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A cylindrical distribution with heavy-tailed linear part

  • Tomoaki ImotoEmail author
  • Kunio Shimizu
  • Toshihiro Abe
Original Paper
  • 45 Downloads

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.

Keywords

Earthquake Generalized Pareto distribution Heavy-tailed distribution Wrapped Cauchy distribution 

Mathematics Subject Classification

60E05 62H11 

1 Introduction

A combination of wind direction and speed is a typical combination of circular and linear observations in environmental science and meteorology that form cylindrical data. Other examples of cylindrical observations include wind direction and concentration of pollutants in environmental science, acrophase and blood plasma in biological rhythms, and turning angles and step lengths in animal movement (Batschelet 1981; Fisher 1993; Mardia and Jupp 2000). More recent studies on animal orientation are Maruotti (2016), Nuñez-Antonio and Gutiérrez-Peña (2014) and Rivest et al. (2016), whereas studies on wave direction and height in marine research include Lagona et al. (2015), Wang and Gelfand (2014) and Wang et al. (2015). Applications to the ecological sciences are found in Aakala et al. (2016) and Abe and Shimatani (2018).

To model cylindrical data, several types of distributions on the cylinder, or cylindrical distributions, have been proposed. Existing classical and recent flexible cylindrical models include: the wrapping approach for the circular part in Johnson and Wehrly (1977), the von Mises-normal type in Mardia and Sutton (1978), the general construction with specified marginals and the von Mises-exponential type in Johnson and Wehrly (1978), the generalized von Mises-normal type in Kato and Shimizu (2008), and the sine-skewed von Mises–Weibull type in Abe and Ley (2017). More details on these models may be found in Sect. 2.4 of the recent book (Ley and Verdebout 2017).

However, almost all existing cylindrical distributions cannot model heavy-tailedness for the linear parts in the sense that the linear moment of some order is infinite (in Crovella and Taqqu 1999, heavy-tailedness is defined by infinite variance), whereas the (symmetric) Abe–Ley distribution with density
$$\begin{aligned} f_{\mathrm{Wei}}(x, \theta ) = \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta -1} \exp \left[ -\left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] \end{aligned}$$
(1)
for \(x > 0\) and \(0 \le \theta < 2\pi\), where \(\sigma > 0\), \(\delta > 0\), \(0 \le \mu < 2\pi\) and \(0 \le \kappa < 1\), can model heavy-tailed situations in the sense that the linear moment generating functions are always infinite (Foss et al. 2011). For linear data, there exist several examples of heavy-tailed phenomena, such as insurance losses and returns in financial data, precipitation in climatological data, and teletransmission and Internet activity in network data. Potential applications are combinations of heavy-tailed linear variable and periodic variable. If light-tailed, or equivalently not heavy-tailed, cylindrical distributions for the linear part are applied to such data, the estimation and test may be biased by large linear observations, thus leading to wrong results.

With this background as motivation, a heavy-tailed cylindrical distribution for the linear part is proposed. One may use specific heavy-tailed marginal distributions for the linear part in the general construction by Johnson and Wehrly (1978). However, in this study, another approach is adopted, namely, a dependence structure of the form \((x/\sigma )^{1/\delta }\{ 1-\kappa \cos (\theta -\mu ) \}\) between the linear variable x and the circular variable \(\theta\) is considered and a Gamma mixture is applied, because a generalized Pareto distribution, as a heavy-tailed distribution, can be obtained by a Gamma mixture of an exponential distribution. Moreover, the proposed cylindrical distribution is a Pareto type for the linear part that contains exponential and Weibull types as submodels.

The remainder of the paper is organized as follows. In Sect. 2, a cylindrical distribution whose linear conditional distribution is a generalized Pareto-type distribution is proposed, its properties, particularly the behavior of the density, are presented, and the marginal and conditional distributions are provided. In Sect. 3, the joint and conditional moments are given, the association between linear and circular variables is studied, and linear–circular and circular–linear regressions are treated. In Sect. 4, random number generation is considered, and a simulation study is conducted. In Sect. 5, illustrative fittings of two data sets are presented. One consists of the direction of motion of blue periwinkles after being experimentally transplanted downshore and the distance traveled by them, and the other consists of the epicenter turning angle and the magnitude during a period of 72 h before the 2011 Great East Japan Earthquake. In the fittings, likelihood techniques are used to compare the fit of the proposed distribution with those of the other cylindrical distributions, including the Abe–Ley distribution. Finally, in Sect. 6, the properties of the proposed model are briefly summarized, and the advantage of the heavy-tailed cylindrical distribution for the linear part is discussed.

Proofs and further details are in Appendix.

2 Generalized Pareto-type cylindrical distribution

2.1 Joint probability density function

For \(\sigma > 0\), \(\delta > 0\), \(\tau > 0\), \(0 \le \mu < 2\pi\), and \(0 \le \kappa < 1\), the following function is the joint probability density function of a random vector \((X, \varTheta )'\) on the cylinder:
$$\begin{aligned} f_{\mathrm{GPar}}(x, \theta ) = \frac{ \sqrt{1-\kappa ^2} }{2\pi \sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta -1} \left[ 1+\frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] ^{-(\delta /\tau +1)} \end{aligned}$$
(2)
for \(x > 0\) and \(0 \le \theta < 2\pi\). This distribution can be obtained through a generalized Gamma mixture of the Abe–Ley distribution, that is, the unconditional distribution \((X, \varTheta )'\) has the density (2) when \((X, \varTheta )'|V \sim\) AL\((V^{-1}, \delta , \mu , \kappa )\) and \(V \sim\) GGa\(\left( \delta /\tau , \delta \sigma ^{1/\delta }/\tau , 1/\delta \right)\), where AL\((\sigma , \delta , \mu , \kappa )\) denotes the Abe–Ley distribution with density (1), and GGa\((\alpha , \beta , \gamma )\) denotes the generalized Gamma distribution with density:
$$\begin{aligned} f_{\mathrm{G}}(v) = \frac{\beta ^{\alpha } \gamma }{\varGamma (\alpha )} v^{\alpha \gamma -1} e^{-\beta v^{\gamma }},\quad v>0. \end{aligned}$$
From this derivation and parameter interpretation of the Abe–Ley distribution (Abe and Ley 2017), the parameter interpretation of the proposed distribution (2) is as follows. The parameter \(\mu\) is a circular location parameter, and the distribution is symmetric about \(\theta =\mu\) for fixed x. The parameter \(\kappa\) plays the role of circular concentration and circular–linear dependence. When \(\kappa = 0\) in (2), X is independent of \(\varTheta\). Then X follows a generalized Pareto-type distribution, and \(\varTheta\) follows a circular uniform distribution. The parameters \(\sigma\) and \(\delta\) control the linear scale and shape, respectively, and \(\tau\) controls the linear shape and tail behavior of the linear part. As \(\tau \rightarrow 0\), the proposed distribution (2) reduces to the Abe–Ley distribution (1). As a special case, when \(\delta =1\), the proposed distribution has a simple density
$$\begin{aligned} f_{\mathrm{Par}}(x, \theta ) = \frac{ \sqrt{1-\kappa ^2} }{2\pi \sigma } \left[ 1+\frac{\tau x}{\sigma } \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] ^{-(1/\tau +1)}, \end{aligned}$$
which is constructed from a Gamma mixture of the von Mises-exponential type distribution in Johnson and Wehrly (1978):
$$\begin{aligned} f_{\mathrm{Exp}}(x, \theta ) = \frac{ \sqrt{1-\kappa ^2} }{ 2\pi \sigma } \exp \left[ -\frac{x}{\sigma }\left\{ 1- \kappa \cos (\theta - \mu ) \right\} \right] . \end{aligned}$$
(3)

2.2 Behavior of the proposed density

The proposed distribution (2) is unimodal, and the mode is taken on the line \(x=0\) when \(\delta \ge 1\) and at \((x, \theta ) = \left( \sigma \left\{ \frac{1-\delta }{(1-\kappa )(1+\tau )} \right\} ^{\delta }, \mu \right)\) when \(\delta < 1\). Contour plots of the density (2) are shown in Fig. 1. The parameters \(\mu\) and \(\kappa\) are fixed at \(\mu =0\) and \(\kappa =\tanh (1)\). For comparison with a symmetric Abe–Ley distribution, Fig. 1 also includes the plot of the density (1) as (a), (d), and (g). From these figures, it can be seen that the tail of the density becomes heavy as \(\tau\) increases.
Fig. 1

Contour plots of the proposed density (2) over \((0, 5) \times [0, 2\pi )\) for \((\sigma , \mu , \kappa )=(1, \pi , \tanh (1))\). The setting of parameters \((\delta , \tau )\) is given in the panels

2.3 Marginal distributions

The marginal density of the circular variable \(\varTheta\) is
$$\begin{aligned} f_{\varTheta }(\theta ) = \frac{ \sqrt{1 - \kappa ^2} }{ 2 \pi \{ 1-\kappa \cos (\theta - \mu ) \} }, \end{aligned}$$
(4)
which is a wrapped Cauchy distribution WC\(\left( \mu ,\kappa /\left( 1+\sqrt{1-\kappa ^2}\right) \right)\) using the usual notation (Mardia and Jupp 2000) with mean direction \(\mu\) and mean resultant length \(\kappa /\left( 1+\sqrt{1-\kappa ^2}\right)\). The marginal density of the linear variable X is expressed by
$$\begin{aligned} f_X(x)= & {} \frac{\sqrt{1-\kappa ^2}}{\sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta - 1} \left\{ 1 + \frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \right\} ^{-(\delta /\tau + 1)} \\&\times \, {}_2F_1 \left( \frac{\delta /\tau + 1}{2}, \frac{\delta /\tau }{2}+1; 1; \left( \frac{\kappa (x /\sigma )^{1/\delta }}{\delta /\tau + (x/\sigma )^{1/\delta }} \right) ^2 \right) , \end{aligned}$$
where \({}_2F_1( \cdot , \cdot ; \cdot ; \cdot )\) denotes the Gauss hypergeometric function. It is confirmed that as \(\tau \rightarrow 0\)
$$\begin{aligned} f_X(x) \rightarrow \frac{\sqrt{1-\kappa ^2}}{\sigma \delta } \left( \frac{x}{\sigma } \right) ^{1/\delta - 1} e^{ - (x/\sigma )^{1/\delta } } I_0 \left( \kappa \left( \frac{x}{\sigma } \right) ^{1/\delta } \right) , \end{aligned}$$
where \(I_p(\cdot )\) is the modified Bessel function of the first kind and order p defined by
$$\begin{aligned} I_p(\kappa )=\frac{1}{2\pi }\int _0^{2\pi } \cos (p\theta ) e^{\kappa \cos \theta } \mathrm{d}\theta . \end{aligned}$$
The limiting distribution corresponds to the linear marginal distribution of the Abe–Ley distribution (1).

2.4 Conditional distributions

By Sect. 2.3, the conditional distribution of X given \(\varTheta = \theta\) is a generalized Pareto-type distribution with density:
$$\begin{aligned} f_{X|\varTheta }(x|\theta )= & {} \frac{1}{\sigma \delta }\left( \frac{x}{\sigma } \right) ^{1/\delta - 1} \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \nonumber \\&\quad \times \left[ 1+\frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] ^{-(\delta /\tau +1)} \end{aligned}$$
(5)
and distribution function:
$$\begin{aligned} F_{X|\varTheta }(x|\theta ) = 1 - \left[ 1 + \frac{\tau }{\delta } \left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] ^{-\delta /\tau }, \end{aligned}$$
(6)
which corresponds to the distribution of a random variable X that is the scale-power transform of the generalized Pareto random variable Y, i.e., \(X=(Y/\sigma )^{1/\delta }\). As will be seen in Sect. 3.3, the moment of a certain order of this distribution is infinite, and thus, this conditional distribution is heavy-tailed. When \(\tau \ge 1\), the conditional mean is infinite; hence, it may be useful to use the mode
$$\begin{aligned} \mathrm{Mode}[X|\varTheta = \theta ] = \left\{ \begin{array}{ll} \displaystyle \sigma \left\{ \frac{1-\delta }{(1+\tau ) (1 - \kappa \cos (\theta - \mu ))} \right\} ^{\delta }, &{}\quad \delta < 1, \\ \displaystyle 0, &{}\quad \delta \ge 1, \end{array} \right. \end{aligned}$$
and the median
$$\begin{aligned} \mathrm{Median}[X|\varTheta = \theta ] = \sigma \left\{ \frac{\delta }{\tau } \frac{2^{\tau /\delta } - 1}{1 - \kappa \cos (\theta -\mu )} \right\} ^{\delta } \end{aligned}$$
(7)
to represent the relation between the linear random variable X and circular random variable \(\varTheta\). Figure 2 shows the curves of \(\mathrm{Median}[X|\varTheta = \theta ]\) for varying \(\tau\). It can be seen that the median increases as \(\tau\) increases.
Fig. 2

Plot of the conditional median \(\mathrm{Median}[X|\varTheta = \theta ]\) when \((\sigma , \delta ,\mu ,\kappa )=(1, 0.8, 0, 0.8)\) with \(\tau =0.5\) (solid), \(\tau =1.0\) (dashed), \(\tau =1.5\) (dot–dashed) and \(\tau =2.0\) (dotted)

It is easily seen that as \(\tau \rightarrow 0\),
$$\begin{aligned} f_{X|\varTheta }(x|\theta ) \rightarrow \frac{1}{\sigma \delta }\left( \frac{x}{\sigma } \right) ^{1/\delta - 1} \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \exp \left[ -\left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta -\mu ) \right\} \right] \end{aligned}$$
and
$$\begin{aligned} F_{X|\varTheta }(x|\theta ) \rightarrow 1 - \exp \left[ - \left( \frac{x}{\sigma } \right) ^{1/\delta } \left\{ 1 - \kappa \cos (\theta - \mu ) \right\} \right] , \end{aligned}$$
which are the conditional density function and distribution function, respectively, of the Weibull distribution. In this case, all moments are finite.
The conditional distribution of \(\varTheta\) given \(X = x\) belongs to the Pearson type VII distribution on the circle (Shimizu and Iida 2002), namely
$$\begin{aligned} f_{\varTheta |X}(\theta |x) = \frac{ \left\{ 1 - \alpha (x) \cos (\theta - \mu ) \right\} ^{-(\delta /\tau + 1)}}{2\pi {}_2F_1 \left( \delta +\tau /(2 \tau ), \delta /(2 \tau ) + 1; 1; \alpha (x)^2 \right) }, \end{aligned}$$
(8)
where \(\alpha (x) = \frac{\kappa \tau (x/\sigma )^{1/\delta }}{\delta + \tau (x/\sigma )^{1/\delta }}\). This distribution reduces to the von Mises distribution:
$$\begin{aligned} f_{\varTheta |X}(\theta |x) \rightarrow \frac{\exp \left\{ \kappa (x/\sigma )^{1/\delta } \cos (\theta -\mu ) \right\} }{2\pi I_0(\kappa (x/\sigma )^{1/\delta })} \end{aligned}$$
as \(\tau \rightarrow 0\), and to the wrapped Cauchy distribution
$$\begin{aligned} f_{\varTheta |X}(\theta |x) = \frac{\sqrt{1-\kappa ^2}}{2\pi \{1 - \kappa \cos (\theta - \mu )\}} \end{aligned}$$
as \(\tau \rightarrow \infty\). It is noted that when \(\tau \rightarrow \infty\), the joint distribution with density (2) tends to the uniform distribution on \((0, \infty ) \times [0, 2\pi )\), which is an improper distribution, but its circular conditional distribution becomes a proper distribution.

3 Moments

3.1 Joint moments

The joint moments for the proposed distribution (2) are expressed by
$$\begin{aligned} \mathrm{E} \left[ X^p e^{i q \varTheta } \right] = A_{p, q} \sigma ^p \left( \frac{\delta }{\tau } \right) ^{p \delta } \frac{\varGamma \left( \delta /\tau - p \delta \right) }{\varGamma (\delta /\tau )} e^{i q \mu },\quad 0 \le p < \frac{1}{\tau }, \end{aligned}$$
(9)
where
$$\begin{aligned} A_{p,q} = \left( \frac{\kappa }{2} \right) ^q \frac{\sqrt{1 - \kappa ^2} \varGamma (p \delta + q + 1)}{\varGamma (q + 1)} {}_2F_1 \left( \frac{p \delta + q + 1}{2}, \frac{p\delta + q}{2} + 1; q+1; \kappa ^2 \right) \end{aligned}$$
and \(i=\sqrt{-1}\). From this expression, the marginal moments of X and \(\varTheta\) are obtained as follows:
$$\begin{aligned} \mathrm{E} [X^p]= & {} \sqrt{1-\kappa ^2} \sigma ^p \left( \frac{\delta }{\tau } \right) ^{p \delta } \frac{\varGamma (\delta /\tau - p \delta ) \varGamma (p \delta + 1)}{\varGamma (\delta /\tau )} {}_2F_1 \left( \frac{p \delta +1}{2}, \frac{p \delta }{2}+1; 1; \kappa ^2 \right) \\&0 \le p <\frac{1}{\tau } \end{aligned}$$
and
$$\begin{aligned} \mathrm{E} [e^{i q \varTheta }]= & {} \sqrt{1-\kappa ^2} \left( \frac{\kappa }{2} \right) ^q {}_2F_1 \left( \frac{q+1}{2},\frac{q}{2}+1;q+1;\kappa ^2 \right) e^{i q \mu } \\= & {} \left( \frac{\kappa }{1+\sqrt{1-\kappa ^2}} \right) ^q e^{i q \mu }. \end{aligned}$$
The last equation follows from the identity \({}_2F_1(a,1/2+a;2a;z)=2^{2a-1}(1-z)^{-1/2}\{ 1+\sqrt{1-z} \}^{1-2a}\). As the \(\left( [1/\tau ]+1\right)\)th marginal moment of X is infinite, the proposed distribution (2) is heavy-tailed for the linear part.

3.2 Circular–linear correlation

As
$$\begin{aligned} \begin{array}{ll} \displaystyle \mathrm{E}[X]=\sigma A_{1,0}\left( \frac{\delta }{\tau }\right) ^{\delta }\frac{\varGamma (\delta /\tau -\delta )}{\varGamma (\delta /\tau )}, &{} \displaystyle \mathrm{E}[X^2]=\sigma ^2 A_{2,0}\left( \frac{\delta }{\tau }\right) ^{2\delta }\frac{\varGamma (\delta /\tau -2\delta )}{\varGamma (\delta /\tau )},\\ \displaystyle \mathrm{E}[e^{i\varTheta }]=A_{0,1} e^{i\mu },\quad \mathrm{E}[e^{2i \varTheta }]=A_{0,2} e^{2i\mu }, &{} \displaystyle \mathrm{E}[Xe^{i\varTheta }]=\sigma A_{1,1}\left( \frac{\delta }{\tau }\right) ^{\delta }\frac{\varGamma (\delta /\tau -\delta )}{\varGamma (\delta /\tau )}e^{i\mu } \end{array} \end{aligned}$$
for \(0<\tau <1/2\), it follows that
$$\begin{aligned} \begin{array}{l} \displaystyle \mathrm{Var}[X]=\frac{\sigma ^2}{\varGamma (\delta /\tau )^2} \left( \frac{\delta }{\tau }\right) ^{2\delta } \left\{ A_{2,0}\varGamma (\delta /\tau )\varGamma (\delta /\tau -2\delta ) - A_{1,0}^2\varGamma (\delta /\tau -\delta )^2 \right\} ,\\ \displaystyle \mathrm{Var}[\cos \varTheta ]=\mathrm{Var}[\sin \varTheta ]=\frac{1-A_{0,2}}{2},\\ \displaystyle \mathrm{Cov}[X, e^{i\varTheta }]=\sigma \left( \frac{\delta }{\tau }\right) ^{\delta }\frac{\varGamma (\delta /\tau -\delta )}{\varGamma (\delta /\tau )} \left( A_{1,1}-A_{1,0}A_{0,1}\right) e^{i\mu },\\ \mathrm{Cov}[\cos \varTheta , \sin \varTheta ]=0. \end{array} \end{aligned}$$
These expressions lead to the correlations:
$$\begin{aligned} r_{XC}= & {} \mathrm{Corr}[X, \cos \varTheta ] \\= & {} \frac{\sqrt{2}\varGamma (\delta /\tau -\delta ) \left( A_{1,1}-A_{1,0}A_{0,1}\right) \cos \mu }{\sqrt{(1-A_{0,2})\left\{ A_{2,0}\varGamma (\delta /\tau )\varGamma (\delta /\tau -2\delta ) - A_{1,0}^2\varGamma (\delta /\tau -\delta )^2 \right\} }},\\ r_{XS}= & {} \mathrm{Corr}[X, \sin \varTheta ] \\= & {} \frac{\sqrt{2}\varGamma (\delta /\tau -\delta ) \left( A_{1,1}-A_{1,0}A_{0,1}\right) \sin \mu }{\sqrt{(1-A_{0,2})\left\{ A_{2,0}\varGamma (\delta /\tau )\varGamma (\delta /\tau -2\delta ) - A_{1,0}^2\varGamma (\delta /\tau -\delta )^2 \right\} }},\\ r_{CS}= & {} \mathrm{Corr}[\cos \varTheta , \sin \varTheta ]=0. \end{aligned}$$
The circular–linear correlation \(R_{X\varTheta }^2\), proposed in Johnson and Wehrly (1977) and Mardia (1976), can then be obtained as
$$\begin{aligned} R_{X\varTheta }^2= & {} \frac{r_{XC}^2+r_{XS}^2-2r_{CS}r_{XC}r_{XS}}{1-r_{CS}^2}\\= & {} \frac{2\varGamma (\delta /\tau -\delta )^2 \left( A_{1,1}-A_{1,0}A_{0,1}\right) ^2}{(1-A_{0,2})\left\{ A_{2,0}\varGamma (\delta /\tau )\varGamma (\delta /\tau -2\delta ) - A_{1,0}^2\varGamma (\delta /\tau -\delta )^2 \right\} }. \end{aligned}$$
The influence of \(\delta\), \(\tau\), and \(\kappa\) is shown by the contour plots in Fig. 3, where it is seen that the correlation \(R_{X\varTheta }^2\) decreases as \(\delta\) or \(\tau\) increases or \(\kappa\) decreases.
Fig. 3

Contour plots of the circular–linear correlation \(R_{X\varTheta }^2\) as a function of \((\delta ,\kappa )\) over \((0,2.5)\times (0,1)\). The setting of the parameter \(\tau\) is given in the panels

3.3 Conditional moments

As the conditional distribution of X given \(\varTheta =\theta\) is the generalized Pareto-type distribution (5), the conditional moments are given by
$$\begin{aligned} \mathrm{E}[X^p |\varTheta =\theta ] = \frac{\sigma ^p \varGamma (\delta /\tau - p \delta ) \varGamma (p \delta + 1) }{ [ \{ 1 - \kappa \cos (\theta - \mu ) \} \tau /\delta ]^{p \delta } \varGamma (\delta /\tau )},\quad 0 \le p <\frac{1}{\tau }. \end{aligned}$$
(10)
It is easily seen that as \(\tau \rightarrow 0\)
$$\begin{aligned} \mathrm{E}[X^p|\varTheta =\theta ] \rightarrow \frac{\sigma ^p \varGamma (p \delta + 1)}{\{ 1-\kappa \cos (\theta -\mu )\}^{p \delta }},\quad p \ge 0. \end{aligned}$$
As the conditional distribution of \(\varTheta\) given \(X=x\) belongs to the Pearson type VII distribution on the circle (8), the conditional moments are given by
$$\begin{aligned} \mathrm{E}[e^{i q \varTheta } |X=x]= & {} \left( \begin{array}{c} \delta /\tau +q \\ \delta /\tau \end{array} \right) \left( \frac{1-\sqrt{1-\alpha (x)^2}}{\alpha (x)} \right) ^{\delta /\tau +q+1} \left( \frac{2}{\alpha (x)}\right) ^{\delta /\tau +1}\\&\times \, \frac{ {}_2F_1 \left( \delta /\tau +1, \delta /\tau +q+1; q+1; (2-2\sqrt{1-\alpha (x)^2})/\alpha (x)^2 -1 \right) }{ {}_2F_1 \left( (\delta +\tau )/(2 \tau ), \delta /(2 \tau )+1; 1; \alpha (x)^2 \right) } e^{i q \mu }, \end{aligned}$$
where \(\alpha (x)\) is the same function as in Sect. 2.4. Using the identity
$$\begin{aligned} \frac{e^{i\theta }}{\{1-\alpha \cos \theta \}^{\beta +1}} = \frac{1}{\alpha \{1-\alpha \cos \theta \}^{\beta +1}} - \frac{1}{\alpha \{1-\alpha \cos \theta \}^{\beta }} +\frac{i\sin \theta }{\{1-\alpha \cos \theta \}^{\beta +1}}, \end{aligned}$$
an alternative expression of the moment for \(q=1\) is obtained as follows:
$$\begin{aligned} \mathrm{E}[e^{i \varTheta } |X=x] = \frac{1}{\alpha (x)} \left( 1-\frac{{}_2F_1[\delta /(2\tau ), \delta /(2\tau )+1/2;1;\alpha (x)]}{{}_2F_1[\delta /(2\tau )+1/2, \delta /(2\tau )+1;1;\alpha (x)]} \right) e^{i\mu }. \end{aligned}$$
(11)

3.4 Regressions

The conditional mean (for \(0< \tau < 1\)) and variance (for \(0< \tau < 1/2\)) of X given \(\varTheta =\theta\) are immediately obtained from (10) as follows:
$$\begin{aligned} \mathrm{E}[X|\varTheta =\theta ] = \frac{\sigma \varGamma (\delta /\tau - \delta ) \varGamma (\delta + 1) }{ [ \{ 1 - \kappa \cos (\theta - \mu ) \}\tau /\delta ]^{\delta } \varGamma (\delta /\tau )} \end{aligned}$$
(12)
and
$$\begin{aligned} \mathrm{Var}[X|\varTheta =\theta ]= & {} \frac{\sigma ^2}{[ \{ 1-\kappa \cos (\theta -\mu ) \} \tau /\delta ]^{2 \delta }\varGamma ^2(\delta /\tau )} \nonumber \\&\times \, \left[ \varGamma \left( \frac{\delta }{\tau } \right) \varGamma \left( \frac{\delta }{\tau } - 2\delta \right) \varGamma \left( 2 \delta + 1\right) - \varGamma \left( \frac{\delta }{\tau } - \delta \right) ^2 \varGamma \left( \delta + 1\right) ^2 \right] . \end{aligned}$$
(13)
The conditional mean direction and circular variance of \(\varTheta\) given \(X=x\) are obtained from (11) as follows:
$$\begin{aligned} \arg \left\{ \mathrm{E}[e^{i\varTheta }|X=x] \right\} = \mu \end{aligned}$$
(14)
and
$$\begin{aligned} \mathrm{V}[\varTheta |X=x]= & {} 1-\Big | \mathrm{E}[e^{i\varTheta }|X=x] \Big | \nonumber \\= & {} 1-\frac{1}{\alpha (x)} \left( 1-\frac{{}_2F_1[\delta /(2\tau ), \delta /(2\tau )+1/2;1;\alpha (x)]}{{}_2F_1[\delta /(2\tau )+1/2, \delta /(2\tau )+1;1;\alpha (x)]} \right) . \end{aligned}$$
(15)
Figures 4 and 5 show the plots of the conditional mean (12), conditional median (7), conditional variance (13), and conditional circular variance (15) for varying \(\delta\) and \(\kappa\). The curve of the conditional mean direction (14) is omitted, because it is constant as a function of x.
Fig. 4

Plots of a conditional mean \(\mathrm{E}[X|\varTheta =\theta ]\), b conditional median \(\mathrm{Median}[X|\varTheta =\theta ]\), c conditional variance \(\mathrm{Var}[X|\varTheta =\theta ]\), and d conditional circular variance \(\mathrm{V}[\theta |X=x]\) when \((\sigma ,\tau ,\mu ,\kappa )=(1, 0.3, 0, 0.8)\) with \(\delta =0.2\) (solid), \(\delta =0.4\) (dashed), \(\delta =0.6\) (dot–dashed), and \(\delta =0.8\) (dotted)

Fig. 5

Plots of a conditional mean \(\mathrm{E}[X|\varTheta =\theta ]\), b conditional median \(\mathrm{Median}[X|\varTheta =\theta ]\), c conditional variance \(\mathrm{Var}[X|\varTheta =\theta ]\), and d conditional circular variance \(\mathrm{V}[\theta |X=x]\) when \((\sigma ,\delta ,\tau ,\mu )=(1, 0.8,0.3, 0)\) with \(\kappa =0.2\) (solid), \(\kappa =0.4\) (dashed), \(\kappa =0.6\) (dot–dashed), and \(\kappa =0.8\) (dotted)

4 Simulation study

To generate a random number \((x,\theta )'\) from the proposed distribution efficiently, the fact that the joint density (2) is expressed by \(f_{\mathrm{GPar}}(x, \theta ) = f_{X|\varTheta }(x|\theta ) f_{\varTheta }(\theta )\) is used, where \(f_{\varTheta }(\theta )\) and \(f_{X|\varTheta }(x|\theta )\) are given by (4) and (5), respectively. As the inverse function of the distribution function (6) has a closed-form expression, namely
$$\begin{aligned} x = F_{X|\varTheta }^{-1}(u|\theta ) = \left\{ \begin{array}{ll} \displaystyle \sigma \left[ \frac{(1 - u)^{-\tau /\delta } - 1 }{ \tau \{ 1- \kappa \cos (\theta -\mu ) \}/\delta } \right] ^{\delta }, &{}\quad \tau > 0, \\ \\ \displaystyle \sigma \left\{ - \frac{ \log (1 - u) }{ 1 - \kappa \cos (\theta -\mu )} \right\} ^{\delta }, &{}\quad \tau = 0, \end{array} \right. \end{aligned}$$
inverse transform sampling may be applied to generate a random number of \(X|(\varTheta =\theta )\). As the marginal distribution of \(\varTheta\) is the wrapped Cauchy distribution (4), a random number of \(\varTheta\) is also generated by the inverse transform sampling.
In the simulation study, the performance of maximum likelihood (ML) estimation was investigated for the proposed distribution with sample sizes \(n = 30, 60\), and 90. The simulation study was conducted for 3000 samples of size n by random number generation using the simple marginal and conditional densities. Then, the bias and mean-squared error (MSE) were computed for 3000 ML estimates of \(\sigma\), \(\delta\), \(\tau\), \(\mu\), and \(\kappa\). Here, although the estimate of \(\mu\) is a circular variable, it was locally regarded as a linear variable; accordingly, the bias and MSE defined for linear variables were used. The results and choices of the parameters are shown in Tables 1, 2, and 3. Simulation studies for \(\tau =0.7\), 0.9 and 1.1 are also conducted. However, the results are very similar to the cases when \(\tau =0.3\) and 0.5, so we omit them.
Table 1

Bias and MSE (in parentheses) of the maximum likelihood estimates of the parameters of the density (2) based on 3000 simulated samples of size n from the proposed distribution with \(\tau = 0.1\), \(\delta = 0.3, 0.4, 0.5\), \(\mu =\pi\), \(\kappa =\tanh (1)\), and \(\sigma = 0.5, 0.75, 1\)

 

n

\(\sigma\)

\(\delta\)

\(\tau\)

\(\mu\)

\(\kappa\)

\(\delta =0.3\)

 \(\sigma =0.5\)

30

0.006

− 0.001

0.010

0.010

− 0.074

(0.004)

(0.006)

(0.013)

(0.086)

(0.039)

60

0.010

0.008

0.002

− 0.003

− 0.069

(0.002)

(0.004)

(0.005)

(0.037)

(0.030)

90

0.011

0.010

0.003

− 0.008

− 0.073

(0.002)

(0.003)

(0.004)

(0.025)

(0.032)

 \(\sigma =0.75\)

30

− 0.009

− 0.010

0.012

0.044

− 0.026

(0.008)

(0.005)

(0.012)

(0.139)

(0.025)

60

− 0.005

0.002

0.010

0.036

− 0.024

(0.005)

(0.003)

(0.005)

(0.089)

(0.018)

90

0.000

0.004

0.007

0.031

− 0.030

(0.004)

(0.002)

(0.004)

(0.064)

(0.018)

 \(\sigma =1\)

30

− 0.010

− 0.010

0.004

0.011

− 0.009

(0.014)

(0.005)

(0.012)

(0.050)

(0.014)

60

− 0.003

0.000

− 0.002

0.005

− 0.007

(0.007)

(0.002)

(0.005)

(0.023)

(0.008)

90

− 0.005

− 0.001

0.002

0.004

− 0.004

(0.005)

(0.002)

(0.004)

(0.018)

(0.005)

\(\delta =0.4\)

 \(\sigma =0.5\)

30

0.002

− 0.005

0.013

− 0.012

− 0.045

(0.006)

(0.009)

(0.017)

(0.046)

(0.026)

60

0.013

0.012

0.002

− 0.010

− 0.069

(0.004)

(0.005)

(0.008)

(0.028)

(0.029)

90

0.019

0.018

0.001

− 0.016

− 0.085

(0.004)

(0.005)

(0.006)

(0.022)

(0.033)

 \(\sigma =0.75\)

30

− 0.020

− 0.015

0.017

0.017

− 0.003

(0.012)

(0.008)

(0.013)

(0.054)

(0.013)

60

− 0.011

− 0.006

0.015

0.004

− 0.003

(0.007)

(0.004)

(0.007)

(0.024)

(0.007)

90

− 0.012

0.000

0.016

0.003

− 0.006

(0.005)

(0.003)

(0.006)

(0.015)

(0.005)

 \(\sigma =1\)

30

− 0.012

− 0.016

0.005

0.007

− 0.002

(0.020)

(0.007)

(0.013)

(0.040)

(0.011)

60

− 0.005

− 0.005

− 0.001

0.002

− 0.002

(0.010)

(0.003)

(0.006)

(0.017)

(0.006)

90

0.000

0.000

− 0.005

− 0.001

− 0.006

(0.007)

(0.003)

(0.005)

(0.012)

(0.004)

\(\delta =0.5\)

 \(\sigma =0.5\)

30

− 0.001

− 0.012

0.008

− 0.005

− 0.032

(0.007)

(0.010)

(0.016)

(0.042)

(0.020)

60

0.010

0.006

0.000

− 0.007

− 0.041

(0.005)

(0.006)

(0.009)

(0.019)

(0.018)

90

0.015

0.015

− 0.004

− 0.006

− 0.052

(0.004)

(0.005)

(0.007)

(0.014)

(0.018)

 \(\sigma =0.75\)

30

− 0.026

− 0.018

0.013

0.008

0.008

(0.015)

(0.009)

(0.013)

(0.040)

(0.009)

60

− 0.014

− 0.009

0.013

0.004

0.001

(0.008)

(0.005)

(0.009)

(0.018)

(0.005)

90

− 0.009

− 0.004

0.008

0.001

− 0.001

(0.006)

(0.004)

(0.006)

(0.012)

(0.003)

 \(\sigma =1\)

30

− 0.020

− 0.021

0.004

0.005

0.004

(0.029)

(0.010)

(0.016)

(0.039)

(0.009)

60

− 0.010

− 0.007

0.000

0.004

0.002

(0.014)

(0.005)

(0.008)

(0.019)

(0.005)

90

− 0.004

− 0.002

− 0.002

0.003

− 0.001

(0.010)

(0.004)

(0.006)

(0.011)

(0.003)

Table 2

Bias and MSE (in parentheses) of the maximum likelihood estimates of the parameters of the density (2) based on 3000 simulated samples of size n from the proposed distribution with \(\tau = 0.3\), \(\delta = 0.3, 0.4, 0.5\), \(\mu =\pi\), \(\kappa =\tanh (1)\), and \(\sigma = 0.5, 0.75, 1\)

 

n

\(\sigma\)

\(\delta\)

\(\tau\)

\(\mu\)

\(\kappa\)

\(\delta =0.3\)

 \(\sigma =0.5\)

30

0.002

− 0.003

− 0.025

− 0.002

− 0.012

(0.006)

(0.009)

(0.027)

(0.061)

(0.015)

60

0.005

0.004

− 0.018

0.002

− 0.007

(0.002)

(0.004)

(0.012)

(0.024)

(0.007)

90

0.001

− 0.001

− 0.010

0.001

− 0.004

(0.002)

(0.002)

(0.007)

(0.013)

(0.004)

 \(\sigma =0.75\)

30

0.003

− 0.003

− 0.027

0.009

− 0.007

(0.013)

(0.008)

(0.026)

(0.061)

(0.014)

60

0.003

0.001

− 0.019

0.000

− 0.003

(0.005)

(0.004)

(0.011)

(0.020)

(0.006)

90

0.001

0.001

− 0.011

0.002

− 0.002

(0.003)

(0.002)

(0.007)

(0.012)

(0.004)

 \(\sigma =1\)

30

0.009

0.000

− 0.036

0.011

− 0.005

(0.023)

(0.008)

(0.028)

(0.053)

(0.012)

60

0.005

0.000

− 0.018

0.000

0.001

(0.009)

(0.003)

(0.012)

(0.020)

(0.005)

90

0.005

0.001

− 0.013

− 0.001

− 0.002

(0.006)

(0.002)

(0.007)

(0.013)

(0.003)

\(\delta =0.4\)

 \(\sigma =0.5\)

30

0.009

− 0.001

− 0.042

0.003

− 0.017

(0.008)

(0.013)

(0.035)

(0.049)

(0.015)

60

0.011

0.008

− 0.031

0.001

− 0.015

(0.004)

(0.007)

(0.018)

(0.020)

(0.009)

90

0.008

0.008

− 0.022

− 0.002

− 0.014

(0.003)

(0.005)

(0.012)

(0.013)

(0.007)

 \(\sigma =0.75\)

30

0.004

− 0.005

− 0.036

0.006

0.002

(0.018)

(0.011)

(0.034)

(0.048)

(0.010)

60

0.001

0.001

− 0.022

− 0.004

0.000

(0.008)

(0.005)

(0.015)

(0.020)

(0.005)

90

0.005

0.001

− 0.016

0.000

− 0.002

(0.005)

(0.003)

(0.009)

(0.013)

(0.004)

 \(\sigma =1\)

30

0.014

0.001

− 0.048

− 0.003

− 0.003

(0.032)

(0.011)

(0.035)

(0.042)

(0.010)

60

0.007

0.004

− 0.030

− 0.002

− 0.001

(0.016)

(0.006)

(0.017)

(0.018)

(0.005)

90

0.003

0.001

− 0.016

0.001

− 0.001

(0.009)

(0.004)

(0.011)

(0.012)

(0.003)

\(\delta =0.5\)

 \(\sigma =0.5\)

30

0.005

0.000

− 0.032

0.005

− 0.017

(0.006)

(0.009)

(0.027)

(0.062)

(0.016)

60

0.003

0.001

− 0.020

0.000

− 0.009

(0.002)

(0.004)

(0.011)

(0.021)

(0.007)

90

0.002

0.000

− 0.012

0.001

− 0.004

(0.002)

(0.002)

(0.007)

(0.013)

(0.004)

 \(\sigma =0.75\)

30

0.001

− 0.003

− 0.027

0.008

− 0.004

(0.013)

(0.008)

(0.028)

(0.055)

(0.013)

60

0.001

0.000

− 0.018

0.007

− 0.002

(0.005)

(0.003)

(0.011)

(0.025)

(0.006)

90

0.002

0.000

− 0.011

0.000

0.000

(0.003)

(0.002)

(0.007)

(0.015)

(0.004)

 \(\sigma =1\)

30

0.007

0.000

− 0.031

− 0.010

− 0.007

(0.022)

(0.008)

(0.028)

(0.046)

(0.011)

60

0.007

0.000

− 0.022

− 0.001

− 0.001

(0.010)

(0.004)

(0.012)

(0.020)

(0.005)

90

0.004

0.001

− 0.014

− 0.001

0.000

(0.006)

(0.002)

(0.007)

(0.012)

(0.003)

Table 3

Bias and MSE (in parentheses) of the maximum likelihood estimates of the parameters of the density (2) based on 3000 simulated samples of size n from the proposed distribution with \(\tau = 0.5\), \(\delta = 0.3, 0.4, 0.5\), \(\mu =\pi\), \(\kappa =\tanh (1)\), and \(\sigma = 0.5, 0.75, 1\)

 

n

\(\sigma\)

\(\delta\)

\(\tau\)

\(\mu\)

\(\kappa\)

\(\delta =0.3\)

  \(\sigma =0.5\)

30

0.004

− 0.005

− 0.039

0.004

− 0.003

(0.008)

(0.011)

(0.044)

(0.062)

(0.012)

60

0.001

− 0.005

− 0.014

0.003

0.000

(0.003)

(0.004)

(0.016)

(0.025)

(0.005)

90

0.001

− 0.002

− 0.012

− 0.002

− 0.001

(0.002)

(0.003)

(0.010)

(0.014)

(0.004)

  \(\sigma =0.75\)

30

0.004

− 0.004

− 0.035

− 0.006

− 0.002

(0.018)

(0.011)

(0.043)

(0.048)

(0.011)

60

0.002

0.000

− 0.018

− 0.005

0.000

(0.007)

(0.004)

(0.018)

(0.021)

(0.005)

90

0.002

− 0.003

− 0.012

0.002

− 0.001

(0.005)

(0.003)

(0.010)

(0.014)

(0.003)

 \(\sigma =1\)

30

0.008

− 0.005

− 0.034

0.000

0.001

(0.033)

(0.011)

(0.043)

(0.049)

(0.011)

60

0.002

− 0.003

− 0.013

0.002

0.000

(0.012)

(0.004)

(0.017)

(0.020)

(0.005)

90

0.002

− 0.001

− 0.012

0.003

0.001

(0.008)

(0.003)

(0.011)

(0.014)

(0.004)

\(\delta =0.4\)

 \(\sigma =0.5\)

30

0.010

− 0.001

− 0.047

0.007

− 0.004

(0.011)

(0.016)

(0.058)

(0.050)

(0.012)

60

0.007

− 0.002

− 0.025

0.002

− 0.004

(0.005)

(0.007)

(0.024)

(0.021)

(0.005)

90

0.003

0.000

− 0.018

0.000

− 0.001

(0.003)

(0.004)

(0.015)

(0.013)

(0.004)

 \(\sigma =0.75\)

30

0.012

− 0.004

− 0.050

− 0.001

− 0.003

(0.026)

(0.016)

(0.055)

(0.044)

(0.011)

60

0.005

− 0.002

− 0.023

− 0.002

0.000

(0.011)

(0.006)

(0.024)

(0.020)

(0.005)

90

0.002

− 0.002

− 0.014

− 0.001

0.000

(0.007)

(0.004)

(0.014)

(0.013)

(0.003)

 \(\sigma =1\)

30

0.023

0.001

− 0.056

− 0.001

− 0.004

(0.047)

(0.016)

(0.059)

(0.045)

(0.011)

60

0.006

− 0.001

− 0.030

− 0.001

0.000

(0.020)

(0.007)

(0.025)

(0.020)

(0.005)

90

0.006

0.000

− 0.017

0.002

− 0.001

(0.012)

(0.004)

(0.015)

(0.013)

(0.003)

\(\delta =0.5\)

 \(\sigma =0.5\)

30

0.004

− 0.003

− 0.037

0.001

− 0.002

(0.008)

(0.011)

(0.043)

(0.051)

(0.012)

60

0.002

− 0.002

− 0.017

− 0.007

− 0.001

(0.003)

(0.004)

(0.018)

(0.022)

(0.005)

90

0.001

− 0.001

− 0.012

0.000

− 0.001

(0.002)

(0.003)

(0.011)

(0.013)

(0.003)

 \(\sigma =0.75\)

30

0.005

− 0.004

− 0.037

− 0.003

− 0.002

(0.018)

(0.011)

(0.043)

(0.054)

(0.012)

60

0.003

− 0.003

− 0.018

− 0.005

0.000

(0.008)

(0.004)

(0.017)

(0.022)

(0.005)

90

− 0.001

− 0.002

− 0.011

− 0.002

0.000

(0.004)

(0.003)

(0.011)

(0.014)

(0.003)

 \(\sigma =1\)

30

0.004

− 0.006

− 0.032

− 0.002

0.001

(0.033)

(0.011)

(0.044)

(0.049)

(0.011)

60

0.000

− 0.005

− 0.016

0.000

0.000

(0.013)

(0.004)

(0.017)

(0.021)

(0.005)

90

0.004

− 0.002

− 0.013

0.001

− 0.001

(0.008)

(0.003)

(0.010)

(0.014)

(0.003)

As expected, it can be seen that the bias and MSE decrease as n increases. Considering the results for the biases in the tables, the estimates of \(\tau\) tend to overestimate when data are generated from a non heavy-tailed model (\(\tau = 0.1\)) and underestimate when those are generated from a heavy-tailed model (\(\tau = 0.3\) and 0.5). The estimates of \(\mu\) and \(\kappa\) do not seem to be influenced by the linear parameters.

5 Illustrative examples

In this section, the proposed distribution (2) is fitted to two real data sets using likelihood techniques. The fitting results are compared with those by distributions defined on \([0, \infty ) \times (0,2\pi )\), namely, the Johnson–Wehrly distribution (3), Abe–Ley distribution (1), and generalized Gamma-von Mises-type distribution in Abe and Ley (2017) with density:
$$\begin{aligned} f_{\mathrm{GGa}} (x, \theta ) = \frac{(x/\sigma )^{\alpha /\delta -1} \exp \left[ - (x/\sigma )^{1/\delta } \left\{ 1-\kappa \cos (\theta -\mu ) \right\} \right] }{2\pi \sigma \delta \varGamma (\alpha ) {}_2F_1[\alpha /2, (\alpha +1)/2;1;\kappa ^2]}, \end{aligned}$$
(16)
which reduces to the Abe–Ley distribution when \(\alpha =1\).
These models are skew with respect to the linear random variable X but symmetric with respect to the circular random variable \(\varTheta\). As actual data may also exhibit asymmetry for circular observations, a sine-skewed (Abe and Pewsey 2011) density \(f_\mathrm{SS*}(x,\theta )\) is derived from the symmetric density \(f_\mathrm{*}(x,\theta )\) about \(\mu\) by adding a skewing parameter \(\lambda\) for \(-1 \le \lambda \le 1\) as follows:
$$\begin{aligned} f_{\mathrm{SS*}} (x, \theta ) = \{ 1 + \lambda \sin (\theta - \mu ) \}f_{\mathrm{*}}(x, \theta ). \end{aligned}$$
(17)
In this section, the abbreviations, ExpCylin, WeiCylin, GGaCylin, and GParCylin, are used for the distributions (3), (1), (16), and (2), respectively, and SSExpCylin, SSWeiCylin, SSGGaCylin, and SSGParCylin are used for their sine-skewed distributions obtained by (17).

5.1 Blue periwinkle

An analysis of data with 31 observations (Table 1 in Fisher and Lee 1992, and B.20 in Fisher 1993) consisting of the direction of motion of blue periwinkles after being experimentally transplanted downshore (Fisher and Lee 1992 for details about the experience) is provided by Abe and Ley (2017). Table 4 shows the ML estimates of the parameters as well as the values of the maximized log-likelihood (MLL) and Akaike’s Information Criterion (AIC) of the fitted symmetric cylindrical distributions and their sine-skewed distributions to the data set. Figure 6 shows the scatter plots of the blue periwinkle data as well as the contour plots of the fitted WeiCylin and SSWeiCylin models with the smaller AIC.
Table 4

Maximum likelihood (ML) estimates, maximized log-likelihood (MLL) and Akaike Information Criterion (AIC) values for ExpCylin, WeiCylin, GGaCylin, GParCylin, and their sine-skewed distributions fitted to the blue periwinkle data

Distributions

ML estimates

MLL

AIC

\({\hat{\lambda }}\)

\({\hat{\alpha }}\)

\({\hat{\sigma }}\)

\({\hat{\delta }}\)

\({\hat{\tau }}\)

\({\hat{\mu }}\)

\({\hat{\kappa }}\)

ExpCylin

9.61

1.44

0.89

\(-\)182.93

371.86

WeiCylin

18.88

0.51

1.35

0.94

\(-\)173.28

354.55

GGaCylin

1.14

16.98

0.56

1.37

0.92

\(-\)173.18

356.36

GParCylin

18.88

0.51

0.00

1.35

0.94

\(-\)173.28

356.55

SSExpCylin

1.00

11.34

1.22

0.87

\(-\)179.13

366.26

SSWeiCylin

1.00

20.27

0.50

1.24

0.93

\(-\)168.57

347.13

SSGGaCylin

1.00

0.98

20.60

0.49

1.24

0.94

\(-\)168.56

349.13

SSGParCylin

1.00

20.27

0.50

0.00

1.24

0.93

\(-\)168.57

349.13

Fig. 6

Scatter plots of the blue periwinkle data and contour plots of the fitted a WeiCylin and b SSWeiCylin. The data are plotted over \((0,125)\times [0,2\pi )\)

The sine-skewed models are better than the original symmetric models in the sense of smaller AIC. This is a natural consequence of the fact that the Pewsey test (Pewsey 2002) rejects symmetry for the circular observations of the data set (p value is 0.006). By comparing SSWeiCylin with SSGGaCylin and SSGParCylin, it is seen that the estimate of \(\alpha\) in GGaCylin is close to 1, and that of \(\tau\) in GParCylin converges to zero. The other estimates in the three models have nearly the same values. That is, the most parsimonious model is the SSWeiCylin, and this indicates that the linear part of this data set is not heavy-tailed.

5.2 Seismic magnitude and sequences of epicenter data

The second example is the magnitude and epicenter (latitude and longitude) data for foreshocks during a period of 72 h before the 2011 Great East Japan Earthquake (off the Pacific coast of Tohoku Earthquake, Japan). The earthquake had a magnitude of 9.0 \(\hbox {M}_{\mathrm{w}}\) and occurred at 14:46 JST (05:46 UTC) on 11 March 2011 with the epicenter at \(38.30^\circ\) of latitude and \(142.37^\circ\) of longitude approximately 70 km east of the coast. The data were taken from the website  http://earthquake.usgs.gov/earthquakes/eqarchives/epic/epic_global.php. Earthquakes whose magnitude M was greater than or equal to 4.0 were chosen, and the method proposed in Banerjee et al. (2005) was used to find clusters of earthquakes that occurred near Japan. The number of observations was 64 including the mainshock. Linear observations (\(x=M-4\)) were obtained, and the epicenter data were transformed into sequences of angles. Thus, the number of combinations of magnitudes and angles is \(n=62\). Considering turning angles between successive moves is not new. Indeed, in Pearson (1905), a random walk problem was proposed, and in Kareiva and Shigesada (1983), the flight sequences of cabbage white butterflies and the movement patterns of pipe-vine swallowtails were studied. Table 5 shows the ML estimates of the parameters, the MLL and AIC values of the fitted symmetric cylindrical distributions and their sine-skewed distributions to the data set. Figure 7 shows the scatter plots of the seismic magnitude and the sequences of epicenter data as well as the contour plots of the fitted SSWeiCylin, SSGGaCylin, and GParCylin models with the smaller AIC.
Table 5

Maximum likelihood (ML) estimates, maximized log-likelihood (MLL), and Akaike Information Criterion (AIC) values for ExpCylin, WeiCylin, GGaCylin, GParCylin, and their sine-skewed distributions fitted to the earthquake data

Distributions

ML estimates

MLL

AIC

\({\hat{\lambda }}\)

\({\hat{\alpha }}\)

\({\hat{\sigma }}\)

\({\hat{\delta }}\)

\({\hat{\tau }}\)

\({\hat{\mu }}\)

\({\hat{\kappa }}\)

ExpCylin

0.86

3.13

0.36

\(-\)170.51

347.03

WeiCylin

1.02

0.65

2.80

0.31

\(-\)161.94

331.89

GGaCylin

3.65

0.18

1.21

3.19

0.16

\(-\)154.95

319.90

GParClin

0.76

0.37

0.29

3.26

0.58

\(-\)151.61

313.21

SSExpCylin

0.58

0.95

2.05

0.17

\(-\)167.38

342.76

SSWeiCylin

0.58

1.07

0.62

2.02

0.23

\(-\)156.92

323.83

SSGGaCylin

0.38

11.97

0.00

2.21

2.80

0.05

\(-\)152.96

317.92

SSGParCylin

0.16

0.76

0.37

0.28

3.10

0.57

\(-\)151.55

315.10

Fig. 7

Scatter plots of the earthquake data and contour plots of the fitted a SSWeiCylin, b SSGGaCylin, and c GParCylin. The data are plotted over \((0,5.5)\times [0,2\pi )\)

Concerning the magnitude in this data set, the linear mean, variance, skewness, and kurtosis are 0.98, 0.70, 3.14, and 15.05, respectively. From these values, it is seen that the fourth moment gives a larger value compared with the first and second moments, and the heavy-tailed cylindrical distributions for the linear part will be suitable to this data set. In fact, Table 5 shows that GParCylin and SSGParCylin, which can model heavy-tailedness for the linear part, result in good fitting in the sense of smaller AIC. Moreover, as the estimates of \(\tau\) are greater than 0.25, the fourth moment is theoretically infinite. An interesting point is that comparing the sine-skewed models with the original symmetric models in terms of AIC shows that GParCylin is better than SSGParCylin, whereas the other models are worse than the sine-skewed models. Indeed, the likelihood ratio test for \(H_0: \lambda =0\) does not reject the hypothesis (p value is 0.740) for GParCylin, whereas it rejects the hypothesis (p value is less than 0.002) for the other models. As can be seen from Fig. 7, particularly (b) SSWeiCylin, the skewness of the light-tailed cylindrical model for the linear part is influenced by the mainshock at \((x,\theta )=(5.0, 1.10)\). The Pewsey test (Pewsey 2002) does not reject symmetry for the circular observations of the data set (p value is 0.86); thus, it is natural to assume symmetry for the circular part of the fitted model. In this sense, GParCylin is robust and flexible for large linear values. The estimate of \(\mu\) in the optimal model, or symmetric GParCylin, is nearly equal to \(\pi\). This shows a tendency to the opposite direction for successive earthquake epicenters, which is difficult to find using the other skewed models.

6 Conclusion

A heavy-tailed cylindrical distribution for the linear part was proposed. It is constructed by a generalized Gamma mixture of the Abe–Ley distribution. The circular marginal and conditional distributions are the wrapped Cauchy and circular Pearson Type VII distributions, whereas the linear marginal and conditional distributions are the heavy-tailed unfamiliar and generalized Pareto-type distributions, respectively. One of the weak points of heavy-tailed distributions is that their means may be infinite and hence unsuitable for regression analysis. However, the proposed model overcomes this shortcoming using the mode and median, which have closed-form expressions, instead of the mean. In our simulation study, we have investigated the cases when the proposed distribution is light-tailed and heavy-tailed. From the study, the estimates of \(\tau\) always tend to underestimate the true value for a heavy-tailed model, but the estimates are reliable because of the small bias and MSE.

The major advantage of the proposed model is its robustness and flexibility for cylindrical data sets with a few large linear observations. As seen in the examples, when light-tailed cylindrical distributions are fitted for such data sets, the circular parts may be strongly influenced by large linear observations. For example, the skewed model is preferable to the symmetric model even when the sample circular skewness shows that the circular observations are symmetric, and this results in different mean direction for the circular variable of the data set. However, the proposed heavy-tailed cylindrical distribution will overcome the issue.

Notes

Acknowledgements

Tomoaki Imoto was supported in part by JSPS KAKENHI Grant Number 18K13459. Toshihiro Abe was supported in part by Nanzan University of Pache Research Subsidy I-A-1 for the 2018 academic year.

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Copyright information

© Japanese Federation of Statistical Science Associations 2019

Authors and Affiliations

  1. 1.School of Management and InformationUniversity of ShizuokaShizuokaJapan
  2. 2.School of Statistical ThinkingThe Institute of Statistical MathematicsTokyoJapan
  3. 3.Department of Systems and Mathematical Science, Faculty of Science and EngineeringNanzan UniversityNagoyaJapan

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