Abstract
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.
Highlights
Flexible modeling is an ongoing study in distribution theory that dates back as long ago as 1879, when Galton pioneered the log-normal distribution [1]
Interpreting the body-tail generalized normal (BTGN) tail parameters, the values of β > α show that all the tail shapes are thinner than a fixed generalized normal distribution (GN) body shape α would accommodate
The departure from the normal distributional shape is evident in Figure 7, where the fitted BTGN density functions are shown
Summary
Flexible modeling is an ongoing study in distribution theory that dates back as long ago as 1879, when Galton pioneered the log-normal distribution [1]. The field has exploded with new distributions and ways of generating them These models include finite mixture models [2], variance-mean mixtures [3], copulas [4], the Box–Cox transformation [5], order-statistics-based distributions [6], probability integral transformations of [7], and the Pearson system of distributions [8], to name but a few. Finite moments: Most real-world measurements require this property To this end, a systematic bottom-up approach is to be taken to specify a new flexible model. The motivation for the body-tail generalization of the normal distribution would be to provide a new symmetric model with the aforementioned desirable traits for symmetric data, and serve as a new base model for further generalizations to accommodate skew data in the future.
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