scipy.stats._continuous_distns:dpareto_lognorm_gen

Notes

The probability density function for dpareto_lognorm is:

where $R(t)=\frac{1-\Phi (t)}{\varphi (t)}$, $\varphi$ and $\Phi$ are the normal PDF and CDF, respectively, $y_1=\alpha \sigma -\frac{\log x-\mu }{\sigma }$, and $y_2=\beta \sigma +\frac{\log x-\mu }{\sigma }$ for real numbers $x$ and $\mu$, $\sigma >0$, $\alpha >0$, and $\beta >0$ ^[1].

dpareto_lognorm takes u as a shape parameter for $\mu$, s as a shape parameter for $\sigma$, a as a shape parameter for $\alpha$, and b as a shape parameter for $\beta$.

A random variable $X$ distributed according to the PDF above can be represented as $X=U\frac{V_1}{V_2}$ where $U$, $V_1$, and $V_2$ are independent, $U$ is lognormally distributed such that $\log U\sim N(\mu ,{\sigma }^2)$, and $V_1$ and $V_2$ follow Pareto distributions with parameters $\alpha$ and $\beta$, respectively ^[2].

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