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4.8 Log Normal, Log Laplace, Log S and Log GN distributions

In addition, it is possible to derive the log-versions of the Normal, \(\mathcal{Laplace}\), \(\mathcal{S}\), and \(\mathcal{GN}\) distributions. The main differences between the original and the log-versions of density functions for these distributions can be summarised as follows: \[\begin{equation} f_{log}(\log(y_t)) = \frac{1}{y_t} f(\log y_t). \tag{4.21} \end{equation}\] They are defined for positive values only and will have different right tail, depending on the location, scale and shape parameters. \(\exp(\mu_{\log y,t})\) in this case represents the geometric mean (and median) of distribution rather than the arithmetic one. The conditional expectation in these distributions is typically higher than \(\exp(\mu_{\log y,t})\) and depends on the value of the scale parameter. It is known for log\(\mathcal{N}\) and is equal to: \[\begin{equation} \mathrm{E}(y_t) = \mathrm{exp}\left(\mu_{\log y,t} + \frac{\sigma^2}{2} \right). \tag{4.22} \end{equation}\] However, it does not have a simple form for the other distributions.