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.