11.7 Skewed Distribution

Motivations:
Incremental data has the following properties:

• Skewed right

• Occasionally negative

11.7.1 Skewed Normal Distribution

From Frühwirth-Schnatter and Pyne (2000)

\[\begin{equation} X = \mu + (\omega \cdot Z) \cdot \delta + (\omega \cdot \varepsilon) \cdot \sqrt{1 - \delta^2} \tag{11.9} \end{equation}\]

• \(\varepsilon \sim \mathcal{N}(0,1)\)

• \(Z \sim Truncated \: Normal_{[0,\infty]} (0,1)\)

• \(\mu\) is the location parameter

• \(\omega\) is the scale parameter, with \(\omega >0\)

• \(\delta\) is the shape parameter, with \(\delta \in (-1,1)\)

  Also weight between \(\varepsilon\) and \(Z\)

Max skewness = 0.995 from the truncated normal

\(\therefore\) Not used by the Meyers as it is not skewed enough

11.7.2 Mixed Lognormal-Normal

Mixed ln-n distribution with parameters \(\delta\), \(\mu\) and \(\sigma\)

\[\begin{equation} \begin{split} X \sim \mathcal{N}(Z, \delta) \\ Z \sim \ln \mathcal{N}(\mu, \sigma) \\ \end{split} \tag{11.10} \end{equation}\]

Pros

• Can create distribution more skew than the skewed normal

• Can also have negative values