0
$\begingroup$

$$F_X(x)=\begin{cases} \quad\dfrac{\alpha}{\alpha+\theta}\left(\dfrac x\omega \right)^\theta &\text{ if } x<\omega \\ \\ 1-\dfrac{\theta}{\alpha+\theta}\left(\dfrac\omega x\right)^{\alpha} &\text{ if } x>\omega \end{cases}\quad\text{ where } 0\le x< +\infty.$$

I derived it by having $F_{X|Y}(x|y)=\left(\dfrac{x}{y}\right)^\theta$, which is $Y$ with a multiplicative "shock/noise" give by a $\mathrm{Beta}(\theta,1)$ (max of $\theta$ uniforms) and $Y$ follows a Pareto distribution $F_Y(y)=1-\left(\dfrac{\omega}{y}\right)^\alpha$.

I was not able to find a classification for it. Does it have a name/family?

$\endgroup$
1
+100
$\begingroup$

Rodney Kreps (a legend among casualty actuaries) calls this the "Split Simple Pareto" For example, see page 3 in Effects of Parameters of Tranformed Beta Distributions, (Venter 2003){pdf}. It is the Transformed Beta distribution as $\tau \to \infty$ (although Krep uses $\theta$ instead of $\omega$.

Some statistics: $$ E(X) = \omega\frac{\alpha\beta}{\alpha-1}\frac{1}{\beta+1}\\ E(X^2) = \omega^2\frac{\alpha\beta}{\alpha-2}\frac{1}{\beta+2}\\ $$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.