On application of the univariate Kotz distribution and some of its extensions

Abstract

Despite a flourishing activity, especially in recent times, for the study of flexible parametric classes of distributions, little work has dealt with the case where the tail weight and degree of peakedness is regulated by two parameters instead of a single one, as it is usually the case. The present contribution starts off from the symmetric distributions introduced by Kotz in 1975, subsequently evolved into the so-called Kotz-type distribution, and builds on their univariate versions by introducing a parameter which allows for the presence of asymmetry. We study some formal properties of these distributions and examine their practical usefulness in some real-data illustrations, considering both symmetric and asymmetric variants of the distributions.

Appendix: Some proofs

Proof of Proposition 2.2

To prove the first equation, consider

$$\begin{aligned} E\left( Z_{\rho ,G,w}^{2k+1}\right)= & {} \int _{-\infty }^{+\infty }2\,z^{2k+1}f_K(z;\rho )G\{w(z)\}\mathrm {d}{z}\nonumber \\= & {} \frac{\sqrt{\pi }2^{1-\rho }}{\Gamma \left( \rho -\frac{1}{2}\right) }\int _{-\infty }^{+\infty }2\,z\left( z^2\right) ^{k+\rho -1}\varphi (z) G\{w(z)\}\mathrm {d}{z}\nonumber \\= & {} \frac{\Gamma (\rho +k-\frac{1}{2})2^k}{\Gamma \left( \rho -\frac{1}{2}\right) } E\left( Z_{\rho +k,G,w}\right) . \end{aligned}$$

(30)

For the second one, we have

$$\begin{aligned} E\left( Z_{\rho +k,G,w}\right)= & {} \frac{\sqrt{\pi }2^{1-\rho -k}}{\Gamma \left( \rho +k-\frac{1}{2}\right) }\int _{-\infty }^{+\infty }2\,z\,|z|^{2\rho +2k-2}\varphi (z) G\{w(z)\}\mathrm {d}{z}\\= & {} \frac{\sqrt{\pi }2^{1-\rho -k}}{\Gamma \left( \rho +k-\frac{1}{2}\right) }\left( \int _{-\infty }^{0}2\,z\,(-z)^{2\rho +2k-2}\varphi (z) G\{w(z)\}\mathrm {d}{z}\right. \\&\left. +\int _{0}^{+\infty }2\,z\,z^{2\rho +2k-2}\varphi (z) G\{w(z)\}\mathrm {d}{z}\right) . \end{aligned}$$

By the change of variable $u=-z$ and the fact that $w(-u)=-w(u)$, we get

$$\begin{aligned} E\left( Z_{\rho +k,G,w}\right)= & {} \frac{\sqrt{\pi }2^{1-\rho -k}}{\Gamma \left( \rho +k-\frac{1}{2}\right) }\left( -\int _{0}^{+\infty }u^{2\rho +2k-1}2\varphi (u)\left( 1-G\{w(u)\}\right) \mathrm {d}{u}\right. \nonumber \\&\left. +\int _{0}^{+\infty }z^{2\rho +2k-1}2\varphi (z) G\{w(z)\}\mathrm {d}{z}\right) \nonumber \\= & {} \frac{\sqrt{\pi }2^{2-\rho -k}}{\Gamma \left( \rho +k-\frac{1}{2}\right) }\left( E\left\{ |U|^{2(\rho +k)-1}G\left( w\left( |U|\right) \right) \right\} -\frac{1}{2}E|U|^{2(\rho +k)-1}\right) \nonumber \\= & {} \frac{\sqrt{\pi }2^{2-\rho -k}}{\Gamma \left( \rho +k-\frac{1}{2}\right) }\left( E\left\{ |U|^{2(\rho +k)-1}G\left( w\left( |U|\right) \right) \right\} {-}\frac{1}{\sqrt{\pi }}\Gamma (k+\rho )2^{k+\rho -3/2}\right) , \nonumber \\ \end{aligned}$$

(31)

where $U\sim N(0,1)$; for the last equality sign, the moments of $U^2\sim \chi ^2_1$ have been used. Finally, substitution of (31) in (30) completes the proof.

Proof of Proposition 2.7

Substituting $F_K(\cdot ;\rho )$ instead of $G(\cdot )$ and $\lambda x$ instead of w(x) in (12) yields

$$\begin{aligned} E\left( Z_{\rho ,\lambda }^{*2k+1}\right)= & {} \frac{\sqrt{\pi }2^{2-\rho }}{\Gamma \left( \rho -\frac{1}{2}\right) }\left( \frac{1}{2}E|U|^{2(\rho +k)-1}+\frac{\mathrm {sign}(\lambda )}{2\Gamma \left( \rho -\frac{1}{2}\right) }\right. \\&\times \left. E\left\{ |U|^{2(\rho +k)-1}\gamma \left( \rho -\frac{1}{2},\frac{\lambda ^2U^2}{2}\right) \right\} \right) -\frac{\Gamma (\rho +k)2^{k+\frac{1}{2}}}{\Gamma \left( \rho -\frac{1}{2}\right) }\\= & {} \frac{\mathrm {sign}(\lambda )\sqrt{\pi }2^{2-\rho }}{2\Gamma ^2\left( \rho -\frac{1}{2}\right) }\int ^{+\infty }_{0}u^{2(\rho +k)-1}\gamma \left( \rho -\frac{1}{2},\frac{\lambda ^2u^2}{2}\right) 2\varphi (u)\mathrm {d}{u}. \end{aligned}$$

By the change of variable $t=\frac{\lambda ^2u^2}{2}$, we conclude that

$$\begin{aligned} E\left( Z_{\rho ,\lambda }^{*2k+1}\right)= & {} \frac{\mathrm {sign}(\lambda )2^{k+\frac{1}{2}}}{\Gamma ^2\left( \rho -\frac{1}{2}\right) \lambda ^{2(\rho +k)}}\int ^{+\infty }_{0}t^{\rho +k-1}\gamma \left( \rho -\frac{1}{2},t\right) e^{-\frac{t}{\lambda ^2}}\mathrm {d}{u}\\= & {} \frac{\mathrm {sign}(\lambda )2^{k+\frac{1}{2}}}{\Gamma ^2\left( \rho -\frac{1}{2}\right) \lambda ^{2(\rho +k)}}\frac{\Gamma \left( 2\rho +k-\frac{1}{2}\right) }{\left( \rho -\frac{1}{2}\right) \left( 1+\frac{1}{\lambda ^2}\right) ^{2\rho +k-\frac{1}{2}}} \\&\times ~{}_{2}F_1\left( 1,2\rho +k-\frac{1}{2};\rho +\frac{1}{2};\frac{\lambda ^2}{1+\lambda ^2}\right) , \end{aligned}$$

which completes the proof.