Abstract
We introduce two families of continuous distribution functions with not-necessarily symmetric densities, which contain a parent distribution as a special case. The two families proposed depend on two parameters and are presented as an alternative to the skew normal distribution and other proposals in the statistical literature. The density functions of these new families are given by a closed expression which allows us to easily compute probabilities, moments and related quantities. The second family can exhibit bimodality and its standardized fourth central moment (kurtosis) can be lower than that of the Azzalini skew normal distribution. Since the second proposed family can be bimodal we fit two well-known data set with this feature as applications. We concentrate attention on the case in which the normal distribution is the parent distribution but some consideration is given to other parent distributions, such as the logistic distribution.
Highlights
There are many situations in which empirical data show slight or marked asymmetry.This is frequently the case, for example, with actuarial and financial data which, in addition to this feature, have heavy tails reflecting the existence of extreme values
This family of distributions has been widely studied as an extension of the normal distribution by means of a shape parameter, λ, which accounts for the skewness
Because it is expressed in a simpler formulation, is the pdf obtained from Equation (6) when g and G are replaced by the pdf and cdf of the standard normal distribution, respectively
Summary
There are many situations in which empirical data show slight or marked asymmetry. This is frequently the case, for example, with actuarial and financial data which, in addition to this feature, have heavy tails reflecting the existence of extreme values. This family of distributions has been widely studied as an extension of the normal distribution by means of a shape parameter, λ, which accounts for the skewness. In this case g and G are replaced in Equation (1) by the pdf and cdf of the standard normal distribution and the resulting distribution is called the skew normal distribution. Due to the importance that the normal distribution plays in numerous problems of applied statistics we dedicate a complete section, Section 3, to the study of this distribution For this purpose, the pdf, which appears in closed form, is shown for the two families. Some conclusions are drawn and promising fields for further research are proposed in the last Section
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