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A New Robust Class of Skew Elliptical Distributions

Abstract

A new robust class of multivariate skew distributions is introduced. Practical aspects such as parameter estimation method of the proposed class are discussed, we show that the proposed class can be fitted under a reasonable time frame. Our study shows that the class of distributions is capable to model multivariate skewness structure and does not suffer from the curse of dimensionality as heavily as other distributions of similar complexity do, such as the class of canonical skew distributions. We also derive a nested form of the proposed class which appears to be the most flexible class of multivariate skew distributions in literature that has a closed-form density function. Numerical examples on two data sets, i) a data set containing daily river flow data recorded in the UK; and ii) a data set containing biomedical variables of athletes collected by the Australian Institute of Sports, are demonstrated. These examples further support the practicality of the proposed class on moderate dimensional data sets.

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Notes

  1. 1.

    obtained from the UK environment agency website; https://environment.data.gov.uk/hydrology/explore

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Acknowledgments

The authors would like to thank the Editor, the Associate Editor and the referee for careful reading and comments which greatly improved the paper.

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Correspondence to Saralees Nadarajah.

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Appendix A:: Proofs

Appendix A:: Proofs

A.1 Proof that (1) is a pdf

Note that

$$ \begin{array}{@{}rcl@{}} {\int}_{-\infty}^{\infty}g(\lambda x)f(x;\phi)dx &=&{\int}_{0}^{\infty}g(\lambda x)f(x;\phi)dx + {\int}_{-\infty}^{0}g(\lambda x)f(x;\phi)dx\\ &=&{\int}_{0}^{\infty}g(\lambda x)f(x;\phi)dx + {\int}_{-\infty}^{0}\left[ 1-g(-\lambda x)\right] f(x;\phi)dx\\ &=&{\int}_{0}^{\infty}g(\lambda x)f(x;\phi)dx + {\int}_{0}^{\infty}\left[ 1-g(\lambda x)\right] f(x;\phi)dx\\ & =& {\int}_{0}^{\infty}f(x;\phi)dx\\ & =& \frac {1}{2} \end{array} $$

and therefore (1) is a valid pdf.

A.2 Proof of (7)

Note that

$$ \begin{array}{@{}rcl@{}} f_{\textbf{X}_{0} |\textbf{X}_{1}}\left( \textbf{x}_{0}|\textbf{x}_{1}\right) &\propto& \left( 1+\frac {q\left( \textbf{x}_{1}\right)+\left( \textbf{x} - {\boldsymbol \mu}_{0.1}\right)^{T}{\boldsymbol {\Omega}}_{0.1}^{-1} \left( \textbf{x} - {\boldsymbol \mu}_{0.1}\right)}{\nu}\right)^{-\left( \nu+k_{0}+k_{1}\right)/2} \\ &\propto& \left( 1+\frac {\left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)^{T}{\boldsymbol {\Omega}}_{0.1}^{-1} \left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)}{\nu+q\left( \textbf{x}_{1}\right)}\right)^{-\left( \nu+k_{0}+k_{1}\right)/2}\\ &\propto& \left( 1+\frac {\left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)^{T} {\boldsymbol {\Omega}}_{0.1}^{-1}\left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)}{\nu+k_{1}}\frac {\nu+k_{1}}{\nu+q \left( \textbf{x}_{1}\right)}\right)^{-\left( \nu+k_{0}+k_{1}\right)/2}\\ &\propto& \left( 1+\frac {\left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)^{T} \left( \frac {\nu+q\left( \textbf{x}_{1}\right)}{\nu+k_{1}}{\boldsymbol {\Omega}}_{0.1}\right)^{-1} \left( \textbf{x}-{\boldsymbol \mu}_{0.1}\right)}{\nu+k_{1}}\right)^{-\left( \nu+k_{0}+k_{1}\right)/2}. \end{array} $$

A.3 Proof of (11)

Note that

$$ \begin{array}{@{}rcl@{}} \phantom{=}&&\underset{k}{{\int}_{-\infty}^{\infty}\cdots{\int}_{-\infty}^{\infty}} \left[{\prod}_{i=1}^{k}g_{k} \left( \lambda_{ii}\right) \right] El\left( \textbf{x}^{(k)}, \textbf{0}, \textbf{I}_{k}, \nu \right) dx_{1} {\cdots} dx_{k} \\ &=& \underset{k-1}{{\int}_{-\infty}^{\infty}\cdots{\int}_{-\infty}^{\infty}} {\int}_{0}^{\infty}g_{k} \left( \lambda_{11}\right) \left[{\prod}_{i=2}^{k}g_{k} \left( \lambda_{ii}\right) \right] El \left( \textbf{x}^{(k)}, \textbf{0}, \textbf{I}_{k}, \nu \right) dx_{1} {\cdots} dx_{k} \\ &&+ \underset{k-1}{{\int}_{-\infty}^{\infty}\cdots{\int}_{-\infty}^{\infty}} {\int}_{0}^{\infty}\left( 1 - g_{k} \left( \lambda_{11}\right)\right) \left[{\prod}_{i=2}^{k} g_{k} \left( \lambda_{ii}\right) \right] El \left( \textbf{x}^{(k)}, \textbf{0}, \textbf{I}_{k}, \nu \right) dx_{1} {\cdots} dx_{k} \\ & =& \underset{k-1}{{\int}_{-\infty}^{\infty}\cdots{\int}_{-\infty}^{\infty}} {\int}_{0}^{\infty}\left[{\prod}_{i=2}^{k} g_{k} \left( \lambda_{ii}\right) \right] El \left( \textbf{x}^{(k)}, \textbf{0}, \textbf{I}_{k}, \nu \right) dx_{1} {\cdots} dx_{k} \\ & =& \frac {1}{2} \underset{k-1}{{\int}_{-\infty}^{\infty}{\cdots} {\int}_{-\infty}^{\infty}} \left[{\prod}_{i=2}^{k} g_{k} \left( \lambda_{ii}\right) \right] El \left( \textbf{x}^{(k-1)}, \textbf{0}, \textbf{I}_{k-1}, \nu \right) dx_{2} {\cdots} dx_{k} \\ & = &2^{-k}. \end{array} $$

A.4 Proof of (12)

By Hölder’s inequality,

$$ \begin{array}{@{}rcl@{}} E_{\textbf{U}}\left[g_{m}\left( {\boldsymbol \lambda} \textbf{u}\right) \right] &=& E_{\textbf{U}} \left[ \left | g_{m}\left( {\boldsymbol \lambda} \textbf{u} \right) \right | \right] \\ & \le& \overset{m}{\underset{i=1}{\prod}} E_{\textbf{U}} \left[ \left | g\left( {\boldsymbol \lambda}_{i} \textbf{u} \right) \right | \right] \\ & =& \overset{m}{\underset{i=1}{\prod}} E_{\textbf{U}} \left[ g\left( {\boldsymbol \lambda}_{i} \textbf{u} \right) \right] \\ & =& 2^{-m}. \end{array} $$

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Kwong, H.S., Nadarajah, S. A New Robust Class of Skew Elliptical Distributions. Methodol Comput Appl Probab (2021). https://doi.org/10.1007/s11009-021-09883-5

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Keywords

  • Maximum likelihood estimation
  • Multivariate distributions
  • Multivariate skew t distributions
  • Robust distributions

Mathematics Subject Classification (2010)

  • 62E99