Appendix A
Derivation of the second-order partial derivatives:
$$ {\displaystyle \begin{array}{c}\frac{\partial^2\ln L}{\partial^2{\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){cA}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right]\\ {}- kc\left(n-{n}_0\right)\left({Y}_c-\uptau \right)\left[\left(c-1\right)\left({Y}_c-\uptau \right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}-\left({Y}_c-\uptau \right){cD}^{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}\left[\left(c-1\right){A}^{c-1}{\left(1+{A}^c\right)}^{-1}-{cA}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right]\\ {}- kc\left(n-{n}_0\right){\left({Y}_c-\uptau \right)}^2\left[\left(c-1\right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}-{cD}^{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}_c-\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-{cD}^{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-{cA}^{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}_c-\uptau \right)\left(n-{n}_0\right){cD}^{c-1}{\left(1+{D}^c\right)}^{-1},\\ {}\frac{\partial^2\ln L}{\partial {c}^2}=-\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
$$ \frac{\partial^2\ln L}{\partial {k}^2}=-\frac{n_0}{k^2}. $$
Appendix B
The determinant of F and its partial derivative with respect to τ. The determinant of F can be written as
$$ \left|F\right|={f}_{11}\left({f}_{22}{f}_{33}-{f}_{23}^2\right)-{f}_{12}\left({f}_{12}{f}_{33}-{f}_{13}{f}_{23}\right)+{f}_{13}\left({f}_{12}{f}_{23}-{f}_{13}{f}_{22}\right), $$
and then
$$ {\displaystyle \begin{array}{c}\frac{\partial \left|F\right|}{\mathrm{\partial \uptau }}={f}_{11}\left({f}_{22}^{\prime }{f}_{33}+{f}_{22}{f}_{33}^{\prime }-2{f}_{23}{f}_{23}^{\prime}\right)+{f}_{11}^{\prime}\left({f}_{22}{f}_{33}-{f}_{23}^2\right)\\ {}-{f}_{12}\left({f}_{12}^{\prime }{f}_{33}+{f}_{12}{f}_{33}^{\prime }-{f}_{13}^{\prime }{f}_{23}-{f}_{13}{f}_{23}^{\prime}\right)-{f}_{12}^{\prime}\left({f}_{12}{f}_{33}-{f}_{12}{f}_{23}\right)\\ {}+{f}_{13}\left({f}_{12}^{\prime }{f}_{33}+{f}_{12}{f}_{23}^{\prime }-{f}_{13}^{\prime }{f}_{22}-{f}_{13}{f}_{22}^{\prime}\right)+{f}_{13}^{\prime}\left({f}_{12}{f}_{23}-{f}_{13}{f}_{22}\right),\end{array}} $$
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}\left[-2\left({y}_i-\uptau \right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}+{\left({y}_i-\uptau \right)}^2\left(1-\upbeta \right)\left(\left(c-2\right){A}^{c-3}{\left(1+{A}^c\right)}^{-1}\right.\right.\\ {}\left.-{cA}^{2c-3}{\left(1+{A}^c\right)}^{-2}\right]\left(k+1\right)c\sum \limits_{i=1}^nc{\updelta}_{2i}{\left[-2\left({y}_i-\uptau \right){A}^{2\left(c-1\right)}\left(1+{A}^c\right)\right.}^{-2}\\ {}+{\left({y}_i-\uptau \right)}^22\left(1-\upbeta \right)\left(\left(c-1\right){A}^{2c-3}{\left(1+{A}^c\right)}^{-2}\left.\left.-{cA}^{3\left(c-1\right)}{\left(1+{A}^c\right)}^{-3}\right)\right]\right.\\ {}+ kc\left(n-{n}_0\right)\left(c-1\right)\left[-2\left({Y}_c-\uptau \right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}+\left(1-\upbeta \right){\left({Y}_c-\uptau \right)}^2\left(\left(c-2\right){D}^{c-3}{\left(1+{D}^c\right)}^{-1}\right.\right.\\ {}\left.\left.-{cD}^{2c-3}{\left(1+{D}^c\right)}^{-2}\right)\right]-{kc}^2\left(n-{n}_0\right)\left[-2\left({Y}_c-\uptau \right){D}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}\right.\\ {}+2{\left({Y}_c-\uptau \right)}^2\left(1-\upbeta \right)\left(\left(c-1\right){D}^{2c-3}{\left(1+{D}^c\right)}^{-2}-{D}^{3\left(c-1\right)}\left.\left.c{\left(1+{D}^c\right)}^{-3}\right)\right],\right.\\ {}{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.{\left(1+{D}^c\right)}^{-1}\right)\right.\right]\\ {}-k\left(n-{n}_0\right)\left(1-\upbeta \right)\left[{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}2\ln D+2c\left(1-\upbeta \right){\left(\ln D\right)}^2\left\{-{D}^{3c-1}{\left(1+{D}^c\right)}^{-3}\right.\right.\\ {}\left.\left.+{D}^{2c-1}{\left(1+{D}^c\right)}^{-2}\right\}\right]+\left(k+1\right)\left(1-\upbeta \right)\sum \limits_{i=1}^n{\updelta}_{2i}\left[2{A}^{c-1}{\left(1+{A}^c\right)}^{-1}\ln A\right.\\ {}+c{\left(\ln A\right)}^2\left.\left\{-{\left(1+{A}^c\right)}^{-2}{A}^{2c-1}+{\left(1+{A}^c\right)}^{-1}\right\}\right]-\left(k+1\right)\left(1-\upbeta \right)\sum \limits_{i=1}^n{\updelta}_{2i}\left[2{A}^{2c-1}{\left(1+{A}^c\right)}^{-2}\ln A\right.\\ {}+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)\left[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}\right]\\ {}+\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)\left[{D}^{c-2}\left(c-1\right){\left(1+{D}^c\right)}^{-1}-c{\left(1+{D}^c\right)}^{-1}-c{\left(1+{D}^c\right)}^{-2}{D}^{2\left(c-1\right)}\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)\left\{-{D}^{2\left(c-1\right)}c{\left(1+{D}^c\right)}^{-2}\ln D\right.\\ {}+{D}^{c-2}{\left(1+{D}^c\right)}^{-1}\left.\left[1+\left(c-1\right)\ln D\right]\right\}+{c}^{2c-1}{\left(1+{D}^c\right)}^{-2}\ln D\\ {}-c\left(1-\upbeta \right)\left({Y}_c-\uptau \right)\left.\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]\right\}\\ {}+\left(k+1\right)\sum \limits_{i=1}^n{\updelta}_{2i}\left({y}_i-\uptau \right)\left\{\left(1-\upbeta \right)\left[\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-{cA}^{2\left(c-1\right)}-{cA}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right.\right.\\ {}-{c}^2{A}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\ln A+{cA}^{c-2}{\left(1+{A}^c\right)}^{-1}\left(1+\left(c-1\right)\ln A\right)-{cA}^{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.{\left(1+{A}^c\right)}^{-2}\ln A\right\}\\ {}-\left[{A}^{c-1}{\left(1+{A}^c\right)}^{-1}+c{\left(1+{A}^c\right)}^{-1}{A}^{c-1}\ln A-{cA}^{2c-1}{\left(1+{A}^c\right)}^{-2}\ln A\right],\\ {}{f}_{13}^{\prime}\sum \limits_{i=1}^n{\updelta}_{2i}c\left\{\left({y}_i-\uptau \right)\left(1-\upbeta \right)\left[\left(c-1\right){A}^{c-2}{\left(1+{A}^c\right)}^{-1}-{cA}^{2\left(c-1\right)}{\left(1+{A}^c\right)}^{-2}\right]\right.\\ {}\left.-{A}^{c-1}{\left(1+{A}^c\right)}^{-1}\right\}+\left(n-{n}_0\right)c\left[\left({Y}_c-\uptau \right)\left(1-\upbeta \right)\left[\left(c-1\right){D}^{c-2}{\left(1+{D}^c\right)}^{-1}-{cD}^{2\left(c-1\right)}{\left(1+{D}^c\right)}^{-2}\right]-{D}^{c-1}{\left(1+{D}^c\right)}^{-1}\right].\end{array}} $$
