Extreme value Birnbaum–Saunders regression models applied to environmental data

Abstract

Extreme value models are widely used in different areas. The Birnbaum–Saunders distribution is receiving considerable attention due to its physical arguments and its good properties. We propose a methodology based on extreme value Birnbaum–Saunders regression models, which includes model formulation, estimation, inference and checking. We further conduct a simulation study for evaluating its performance. A statistical analysis with real-world extreme value environmental data using the methodology is provided as illustration.

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Acknowledgments

The authors wish to thank the Editors and three anonymous referees for their constructive comments on an earlier version of this manuscript, which resulted in this improved version. This study was partially supported by the Chilean Council for Scientific and Technological Research under the project grant FONDECYT 1120879, by FEDER Funds through “Programa Operacional de Factores de Competitividade-COMPETE” and by Portuguese Funds through “Fundação para a Ciência e a Tecnologia” (FCT) under the project Grants PEst-OE/MAT/UI0006/2014 (CEAUL) and PEst-OE/MAT/UI0013/2014.

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Correspondence to Víctor Leiva.

Appendices

Appendix 1: Hessian matrix

For the case \(\xi \ne 0\), \({\varvec{V}} = {{\text{ diag }}(v_1,\ldots ,v_n)}\), \({\varvec{k}} = (k_1,\ldots ,k_n)^\top \) and \({\varvec{u}} = (u_1,\ldots ,u_n)^\top \) defined in (7) have elements

$$\begin{aligned} v_i= & {} -\frac{\text { sech}^2(\zeta _{i1})(1+\zeta _{i2}\xi )^{{{-2}}-\frac{1}{\xi }}}{8\alpha ^2} \left( 3+4\xi +4\cosh (2\zeta _{i1})(1+\xi )\right. \\&\left. -\cosh (4\zeta _{i1}) +\frac{\alpha ^2}{2}\zeta _{i2}+\alpha \sinh (3\zeta _{i1})\right. \\&\left. +(1+\zeta _{i2}\xi )^{\frac{1}{\xi }} \big (2(\alpha ^2+2\xi )+4\xi (1+2\xi )\cosh (2\zeta _{i1}) +\frac{\alpha ^2\zeta _{i2}(1-7\xi )}{2}\big .\right. \\&\left. \left. -\alpha (1+\xi )\sinh (3\zeta _{i1})\right) \right) ,\\ k_i= & {} \displaystyle \frac{\cosh (\zeta _{i1})}{\alpha ^2}(1+\zeta _{i2}\xi )^{-2-\frac{1}{\xi }} \left( 1-\zeta _{i2}-(1+\xi )(1+\xi \zeta _{i2})^{\frac{1}{\xi }}\right) ,\\ u_i= & {} \displaystyle \frac{\cosh (\zeta _{i1})(1+\zeta _{i2}\xi )^{-2-\frac{1}{\xi }}}{\xi ^2\alpha } \Big (2\xi -\xi \log (1+\zeta _{i2}\xi )-\,\log (1+\zeta _{i2}\xi )\zeta _{i2}\Big .\\&\Big . +\left( -\xi +(\xi -1)\log (1+\zeta _{i2}\xi )-\,\xi \zeta _{i2}\right) (1+\zeta _{i2}\xi )^{\frac{1}{\xi }}\Big ). \end{aligned}$$

In addition, we have

$$\begin{aligned} \displaystyle \ddot{\ell }_{\alpha \alpha }\,=\, & {} \displaystyle \frac{n}{\alpha ^2}+{\sum _{i=1}^{n}} \frac{2\zeta _{i2}(\xi -1+(1+\zeta _{i2}\xi )^{-\frac{1}{\xi }})}{{(1+\zeta _{i2}\xi )}}\\&+{\sum _{i=1}^{n}}\frac{2(1+\xi )}{\alpha ^4}(\cosh (2\zeta _{i1})-1)(1+\zeta _{i2}\xi )^{-2-\frac{1}{\xi }}(\xi (1+\zeta _{i2}\xi )^{\frac{1}{\xi }}-1),\\ \displaystyle \ddot{\ell }_{\alpha \xi }\,=\,& {} \displaystyle \ddot{\ell }_{\xi \alpha }\,=\, \displaystyle {\sum _{i=1}^{n}} \frac{\zeta _{i2}(1+\zeta _{i2}\xi )^{-2-\frac{1}{\xi }}}{\alpha \xi ^2} \Big (\log (1+\zeta _{i2}\xi )+\xi \Big ((1+\xi -\log (1+\zeta _{i2}\xi ))\zeta _{i2} \Big .\Big .,\\&\Big .\Big .\displaystyle +\xi (1-\zeta _{i2})(1+\zeta _{i2}\xi )^{\frac{1}{\xi }}\Big )\Big ),\\ {\displaystyle \ddot{\ell }_{\xi \xi }}= & {} \displaystyle {\sum _{i=1}^{n}} \frac{(1+\zeta _{i2}\xi )^{-2-\frac{1}{\xi }}}{\xi ^4\alpha ^2} \Big (\alpha ^{2}\left( 2\xi - \log (1+\zeta _{i2}\xi )\right) \log (1+\zeta _{i2}\xi ) -2\alpha ^{2}\xi \left( 1+\zeta _{i2}\xi \right) ^{\frac{1}{\xi }} \Big . \\&\times \left( \log (1+\zeta _{i2}\xi ) + \xi \left( \left( \cosh (2{\zeta _{i1}}) - 1 \right) \xi \alpha ^{-2} (2\log (1+\zeta _{i2}\xi ) - \xi - 3) + {\zeta _{i2}}(2\log (1+\zeta _{i2}\xi ) - 1) \right) \right) \\&+2\xi \left( \xi \left( \cosh (2{\zeta _{i1}})- 1\right) \left( -\log (1+\zeta _{i2}\xi )\left( \log (1+\zeta _{i2}\xi ) - 2\xi - 2 \right) - 3\xi - 1 \right) \right. \\&\Big .\left. + {\zeta _{i2}}\alpha ^{2}\left( \log (1+\zeta _{i2}\xi )\left( -\log (1+\zeta _{i2}\xi ) + 2\xi + 1 \right) -\xi \right) \right) \Big ). \end{aligned}$$

For the case \(\xi = 0\), \(\varvec{V}_0 = {{\text{ diag }}(v_{0,1},\ldots ,v_{0,n})}\) and \(\varvec{k}_0 = {(k_{0,1},\ldots ,k_{0,n})}^\top \) defined in (8) have elements

$$\begin{aligned} v_{0,i}= & {} \displaystyle \frac{1}{4\alpha ^2}\left( \alpha (\alpha \text {sech}^2(\zeta _{i1})-\alpha \zeta _{i2}) -2\exp (-\zeta _{i2})\left( 1+\cosh (2\zeta _{i1})-\frac{\alpha ^2\zeta _{i2}}{2}\right) \right) ,\\ k_{0,i}= & {} \displaystyle \frac{\cosh (\zeta _{i1})}{\alpha ^2}(-1+\exp (-\zeta _{i2})(1-\zeta _{i2})). \end{aligned}$$

In addition, we have

$$\begin{aligned} {\displaystyle \ddot{\ell }}_{\alpha \alpha ,0}= & {} \frac{n}{\alpha ^2}-{\sum _{i=1}^{n}}\frac{2\zeta _{i2}}{\alpha ^2}(1-\exp (-\zeta _{i2})) -{\sum _{i=1}^{n}}\frac{\zeta _{i2}^2}{\alpha ^2}\exp (-\zeta _{i2}). \end{aligned}$$

Appendix 2: Data set

Daily maximum ozone concentration data (\(T\), in ppb): 14.290, 40.820, 40.310, 45.920, 8.163, 13.270, 11.220, 21.940, 14.800, 20.920, 38.270, 32.650, 41.330, 46.430, 42.860, 23.980, 27.040, 34.180, 20.410, 11.730, 36.220, 11.220, 31.120, 13.270, 13.270, 16.330, 19.900, 14.800, 15.820, 14.800, 14.800, 18.370, 8.673.

Daily maximum temperature data (\(X\), in \(^{\circ }\)C): 16.8, 24.2, 22.9, 22.5, 15.8, 17.2, 14.8, 18.5, 16.4, 19.8, 26.5, 18.1, 19.1, 22.2, 21.4, 21.3, 25.5, 20.3, 15.7, 16.2, 22.7, 16.8, 22.5, 17.0, 16.5, 17.6, 16.8, 17.9, 18.4, 18.7, 19.9, 17.1, 15.4.

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Leiva, V., Ferreira, M., Gomes, M.I. et al. Extreme value Birnbaum–Saunders regression models applied to environmental data. Stoch Environ Res Risk Assess 30, 1045–1058 (2016). https://doi.org/10.1007/s00477-015-1069-6

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Keywords

  • Data analysis
  • Maximum likelihood method
  • Monte Carlo simulation
  • Residuals
  • Statistical modeling