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.

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.