Appegesndix A.
Derivation of the second-order partial derivatives:
$$ {\displaystyle \begin{array}{c}\frac{\partial^2\ln L}{\partial {\upbeta}^2}=-\frac{n_a}{\upbeta^2}-\left(c-1\right)\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right)\frac{\left({y}_i-\uptau \right)}{{\left(\uptau +\upbeta \left({y}_i-\uptau \right)\right)}^2}\\ {}-\left(k+1\right)c\sum \limits_{i=1}^n\left({y}_i-\uptau \right){\updelta}_{2i}\left[\left({y}_i-\uptau \right)\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-\left({y}_i-\uptau \right)c{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right]\\ {}- kc\left(n-{n}_0\right)\left({y}_{(r)}-\uptau \right)\left[\left(c-1\right)\left({y}_{(r)}-\uptau \right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}-\left({y}_{(r)}-\uptau \right)c{D}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}\right]\\ {}=-\frac{n_a}{\upbeta^2}-\left(c-1\right)\sum \limits_{i=1}^n{\updelta}_{2i}{\left({y}_i-\uptau \right)}^2{A}^{-2}-\left(k+1\right)c\sum \limits_{i=1}^n{\left({y}_i-\uptau \right)}^2{\updelta}_{2i}\Big[\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-c{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\\ {}- kc\left(n-{n}_0\right){\left({y}_{(r)}-\uptau \right)}^2\left[\left(c-1\right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}-c{D}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}\right],\\ {}\frac{\partial^2\ln L}{\mathrm{\partial \upbeta \partial }c}=\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right){A}^{-1}-k\left({y}_{(r)}-\uptau \right)\left(n-{n}_0\right)\\ {}\times \left[{D}^{c-1}{\left(1+{D}^c\right)}^{-1}+c{\left(1+{D}^c\right)}^{-1}{D}^{c-1}\ln D-c{D}^{c-1}{\left(1+{D}^c\right)}^{-2}{D}^c\ln D\right]\\ {}-\left(k+1\right)\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right)\left[{A}^{c-1}{\left(1+{A}^c\right)}^{-1}+c{\left(1+{A}^c\right)}^{-1}{A}^{c-1}\ln A-c{A}^{c-1}{\left(1+{A}^c\right)}^{-2}{A}^c\ln A\right],\\ {}\frac{\partial^2\ln L}{\mathrm{\partial \upbeta \partial }k}=-c\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right){A}^{c-1}{\left(1+{A}^c\right)}^{-1}-\left({y}_{(r)}-\uptau \right)\left(n-{n}_0\right)c{D}^{c-1}{\left(1+ Dc\right)}^{-1},\\ {}\frac{\partial^2\ln L}{\partial c}=-\frac{n_0}{c^2}-k\left(n-{n}_0\right)\ln D\left[{\left(1+{D}^c\right)}^{-1}{D}^c\ln D-{\left(1+{D}^c\right)}^{-2}{D}^{2c}\ln D\right]\\ {}-\left(k+1\right)\left[\sum \limits_{i=1}^n{\updelta}_{1i}\ln {y}_i\left\{{\left(1+{y}_i^c\right)}^{-1}{y}_i^c\ln {y}_i-{y}_i^{2c}{\left(1+{y}_i^c\right)}^{-2}\ln {y}_i\right\}\right]\\ {}-\left(k+1\right)\left[\sum \limits_{i=1}^n{\updelta}_{2i}\ln A\left\{{\left(1+{A}^c\right)}^{-1}{A}^c\ln A-{A}^{2c}{\left(1+{A}^c\right)}^{-2}\ln A\right\}\right],\\ {}\frac{\partial^2\ln L}{\partial c\partial k}=-\sum \limits_{i=1}^n{\updelta}_{1i}\ln {y}_i\left\{{\left(1+{y}_i^c\right)}^{-1}{y}_i^c\ln {y}_i\right\}-\sum \limits_{i=1}^n{\updelta}_{2i}\ln A{\left(1+{A}^c\right)}^{-1}{A}^c\ln A-\left(n-{n}_0\right){\left(1+{D}^c\right)}^{-1}{D}^c\ln D,\end{array}} $$
and
$$ \frac{\partial^2\ln k}{\partial {k}^2}=-\frac{n_0}{k^2}. $$
Appendix B.
The determinant of F and its partial derivative w.r.t. τ. The determinant of F is given by
$$ \mid F\mid ={f}_{11}\left({f}_{22}{f}_{33}-{f}_{23}^2\right)-{f}_{12}\left({f}_{12}{f}_{33}-{f}_{13}{f}_{23}\right)+{f}_{13}\left({f}_{12}{f}_{23}-{f}_{13}{f}_{22}\right). $$
Its partial derivative w.r.t. τ is obtained as
$$ {\displaystyle \begin{array}{c}\frac{\partial \mid F\mid }{\mathrm{\partial \uptau }}={f}_{11}\left({f}_{22}^{\prime }{f}_{33}^{\prime }+{f}_{22}{f}_{33}^{\prime }-2{f}_{23}{f}_{33}^{\prime}\right)+{f}_{11}^{\prime}\left({f}_{22}{f}_{33}-{f}_{23}^2\right)\\ {}-{f}_{12}\left({f}_{12}^{\prime }{f}_{33}+{f}_{12}{f}_{33}^{\prime }-{f}_{13}^{\prime }{f}_{23}-{f}_{13}{f}_{23}^{\prime}\right)-{f}_{12}^{\prime}\left({f}_{12}{f}_{33}-{f}_{13}{f}_{23}\right)\\ {}+{f}_{13}\left({f}_{12}^{\prime }{f}_{23}+{f}_{12}{f}_{23}^{\prime }-{f}_{13}^{\prime }{f}_{22}-{f}_{13}{f}_{22}^{\prime}\right)-{f}_{13}^{\prime}\left({f}_{12}{f}_{23}-{f}_{13}{f}_{22}\right),\end{array}} $$
where
$$ {\displaystyle \begin{array}{c}{f}_{11}^{\prime }=\left(c-1\right)\sum \limits_{i=1}^n{\updelta}_{2i}\left[-2\left({y}_i-\uptau \right){A}^{-2}-2{\left({y}_i-\uptau \right)}^2{A}^{-3}\left(1-\upbeta \right)\right]\\ {}+\left(k+1\right)c\sum \limits_{i=1}^n\left(c-1\right){\updelta}_{2i}\Big[-2\left({y}_i-\uptau \right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}+{\left({y}_i-\uptau \right)}^2\\ {}\times \left(1-\upbeta \right)\left(\left(c-2\right){A}^{c-3}{\left(1+{A}^c\right)}^{-1}-c{A}^{2c-3}{\left(1+{A}^c\right)}^{-2}\right)\Big]\\ {}+\left(k+1\right)c\sum \limits_{i=1}^nc{\updelta}_{2i}\Big[-2\left({y}_i-\uptau \right){A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}+{\left({y}_i-\uptau \right)}^2\\ {}\times 2\left(1-\upbeta \right)\left(\left(c-1\right){A}^{2c-3}{\left(1+{A}^c\right)}^{-2}-c{A}^{3\left(c-1\right)}{\left(1+{A}^c\right)}^{-3}\right)\Big]\\ {}+ kc\left(n-{n}_0\right)\left(c-1\right)\Big[-2\left({y}_{(r)}-\uptau \right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}+\left(1-\upbeta \right){\left({Y}_c-\uptau \right)}^2\\ {}\times \left(\left(c-2\right){D}^{c-3}{\left(1+{D}^c\right)}^{-1}-c{D}^{2c-3}{\left(1+{D}^c\right)}^{-2}\right)\Big]\\ {}+ kc\left(n-{n}_0\right)\Big[-2\left({y}_{(r)}-\uptau \right){D}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}+2{\left({y}_{(r)}-\uptau \right)}^2\left(1-\upbeta \right)\\ {}\times \left(\left(c-1\right){D}^{2c-3}{\left(1+{D}^c\right)}^{-2}-{D}^{3\left(c-1\right)}c{\left(1+{D}^c\right)}^{-3}\right)\Big],\\ {}{f}_{22}^{\prime }=k\left(n-{n}_0\right)\left(1-\upbeta \right)\left[2{D}^{c-1}{\left(1+{D}^c\right)}^{-1}\ln D-c{\left(\ln D\right)}^2\left({D}^{2c-1}{\left(1+{D}^c\right)}^2+{D}^{c-1}{\left(1+{D}^c\right)}^{-1}\right)\right]\\ {}-k\left(n-{n}_0\right)\left(1-\upbeta \right)\Big[{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}2\ln D+2c\left(1-\upbeta \right){\left(\ln D\right)}^2\\ {}\times \left\{-{D}^{3c-1}{\left(1+{D}^c\right)}^{-3}+{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}\right\}\Big]\\ {}+\left(k+1\right)\left(1-\upbeta \right)\sum \limits_{i=1}^n{\updelta}_{2i}\Big[2{A}^{c-1}{\left(1+{A}^c\right)}^{-1}\ln A+c{\left(\ln A\right)}^2\\ {}\times \left\{-{\left(1+{A}^c\right)}^{-2}{A}^{2c-1}+{\left(1+{A}^c\right)}^{-1}{A}^{c-1}\right\}\Big]\\ {}-\left(k+1\right)\left(1-\upbeta \right)\sum \limits_{i=1}^n{\updelta}_{2i}\Big[2{A}^{2c-1}{\left(1+{A}^c\right)}^{-2}\ln A+2c{\left(\ln A\right)}^2\left\{{\left(1+{A}^c\right)}^{-2}{A}^{2c-1}-{\left(1+{A}^c\right)}^{-3}{A}^{3c-1}\right\},\\ {}{f}_{33}^{\prime }=0,\\ {}{f}_{23}^{\prime }=\sum \limits_{i=1}^n{\updelta}_{2i}\left(1-\upbeta \right)\Big[2{A}^{c-1}{\left(1+{A}^c\right)}^{-1}\ln A+c{\left(\ln A\right)}^2{\left(1+{A}^c\right)}^{-1}{A}^{c-1}-{\left(1+{A}^c\right)}^{-1}{A}^{2c-1}\\ {}+\left(n-{n}_0\right)\left(1-\upbeta \right)\left[-{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}c\ln D+{\left(1+{D}^c\right)}^{-1}{D}^{c-1}\left\{c\ln D+1\right\}\right],\\ {}{f}_{12}^{\prime }=\sum \limits_{i=1}^n{\updelta}_{2i}\left[{A}^{-1}+\left(1-\upbeta \right)\left({y}_i-\uptau \right){A}^{-2}\right]+k\left(n-{n}_0\right)\left\{\left(1-\upbeta \right)\left({Y}_c-\uptau \right)\right\}\\ {}\times \left[{D}^{c-2}\left(c-1\right){\left(1+{D}^c\right)}^{-1}-c{\left(1+{D}^c\right)}^{-2}{D}^{2\left(c-1\right)}\right]-{D}^{c-1}{\left(1+{D}^c\right)}^{-1}\\ {}-c{\left(1+{D}^c\right)}^{-1}{D}^{c-1}\ln D+c\left({Y}_c-\uptau \right)\left(1-\upbeta \right)\Big\{-{D}^{2\left(c-1\right)}c{\left(1+{D}^c\right)}^{-2}\ln D\\ {}+{D}^{c-2}{\left(1+{D}^c\right)}^{-1}\left[1+\left(c-1\right)\ln D\right]\Big\}+c{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}\ln D-c\left(1-\upbeta \right)\left({Y}_c-\uptau \right)\\ {}\times \left[{D}^{2c-2}{\left(1+{D}^c\right)}^{-2}-2c{\left(1+{D}^c\right)}^{-3}{D}^{3c-2}\ln D+\left(2c-1\right){\left(1+{D}^c\right)}^{-2}{D}^{2c-2}\ln D\right]\Big\}\\ {}+\left(k+1\right)\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right)\left\{\left(1-\upbeta \right)\right[\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-c{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}-{c}^2{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\ln A\\ {}+c{A}^{c-2}{\left(1+{A}^c\right)}^{-1}\left(1+\left(c-1\right)\ln A\right)-c{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\\ {}+2{c}^2{A}^{3c-2}{\left(1+{A}^c\right)}^{-3}\ln A-c\left(2c-1\right){A}^{2c-2}{\left(1+{A}^c\right)}^{-2}\ln A\Big\}\\ {}-\left[{A}^{c-1}{\left(1+{A}^c\right)}^{-1}+c{\left(1+{A}^c\right)}^{-1}{A}^{c-1}\ln A-c{A}^{2c-1}{\left(1+{A}^c\right)}^{-2}\ln A\right],\\ {}{f}_{13}^{\prime }=\sum \limits_{i=1}^n{\updelta}_{2i}c\Big\{\left({y}_i-\uptau \right)\left(1-\upbeta \right)\left[\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-c{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right]\\ {}-{A}^{c-1}{\left(1+{A}^c\right)}^{-1}\left\}+\left(n-{n}_0\right)c\right[\left({y}_{(r)}-\uptau \right)\left(1-\upbeta \right)\Big[\left(c-1\right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}\\ {}-c{D}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}\left]-{D}^{c-1}{\left(1+{D}^c\right)}^{-1}\right].\end{array}} $$
