple disciplines, for example the joint modeling of wind direction and another climatological variable such as wind speed or air temperature, the direction an animal moves and the distance moved, or wave direction and wave height. Hence, there is a need for developing flexible distributions on the hyper-disc, which has support of the interior of the hyper-sphere, as it allows for modeling the combination of angular and linear observations. This paper addresses this need by developing flexible distributions for the disc that have the ability to capture any inherent bimodality present in the data. A new class of bivariate distributions is proposed which has support on the unit disc in two dimensions that includes, as a special case, the existing Möbius distribution on the disc. This class is obtained by expressing the density function in a general form using a measurable function termed as generator. Special cases of this generator are considered to demonstrate the flexibility. By applying a conformal mapping to the generator function a new Möbius distribution class emanates. This class of bivariate distributions on the disc is the first to account for bimodality and skewness present in the data. The flexible behavior of the proposed models in terms of bimodality and skewness is graphically demonstrated. Preliminary evidential analysis of the wind data observed at Marion Island reveals the absence of unimodality in the data. The fit of the proposed models, which account for bimodality, to the Marion Island wind data were evaluated analytically and visually. [ABSTRACT FROM AUTHOR]