chapter  7
18 Pages

## From Symmetric to Asymmetric Distributions: A Uniﬁed Approach

WithReinaldo B. Arellano-Valle and Guido E. del Pino

Asymmetric (skewed) distributions are useful in probability modeling, statistical analysis, and robustness studies. This chapter presents a uniﬁed approach to generating them, starting from a given family of symmetric distributions, either univariate or multivariate. The emphasis here lies on general methods to achieve the following aims:

• Deﬁne a skewed distribution. • Find its probability density function (pdf). • Provide a stochastic representation if possible. • Simulate random variables from the given distribution. Location and scale families occupy a predominant place in statistical

modeling and analysis, and this will also be the case here. Any such family may be generated by applying the following steps: • Pick a cumulative distribution function (cdf) F and let X be a random

variable with X ∼ F. • Let Y = θ0 + θ1X, where −∞ < θ0 <∞, θ1 > 0. • Let Fθ0,θ1 be the cdf of Y. • Deﬁne the family F = {Fθ0,θ1 : −∞ < θ0 <∞, θ1 > 0}. • With this notation, the generating distribution F is identiﬁed with F0,1. When the distributions in F are symmetric, θ0 may always be chosen as the center of symmetry, which is equivalent to F being symmetric about the origin. In order to simplify the writing, a symmetric distribution will be understood as symmetric about the origin, unless explicitly stated otherwise.