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.
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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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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:
The substitution \(v=w^{\delta }\) yields
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
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
and
The second derivatives of the logarithm of the density (2) are given by
and thus, the Hessian matrices at \((x,\theta )=(x_1,\theta _1)\) and at \((x,\theta )=(x_2,\theta _2)\) are given by
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
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:
The substitution \(x=\sigma (\delta y/\tau )^{\delta }\) yields
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
and the combination of Equations 9.112 and 9.134.2 in Gradshteyn and Ryzhik (2007) yields
the marginal density \(f_X(x)\) of X is
As
it is confirmed that
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
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).
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Imoto, T., Shimizu, K. & Abe, T. A cylindrical distribution with heavy-tailed linear part. Jpn J Stat Data Sci 2, 129–154 (2019). https://doi.org/10.1007/s42081-019-00031-5
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DOI: https://doi.org/10.1007/s42081-019-00031-5