A cylindrical distribution with heavy-tailed linear part

Abstract

A cylindrical distribution whose linear part models heavy-tailedness is proposed. The conditional distribution of the linear variable given the circular variable is a generalized Pareto-type distribution. Therefore, it may not have any conditional moments; however, the mode and median have closed-form expressions. The circular marginal distribution is a wrapped Cauchy distribution, and the conditional distribution of the circular variable given the linear variable belongs to a family of symmetric distributions. These properties allow its application to cylindrical data, whose linear observations may take large values and whose circular observations are symmetric. As illustrative examples, the proposed distribution is fitted to two data sets, and the results are compared with those by other cylindrical distributions that cannot model heavy-tailedness for the linear parts.

Appendices

Appendix

A Generalized Gamma mixture

It is assumed that $(X, \varTheta )'|V \sim$ AL$(V^{-1}, \delta , \mu , \kappa )$ and $V \sim$ GGa$\left( \alpha , \beta , 1/\delta \right)$. Then the unconditional density $g(x, \theta )$ of $(X, \varTheta )'$ is derived as follows:

$$\begin{aligned} g(x, \theta )= & {} \int _0^{\infty } f_{\mathrm{Wei}}(x, \theta |v)f_\mathrm{GGa}(v) \mathrm{d}v\\= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \delta ^2} \frac{\beta ^{\alpha }}{\varGamma (\alpha )}x^{1/\delta -1} \int _{0}^{\infty } v^{\alpha /\delta + 1/\delta - 1} e^{ -v^{1/\delta } \left[ \beta + x^{1/\delta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] } \mathrm{d}v. \end{aligned}$$

The substitution $v=w^{\delta }$ yields

$$\begin{aligned} g(x, \theta )= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \delta ^2} \frac{\beta ^{\alpha }}{\varGamma (\alpha )}x^{1/\delta -1} \int _{0}^{\infty } w^{\alpha - \delta + 1} e^{ -w \left[ \beta + x^{1/\delta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] } \delta w^{\delta -1}\mathrm{d}w\\= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \delta } \frac{\beta ^{\alpha }}{\varGamma (\alpha )}x^{1/\delta -1} \int _{0}^{\infty } w^{\alpha } e^{ -w \left[ \beta + x^{1/\delta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] } \mathrm{d}w\\= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \delta } \frac{\beta ^{\alpha }}{\varGamma (\alpha )}x^{1/\delta -1} \frac{\varGamma (\alpha +1)}{\left[ \beta + x^{1/\delta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] ^{\alpha +1}}\\= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \delta } \frac{\alpha }{\beta } x^{1/\delta -1} \left[ 1 + \frac{x^{1/\delta }}{\beta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] ^{-(\alpha +1)}. \end{aligned}$$

Here, by replacing $\alpha =\delta /\tau$ and $\beta =\delta \sigma ^{1/\delta }/\tau$, it follows that $g(x, \theta )$ is equal to the proposed density (2).

B Modality

The first derivatives of the logarithm of the density (2) are given by

$$\begin{aligned} \begin{array}{l} \displaystyle \frac{\partial \log f_{\mathrm{GPar}}(x, \theta )}{\partial x} = \frac{1 - \delta }{ \delta x} - \frac{ x^{1/\delta -1} (\delta + \tau ) \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} }{ \delta \left[ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] }, \\ \\ \displaystyle \frac{\partial \log f_{\mathrm{GPar}}(x, \theta )}{\partial \theta } = - \frac{ \kappa x^{1/\delta } (\delta + \tau ) \sin (\theta - \mu ) }{ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} }. \end{array} \end{aligned}$$

When $\delta >1$, as there is no stationary point on $(0,\infty )\times [0,2\pi )$, and $\displaystyle \lim _{x \rightarrow 0} f_{\mathrm{GPar}}(x,\theta ) = \infty$, the density (2) is unimodal and the mode is taken on the line $x=0$.

When $\delta <1$, the stationary points of $\log f_{\mathrm{GPar}}(x, \theta )$ on $(0, \infty )\times [0, 2\pi )$ are

$$\begin{aligned} (x,\theta )=\left( \sigma \left\{ \frac{1-\delta }{(1-\kappa )(1+\tau )} \right\} ^{\delta }, \mu \right) =(x_1,\theta _1) \end{aligned}$$

and

$$\begin{aligned} (x,\theta )=\left( \sigma \left\{ \frac{1-\delta }{(1+\kappa )(1+\tau )} \right\} ^{\delta }, \mu +\pi \ (\mathrm{mod}\ 2\pi ) \right) =(x_2,\theta _2). \end{aligned}$$

The second derivatives of the logarithm of the density (2) are given by

$$\begin{aligned} \frac{\partial ^2 \log f_{\mathrm{GPar}}(x, \theta )}{\partial x^2}= & {} -\frac{1-\delta }{\delta ^2 x^2} \Biggl ( \delta + \frac{ x^{1/\delta } (\delta + \tau ) \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} }{ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} }\\&-\,\frac{ x^{2/\delta } \tau (\delta + \tau ) \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} ^2 }{ (1-\delta )\left[ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] ^2 } \Biggl ), \\ \frac{\partial ^2 \log f_{\mathrm{GPar}}(x, \theta )}{\partial \theta ^2}= & {} - \frac{ \kappa x^{1/\delta } (\delta + \tau ) \cos (\theta - \mu ) }{ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} }\\&+\, \frac{ \kappa ^2 x^{2/\delta } \tau (\delta + \tau ) \sin ^2 (\theta - \mu ) }{ \left[ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] ^2},\\ \frac{\partial ^2 \log f_{\mathrm{GPar}}(x, \theta )}{\partial x \partial \theta }= & {} - \frac{\kappa x^{1/\delta -1} (\delta + \tau ) \sin (\theta - \mu )}{ \delta \left[ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] }\\&+\, \frac{ \kappa x^{2/\delta -1} \tau (\delta + \tau ) \sin (\theta - \mu )\left\{ 1-\kappa \cos (\theta -\mu )\right\} }{ \delta \left[ \sigma ^{1/\delta } \delta + \tau x^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] ^2}, \\ \end{aligned}$$

and thus, the Hessian matrices at $(x,\theta )=(x_1,\theta _1)$ and at $(x,\theta )=(x_2,\theta _2)$ are given by

$$\begin{aligned} H_1= \left( \begin{array}{cc} -\frac{(1-\delta )(1+\tau )}{\delta (\delta +\tau ) x_1^2} &{} 0 \\ 0 &{} -\frac{\kappa (1-\delta )}{1-\kappa } \end{array}\right) \quad \text{ and }\quad H_2= \left( \begin{array}{cc} -\frac{(1-\delta )(1+\tau )}{\delta (\delta +\tau ) x_2^2} &{} 0 \\ 0 &{} \frac{\kappa (1-\delta )}{1+\kappa } \end{array}\right) , \end{aligned}$$

respectively. As $\sigma >0$, $\tau >0$, and $0 \le \kappa <1$, the matrix $H_1$ is negative-definite and the matrix $H_2$ is positive-definite. Therefore, the density (2) is unimodal, and the mode is taken at $(x, \theta ) = (x_1,\theta _1)$.

When $\delta =1$, as there is no stationary point, and $f_\mathrm{GPar}(0, \theta _1)=f_{\mathrm{GPar}}(0, \theta _2)$ and

$$\begin{aligned} f_{\mathrm{GPar}}(0,\theta _1)-f_\mathrm{GPar}(x,\theta _2)=\frac{\sqrt{1-\kappa ^2}}{2 \pi \sigma } \left[ 1-\left[ \frac{1}{1+\tau x\left\{ 1-\kappa \cos (\theta _2-\mu )/\sigma \right\} } \right] ^{1/\tau +1}\right] >0 \end{aligned}$$

for arbitrary $\theta _1$, $\theta _2$, and $x>0$, the density (2) is unimodal, and the mode is taken on the line $x=0$.

C Calculations about marginal distributions

The marginal density $f_{\varTheta }(\theta )$ of $\varTheta$ is obtained as follows:

$$\begin{aligned} f_{\varTheta }(\theta )= & {} \int _{0}^{\infty } f_{\mathrm{GPar}}(x,\theta ) \mathrm{d}x \\= & {} \frac{ \sqrt{1-\kappa ^2} }{2\pi \sigma \delta } \int _{0}^{\infty } \left( \frac{x}{\sigma } \right) ^{1/\delta -1} \left[ 1+\frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] ^{-(\delta /\tau +1)} \mathrm{d}x. \end{aligned}$$

The substitution $x=\sigma (\delta y/\tau )^{\delta }$ yields

$$\begin{aligned} f_{\varTheta }(\theta ) = \frac{ \sqrt{1 - \kappa ^2} }{2\pi } \int _{0}^{\infty } \left[ 1+y \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] ^{-(\delta /\tau +1)} \mathrm{d}y = \frac{ \sqrt{1 - \kappa ^2} }{ 2 \pi \{ 1-\kappa \cos (\theta - \mu ) \} } \end{aligned}$$

by the integral formula $\int _0^{\infty }(1+\alpha y)^{-(\beta +1)}\mathrm{d}y=1/(\alpha \beta )$ for $\alpha , \beta >0$.

As the density (2) is expressed by

$$\begin{aligned} f_{\mathrm{GPar}}(x, \theta )= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2 \pi \sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta -1} \left\{ 1 + \frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \right\} ^{-(\delta /\tau +1)} \\&\times \, \left\{ 1 - \frac{ \kappa \tau (x/\sigma )^{1/\delta } }{ \delta + \tau (x/\sigma )^{1/\delta } } \cos (\theta -\mu ) \right\} ^{-(\delta /\tau +1)} \end{aligned}$$

and the combination of Equations 9.112 and 9.134.2 in Gradshteyn and Ryzhik (2007) yields

$$\begin{aligned} \frac{1}{2\pi }\int _0^{2\pi }\frac{\mathrm{d}t}{(1-z \cos t)^p} ={}_2F_1\left( \frac{p}{2},\frac{p+1}{2};1;z^2 \right) , \quad |z|<1, \end{aligned}$$

the marginal density $f_X(x)$ of X is

$$\begin{aligned} f_{X}(x)= & {} \int _{0}^{2\pi } f_{\mathrm{GPar}}(x,\theta ) \mathrm{d}\theta \\= & {} \frac{ \sqrt{1-\kappa ^2} }{ 2 \pi \sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta -1} \left\{ 1 + \frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \right\} ^{-(\delta /\tau +1)}\\&\times \, \int _{0}^{2\pi }\left\{ 1 - \frac{ \kappa \tau (x/\sigma )^{1/\delta } }{ \delta + \tau (x/\sigma )^{1/\delta } } \cos (\theta -\mu ) \right\} ^{-(\delta /\tau +1)} \mathrm{d}\theta \\= & {} \frac{\sqrt{1-\kappa ^2}}{\sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta - 1} \left\{ 1 + \frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \right\} ^{-(\delta /\tau + 1)} \\&\times \, {}_2F_1 \left( \frac{\delta /\tau + 1}{2}, \frac{\delta /\tau }{2}+1; 1; \left( \frac{\kappa (x /\sigma )^{1/\delta }}{\delta /\tau + (x/\sigma )^{1/\delta }} \right) ^2 \right) . \end{aligned}$$

As

$$\begin{aligned}&{}_2F_1 \left( \frac{\delta /\tau + 1}{2}, \frac{\delta /\tau }{2}+1; 1; \left( \frac{\kappa (x/\sigma )^{1/\delta }}{\delta /\tau + (x/\sigma )^{1/\delta }} \right) ^2 \right) \\&\quad = 1 + \sum _{k=1}^{\infty } \frac{\{\delta /(2\tau ) + k - 1/2\} \cdots \{\delta /(2\tau ) + 1/2\}}{ \{\delta /\tau + (x/\sigma )^{1/\delta }\}^k } \frac{\{\delta /(2\tau ) + k - 1\} \cdots \{\delta /(2\tau ) + 1\}}{ \{\delta /\tau + (x/\sigma )^{1/\delta }\}^k } \frac{ \{\kappa (x/\sigma )^{1/\delta }\}^{2k} }{(k!)^2} \\&\quad \rightarrow \sum _{k=0}^{\infty } \frac{ \{\kappa (x/\sigma )^{1/\delta }\}^{2k} }{2^{2k} (k!)^2} = I_0 \left( \kappa \left( \frac{x}{\sigma } \right) ^{1/\delta } \right) ,\quad \text{ as }\quad \tau \rightarrow 0, \end{aligned}$$

it is confirmed that

$$\begin{aligned} f_X(x) \rightarrow \frac{\sqrt{1-\kappa ^2}}{\sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta - 1} e^{ - (x/\sigma )^{1/\delta } } I_0 \left( \kappa \left( \frac{x}{\sigma } \right) ^{1/\delta } \right) , \quad \text{ as }\quad \tau \rightarrow 0. \end{aligned}$$

D Joint moments

It is assumed that $(X, \varTheta )'|V \sim$ AL$(V^{-1}, \delta , \mu , \kappa )$ and $V \sim$ GGa$\left( \alpha , \beta , 1/\delta \right)$. Then, as $\mathrm{E} \left[ X^p e^{i q \varTheta } | V=v \right] =v^{-p} A_{p, q} e^{i q \mu }$ and the unconditional distribution of $(X, \varTheta )'$ is the proposed distribution (2), the joint moments are obtained by

$$\begin{aligned} \mathrm{E} \left[ X^p e^{i q \varTheta } \right]= & {} \int _0^{\infty } \mathrm{E} \left[ X^p e^{i q \varTheta } | V=v \right] f_{\mathrm{GGa}}(v) \mathrm{d}v \\= & {} A_{p, q} e^{i q \mu } \int _0^{\infty } \frac{\beta ^{\alpha }}{\varGamma (\alpha ) \delta } v^{\alpha /\delta - p -1} e^{-\beta v^{1/\delta }} \mathrm{d}v \\= & {} A_{p, q} e^{i q \mu } \int _0^{\infty } \frac{\beta ^{\alpha }}{\varGamma (\alpha )} w^{\alpha -p \delta -1} e^{-\beta w} \mathrm{d}w \\= & {} A_{p, q} \beta ^{p\delta }\frac{\varGamma \left( \alpha - p \delta \right) }{\varGamma (\alpha )} e^{i q \mu },\quad \alpha -p\delta >0. \end{aligned}$$

In the third equation, the substitution $v=w^{\delta }$ was made. By replacing $\alpha =\delta /\tau$ and $\beta =\delta \sigma ^{1/\delta }/\tau$, the above joint moments yield (9).