Appendix
1.1 Lehmann Family of Distributions
1.1.1 Normal Equations for the Type-II Censoring Case
The normal equations associated with the log-likelihood function (13) are given by
$$\displaystyle \begin{aligned} \frac{\partial l^{II}}{\partial \alpha_{1}} = & \frac{n_{1}}{\alpha_{1}} + \sum_{k=1}^{n_{1}}\ln G_0(t_{k:n};\lambda_1) -\frac{(n-n_1)\{G_0(\tau;\lambda_1)\}^{\alpha_1}\ln G_0(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}}=0 ,{} \end{aligned} $$
(A.1)
$$\displaystyle \begin{aligned} \\ \frac{\partial l^{II}}{\partial \lambda_{1}} = & (\alpha_1-1)\sum_{k=1}^{n_1}\frac{m_0(t_{k:n};\lambda_1)}{G_0(t_{k:n};\lambda_1)} -\frac{(n-n_1)\alpha_1 \{G_0(\tau;\lambda_1)\}^{\alpha_1-1}m_0(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}} \\ & +\sum_{k=1}^{n_1}\frac{n_0(t_{k:n};\lambda_1)}{g_0(t_{k:n};\lambda_1)}=0, {} \end{aligned} $$
(A.2)
$$\displaystyle \begin{aligned} \\ \frac{\partial l^{II}}{\partial \alpha_{2}} = & \frac{n_{2}}{\alpha_{2}} - \sum_{k=n_1+1}^{r}\ln G_0(t_{k:n};\lambda_1) +\frac{(n-n_1)\{G_0(\tau;\lambda_2)\}^{\alpha_2}\ln G_0(\tau;\lambda_2)}{1-\{G_0(\tau;\lambda_2)\}^{\alpha_2}} \\ & -\frac{(n-r)\{G_0(t_{r:n};\lambda_2)\}^{\alpha_2}\ln G_0(t_{r:n};\lambda_2)}{1-\{G_0(t_{r:n};\lambda_2)\}^{\alpha_2}}=0 ,{} \end{aligned} $$
(A.3)
$$\displaystyle \begin{aligned} \\ \frac{\partial l^{II}}{\partial \lambda_{2}} = & (\alpha_2-1)\sum_{k=n_1+1}^{r}\frac{m_0(t_{k:n};\lambda_2)}{G_0(t_{k:n};\lambda_2)} +\frac{(n-n_1)\alpha_2 \{G_0(\tau;\lambda_2)\}^{\alpha_2-1}m_0(\tau;\lambda_2)}{1-\{G_0(\tau;\lambda_2)\}^{\alpha_2}} \\ & +\sum_{k=n_1+1}^{r}\frac{n_0(t_{k:n};\lambda_2)}{g_0(t_{k:n};\lambda_2)}-\frac{(n-r)\alpha_2 \{G_0(t_{r:n};\lambda_2)\}^{\alpha_2-1}m_0(t_{r:n};\lambda_2)}{1-\{G_0(t_{r:n};\lambda_2)\}^{\alpha_2}}=0, {} \end{aligned} $$
(A.4)
where
$$\displaystyle \begin{aligned} m_0(.;\lambda_1) &= \frac{\partial }{\partial \lambda_{1}}G_0(.;\lambda_1) ,\:\:\: n_0(.;\lambda_1) = \frac{\partial }{\partial \lambda_{1}}g_0(.;\lambda_1), \\ m_0(.;\lambda_2) &= \frac{\partial }{\partial \lambda_{2}}G_0(.;\lambda_2) ,\:\:\: n_0(.;\lambda_2) = \frac{\partial }{\partial \lambda_{2}}g_0(.;\lambda_2) . \end{aligned} $$
Now multiplying (A.1) by \(\dfrac {\alpha _1 m_0(\tau ;\lambda _1)}{G_0(\tau ;\lambda _1)}\) and (A.2) by \(\ln G_0(\tau ;\lambda _1),\) respectively, we have
$$\displaystyle \begin{aligned} \frac{n_{1}m_0(\tau;\lambda_1)}{G_0(\tau;\lambda_1)} &-\frac{(n-n_1)\alpha_1\{G_0(\tau;\lambda_1)\}^{\alpha_1-1}m_0(\tau;\lambda_1) \ln G_0(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}} \\ &+ \frac{\alpha_{1}m_0(\tau;\lambda_1)}{G_0(\tau;\lambda_1)} \sum_{k=1}^{n_{1}}\ln G_0(t_{k:n};\lambda_1)= 0 , \end{aligned} $$
(A.5)
$$\displaystyle \begin{aligned} &-\frac{(n-n_1)\alpha_1\{G_0(\tau;\lambda_1)\}^{\alpha_1-1}m_0(\tau;\lambda_1) \ln G_0(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}}\\ &+ \ln G_0(\tau;\lambda_1)\sum_{k=1}^{n_1}\frac{n_0(t_{k:n};\lambda_1)}{g_0(t_{k:n};\lambda_1)} +(\alpha_1-1)\ln G_0(\tau;\lambda_1)\sum_{k=1}^{n_1}\frac{m_0(t_{k:n};\lambda_1)}{G_0(t_{k:n};\lambda_1)} = 0. \end{aligned} $$
(A.6)
Subtracting (A.6) from (A.5), and after little simplification, finally we establish the following relation and \(\widehat {\alpha }_{1}(\lambda _{1})\) is
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} {} & &\displaystyle \dfrac{ \ln [G_0(\tau;\lambda_1)] \sum_{k=1}^{n_{1}}\dfrac{n_o(t_{k:n};\lambda_1)}{g_0(t_{k:n};\lambda_1)} - \dfrac{n_{1}m_o(\tau;\lambda_1)}{G_0(\tau;\lambda_1)}- \ln [G_0(\tau;\lambda_1)] \sum_{k=1}^{n_{1}}\dfrac{m_o(t_{k:n};\lambda_1)}{G_0(t_{k:n};\lambda_1)}}{\dfrac{ m_o(\tau;\lambda_1)}{G_0(\tau;\lambda_1)} -\ln [G_0(\tau;\lambda_1)] \sum_{k=1}^{n_{1}}\dfrac{m_o(t_{k:n};\lambda_1)}{G_0(t_{k:n};\lambda_1)} }. \\ \end{array} \end{aligned} $$
(A.7)
1.1.2 Normal Equations for the Complete Sample Case
$$\displaystyle \begin{aligned} \frac{\partial l^{c}}{\partial \alpha_{1}} &=\frac{n_{1}}{\alpha_{1}} + \sum_{k=1}^{n_{1}}\ln G_0(t_{k:n};\lambda_1) -\frac{(n-n_1)\{G_0(\tau;\lambda_1)\}^{\alpha_1}\ln G_0(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}}=0 , \\ \\ \frac{\partial l^{c}}{\partial \lambda_{1}} &=(\alpha_1-1)\sum_{k=1}^{n_1}\frac{m_o(t_{k:n};\lambda_1)}{G_0(t_{k:n};\lambda_1)}-\frac{(n-n_1)\alpha_1 \{G_0(\tau;\lambda_1)\}^{\alpha_1-1}m_o(\tau;\lambda_1)}{1-\{G_0(\tau;\lambda_1)\}^{\alpha_1}} \\ &\quad +\sum_{k=1}^{n_1}\frac{n_o(t_{k:n};\lambda_1)}{g_0(t_{k:n};\lambda_1)} =0, \\ \\ \frac{\partial l^{c}}{\partial \alpha_{2}} &=\frac{n_{2}}{\alpha_{2}} - \sum_{k=n_1+1}^{r}\ln G_0(t_{k:n};\lambda_1) +\frac{(n-n_1)\{G_0(\tau;\lambda_2)\}^{\alpha_2}\ln G_0(\tau;\lambda_2)}{1-\{G_0(\tau;\lambda_2)\}^{\alpha_2}} =0 ,\\ \\ \frac{\partial l^{c}}{\partial \lambda_{2}} &=(\alpha_2-1)\sum_{k=n_1+1}^{r}\frac{m_o(t_{k:n};\lambda_2)}{G_0(t_{k:n};\lambda_2)} +\frac{(n-n_1)\alpha_2 \{G_0(\tau;\lambda_2)\}^{\alpha_2-1}m_o(\tau;\lambda_2)}{1-\{G_0(\tau;\lambda_2)\}^{\alpha_2}} \\ & \quad +\sum_{k=n_1+1}^{r}\frac{n_o(t_{k:n};\lambda_2)}{g_0(t_{k:n};\lambda_2)}=0. \end{aligned} $$
1.2 Special Case: GE Distribution
1.2.1 Normal Equations for the Type-II Censoring Case
$$\displaystyle \begin{aligned} \frac{\partial l_{\mathrm{GE}}}{\partial \alpha_{1}} & = \frac{n_{1}}{\alpha_{1}} + \sum_{k=1}^{n_{1}}\ln (1-e^{-\lambda_{1} t_{k:n}}) - A(\alpha_{1},\lambda_{1})= 0, \end{aligned} $$
(A.8)
$$\displaystyle \begin{aligned} \frac{\partial l_{\mathrm{GE}}}{\partial \lambda_{1}} &=\frac{n_{1}}{\lambda_{1}} - \sum_{k=1}^{n_{1}}t_{k:n} +(\alpha_{1}-1)\sum_{k=1}^{n_{1}}\frac{t_{k:n}e^{-\lambda_{1}t_{k:n}}}{1-e^{-\lambda_{1}t_{k:n}}} - B(\alpha_{1},\lambda_{1})=0 , \end{aligned} $$
(A.9)
$$\displaystyle \begin{aligned} \frac{\partial l_{\mathrm{GE}}}{\partial \alpha_{2}} &= \frac{n_{2}}{\alpha_{2}} + \sum_{k=n_{1} +1}^{n_{1} + n_{2}}\ln (1-e^{-\lambda_{2} t_{k:n}}) + C_{1}(\alpha_{2},\lambda_{2}) -C_{2}(\alpha_{2},\lambda_{2}) =0, \end{aligned} $$
(A.10)
$$\displaystyle \begin{aligned} \frac{\partial l_{\mathrm{GE}}}{\partial \lambda_{2}} &= \frac{n_{2}}{\lambda_{2}}-\sum_{k=n_{1} +1}^{n_{1} + n_{2}}t_{k:n} +(\alpha_{2}-1)\sum_{k=n_{1} +1}^{n_{1} + n_{2}}\frac{t_{k:n}e^{-\lambda_{2}t_{k:n}}}{1-e^{-\lambda_{2}t_{k:n}}} + D_{1}(\alpha_{2},\lambda_{2}) \\ & \quad -D_{2}(\alpha_{2},\lambda_{2}) =0 , \end{aligned} $$
(A.11)
where
$$\displaystyle \begin{aligned} A(\alpha_{1},\lambda_{1}) &=\dfrac{(n-n_{1})\big(1-e^{-\lambda_{1} \tau}\big)^{\alpha_{1}}}{1-\big(1-e^{-\lambda_{1} \tau}\big)^{\alpha_{1}}}\ln \big(1-e^{-\lambda_{1} \tau}\big), \\ B(\alpha_{1},\lambda_{1})&=\dfrac{(n-n_{1})\alpha_{1}\tau e^{-\lambda_{1}\tau}\big(1-e^{-\lambda_{1} \tau}\big)^ {\alpha_{1}-1}}{1-\big(1-e^{-\lambda_{1} \tau}\big)^ {\alpha_{1}}}, \\ C_{1}(\alpha_{2},\lambda_{2})& =\dfrac{(n-n_{1})\big(1-e^{-\lambda_{2} \tau}\big)^{\alpha_{2}}}{1-\big(1-e^{-\lambda_{2} \tau}\big)^{\alpha_{2}}}\ln \big(1-e^{-\lambda_{2} \tau}\big), \\ C_{2}(\alpha_{2},\lambda_{2}) & = \dfrac{(n-r)\big(1-e^{-\lambda_{2} t_{r:n}}\big)^{\alpha_{2}}}{1-\big(1-e^{-\lambda_{2} t_{r:n}}\big)^{\alpha_{2}}}\ln \big(1-e^{-\lambda_{2} t_{r:n}}\big), \\ D_{1}(\alpha_{2},\lambda_{2})&= \dfrac{(n-n_{1})\alpha_{2}\tau\big(1-e^{-\lambda_{2} \tau}\big)^{\alpha_{2}-1}e^{-\lambda_{2}\tau}}{1-\big(1-e^{-\lambda_{2} \tau}\big)^{\alpha_{2}}}, \\ D_{2}(\alpha_{2},\lambda_{2})&=\dfrac{(n-r)\alpha_{2}t_{r:n}\big(1-e^{-\lambda_{2} t_{r:n}}\big)^{\alpha_{2}-1}e^{-\lambda_{2}t_{r:n}}}{1-\big(1-e^{-\lambda_{2} t_{r:n}}\big)^{\alpha_{2}}} . \end{aligned} $$
1.3 Elements of the Fisher Information Matrix
1.3.1 GE Distribution
The Fisher information matrix I
GE(α
1, λ
1, α
2, λ
2) can be expressed using two block diagonal matrices, viz., \( I^{\mathrm {GE}}_{1}(\alpha _{1},\lambda _{1})\) and \( I^{\mathrm {GE}}_{2}(\alpha _{2},\lambda _{2})\). Thus, we have
$$\displaystyle \begin{aligned} I^{\mathrm{GE}}(\alpha_{1},\lambda_{1},\alpha_{2},\lambda_{2})= \begin{bmatrix} I^{\mathrm{GE}} _{1}(\alpha_{1},\lambda_{1}) & \textbf{0} \\ \textbf{0} & I^{\mathrm{GE}}_{2}(\alpha_{2},\lambda_{2}) \\ \end{bmatrix} \end{aligned}$$
The elements of \(\:\:I^{\mathrm {GE}}_{1}(\alpha _{1},\lambda _{1})= \begin {bmatrix} -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{1}^2} & -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{1} \partial \lambda _{1}}\\ \\ -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{1} \partial \lambda _{1}} & -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \lambda _{1}^2} \\ \end {bmatrix}\) are
$$\displaystyle \begin{aligned} \frac{\partial^2 l^{\mathrm{GE}}}{\partial \alpha_{1}^2} & = -\bigg( \frac{n_{1}}{\alpha_{1}^{2}} + \frac{(n-n_{1})\big(1-e^{-\lambda_{1}\tau}\big)^{\alpha_{1}}\{\ln[1-e^{-\lambda_{1}\tau}]\}^2}{\{1-\big(1-e^{-\lambda_{1}\tau}\big)^{\alpha_ {1}}\}^{2}} \bigg), \\ {} \frac{\partial^2 l^{\mathrm{GE}}}{\partial \alpha_{1} \partial \lambda_{1}} & = \sum_{k=1}^{n_{1}}\frac{t_{k:n}e^{-\lambda_{1}t_{k:n}}}{1-e^{-\lambda_{1}t_{k:n}}} - (n-n_{1}) \tau e^{-\lambda_{1}\tau}\varrho_1(\alpha_{1},\lambda_{1}), \\ {} \frac{\partial^2 l^{\mathrm{GE}}}{\partial \lambda_{1}^2} & = -\bigg( \frac{n_{1}}{\lambda_{1}^{2}} - (\alpha_{1}-1) \psi_1(\lambda_{1}) + (n-n_{1})\alpha_{1} \tau \kappa_1(\alpha_{1},\lambda_{1},\tau) \bigg), \end{aligned} $$
where
$$\displaystyle \begin{aligned} \varrho_1(\alpha_{1},\lambda_{1}) & = \frac{\{1-r(\lambda_{1})^{\alpha_{1}}\}\big[r(\lambda_{1})^{\alpha_{1}}\{\alpha_{1}\ln(r(\lambda_{1}) + 1\}\big]+ \alpha_{1}r(\lambda_{1})^{2\alpha_{1}}\ln(r(\lambda_{1})} {r(\lambda_{1})\{1-r(\lambda_{1})^{\alpha_{1}}\}^2}, &&\\ \psi_1(\lambda_{1}) & = - \sum_{k=1}^{n_{1}}\frac{t_{k:n}^{2}e^{-\lambda_{1}t_{k:n}}}{(1-e^{-\lambda_{1}t_{k:n}})^{2}}, &&\\ \kappa_1(\alpha_{1},\lambda_{1}) & = \frac{\{1-r(\lambda_{1})^{\alpha_{1}}\}\big[r(\lambda_{1})^{\alpha_{1}-2}\tau e^{-\lambda_{1}\tau} \{\alpha_{1} e^{-\lambda_{1} \tau} - 1\}\big]+ \alpha_{1}\tau e^{-2 \lambda_{1} \tau}r(\lambda_{1})^{2\alpha_{1}-2}} {\{1-r(\lambda_{1})^{\alpha_{1}}\}^2},&&\\ r_1(\lambda_{1}) &= \big(1-e^{-\lambda_{1}\tau}\big). \end{aligned} $$
The elements of \(\:\:I^{\mathrm {GE}}_{2}(\alpha _{2},\lambda _{2})= \begin {bmatrix} -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{2}^2} & -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{2} \partial \lambda _{2}}\\ {} -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \alpha _{1} \partial \lambda _{1}} & -\dfrac {\partial ^2 l^{\mathrm {GE}}}{\partial \lambda _{2}^2} \\ \end {bmatrix} \) are
$$\displaystyle \begin{aligned} \frac{\partial^2 l^{\mathrm{GE}}}{\partial \alpha_{2}^2} & = -\bigg( \frac{n_{2}}{\alpha_{2}^{2}} -(n-n_{1})\beta_1(\alpha_{2},\lambda_{2}) + (n-r)\eta_1(\alpha_{2},\lambda_{2})\bigg), \\ \frac{\partial^2 l^{\mathrm{GE}}}{\partial \alpha_{2} \partial \lambda_{2}} & = \sum_{k=n_{1}+1}^{n_{1}+n_{2}}\frac{t_{k:n}e^{-\lambda_{2}t_{k:n}}}{1-e^{-\lambda_{2}t_{k:n}}} +(n-n_{1})\xi_1(\alpha_{2},\lambda_{2}) - (n-r)\Upsilon_1(\alpha_{2},\lambda_{2}),\\ \frac{\partial^2 l^{\mathrm{GE}}}{\partial \lambda_{2}^2} & = -\Big( \frac{n_{2}}{\lambda_{2}^{2}} - (n-n_{1})\sigma_1(\alpha_{2},\lambda_{2}) - (\alpha_{2}-1)\delta_1(\lambda_{2})+ (n-r)\zeta_1(\alpha_{2},\lambda_{2})\Big), \end{aligned} $$
where
$$\displaystyle \begin{aligned} \beta_1(\alpha_{2},\lambda_{2}) & =\frac{\big(1-e^{-\lambda_{2}\tau}\big)^{\alpha_{2}}\{\ln[1-e^{-\lambda_{2}\tau}]\}^2}{\{1-(1-e^{-\lambda_{2}\tau})^{\alpha_{2}}\}^2}, \\ \\ \eta_1(\alpha_{2},\lambda_{2}) & = \frac{\big(1-e^{-\lambda_{2} t_{r:n}}\big)^{\alpha_{2}}\{\ln[1-e^{-\lambda_{2}t_{r:n}}]\}^2}{\{1-\big(1-e^{-\lambda_{2}t_{r:n}}\big)^{\alpha_{2}}\}^2}, \\ \\ \xi_1(\alpha_{2},\lambda_{2}) & = \tau e^{-\lambda_{2}\tau} \frac{\{1-q_1(\lambda_{2})^{\alpha_{2}}\}\big[q_1(\lambda_{2})^{\alpha_{2}}\{\alpha_{2}\ln(q_1(\lambda_{2}) + 1\}\big]} {q_1(\lambda_{2})\{1-q_1(\lambda_{2})^{\alpha_{2}}\}^2} &&\\ \\ & \:\:\:\:+ \tau e^{-\lambda_{2}\tau} \frac{ \alpha_{2}q_1(\lambda_{2})^{2\alpha_{2}}\ln(q_1(\lambda_{2})} {q_1(\lambda_{2})\{1-q_1(\lambda_{2})^{\alpha_{2}}\}^2}, &&\\ \\ \Upsilon_1(\alpha_{2},\lambda_{2}) & = t_{r:n} e^{-\lambda_{2}t_{r:n}} \frac{\{1-s_1(\lambda_{2})^{\alpha_{2}}\}\big[s_1(\lambda_{2})^{\alpha_{2}}\{\alpha_{2}\ln(s_1(\lambda_{2}) + 1\}\big]} {s_1(\lambda_{2})\{1-s_1(\lambda_{2})^{\alpha_{2}}\}^2} &&\\ \\ & \:\:\:\:+ t_{r:n} e^{-\lambda_{2}t_{r:n}} \frac{ \alpha_{2}s_1(\lambda_{2})^{2\alpha_{2}}\ln(s_1(\lambda_{2})} {s_1(\lambda_{2})\{1-s_1(\lambda_{2})^{\alpha_{2}}\}^2}, &&\\ \sigma_1(\alpha_{2},\lambda_{2}) & = \alpha_{2} \tau \frac{\{1-q_1(\lambda_{2})^{\alpha_{2}}\}\big[q_1(\lambda_{2})^{\alpha_{2}-2}\tau e^{-\lambda_{2}\tau} \{\alpha_{2} e^{-\lambda_{2} \tau} - 1\}\big]} {\{1-q_1(\lambda_{2})^{\alpha_{2}}\}^2}&&\\ \\ & \:\:\:\:+ \alpha_{2} \tau \frac{ \alpha_{2}\tau e^{-2 \lambda_{2} \tau}q_1(\lambda_{2})^{2\alpha_{2}-2}} {\{1-q_1(\lambda_{2})^{\alpha_{2}}\}^2},&&\\ \\ \delta(\lambda_{2}) & = - \sum_{k=n_{1}+1}^{n_{1}+n_{2}}\left[\frac{t_{k:n}^{2}e^{-\lambda_{2}t_{k:n}}}{\big(1-e^{-\lambda_{2}t_{k:n}}\big)^{2}} \right], &&\\ \\ q_1(\lambda_{2}) &= \big(1-e^{-\lambda_{2}\tau}\big), \: s_1(\lambda_{2}) = \big(1-e^{-\lambda_{2}t_{r:n}}\big). \\ \zeta_1(\alpha_{2},\lambda_{2}) & = \alpha_{2} t_{r:n} \frac{\{1-s_1(\lambda_{2})^{\alpha_{2}}\}\big[s_1(\lambda_{2})^{\alpha_{2}-2}t_{r:n} e^{-\lambda_{2}t_{r:n}} \{\alpha_{2} e^{-\lambda_{2} t_{r:n}} - 1\}\big]} {\{1-s_1(\lambda_{2})^{\alpha_{2}}\}^2}&&\\ \\ & \:\:\:\:+ \alpha_{2} t_{r:n} \frac{ \alpha_{2}t_{r:n} e^{-2 \lambda_{2} t_{r:n}}s_1(\lambda_{2})^{2\alpha_{2}-2}} {\{1-s_1(\lambda_{2})^{\alpha_{2}}\}^2}.&&\\ \\ \end{aligned} $$
1.3.2 GR Distribution with Different Shape and Common Scale Parameter
Let the Fisher information matrix associated with the parameters α
1, λ, α
2, respectively, be
$$\displaystyle \begin{aligned} \:\:I_{\mathrm{sc}}^{\mathrm{GR}}(\alpha_{1},\lambda,\alpha_2)= \begin{bmatrix} -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1}^2} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \lambda} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \alpha_2}\\ {} -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \lambda} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \lambda^2} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{2} \partial \lambda} \\ {} -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \alpha_2} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{2} \partial \lambda} & -\dfrac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{ \partial \alpha_{2}^2}\\ \end{bmatrix}\end{aligned} $$
The corresponding elements are
$$\displaystyle \begin{aligned} \frac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1}^2} & = -\bigg( \frac{n_{1}}{\alpha_{1}^{2}} + \frac{(n-n_{1})\big(1-e^{-\lambda\tau^2}\big)^{\alpha_{1}}\{\ln[1-e^{-\lambda\tau^2}]\}^2}{\{1-\big(1-e^{-\lambda\tau^2}\big)^{\alpha_ {1}}\}^{2}} \bigg), \\ {} \frac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \lambda} & = \sum_{k=1}^{n_{1}}\frac{t_{k:n}^{2}e^{-\lambda t_{k:n}^{2}}}{1-e^{-\lambda t_{k:n}^{2}}} - (n-n_{1})\varrho_3(\alpha_{1},\lambda), \: \: \: \: \frac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{1} \partial \alpha_2} = 0, \\ {} \frac{\partial^2 l^{\mathrm{GR}}}{\partial \lambda^2} & = -\bigg( \frac{n}{\lambda^{2}} - (\alpha_{1}-1) \psi_3(\lambda) + (n-n_{1})\kappa_3(\alpha_{1},\lambda) - (n-n_{1})\sigma_3(\alpha_{2},\lambda) \\ & \:\:\quad - (\alpha_{2}-1)\delta_3(\lambda) \bigg), \\ {} \frac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{2} \partial \lambda} & = \sum_{k=n_{1}+1}^{n_{1}+n_{2}}\frac{t_{k:n}^{2}e^{-\lambda t_{k:n}^{2}}}{1-e^{-\lambda t_{k:n}^{2}}} +(n-n_{1})\xi_3(\alpha_{2},\lambda), \\ \frac{\partial^2 l_{\mathrm{sc}}^{\mathrm{GR}}}{\partial \alpha_{2}^2} & = -\bigg( \frac{n_{2}}{\alpha_{2}^{2}} + \frac{(n-n_{1})\big(1-e^{-\lambda\tau^2}\big)^{\alpha_{2}}\{\ln[1-e^{-\lambda\tau^2}]\}^2}{\{1-(1-e^{-\lambda\tau^2})^{\alpha_ {2}}\}^{2}} \bigg), \end{aligned} $$
where
$$\displaystyle \begin{aligned} \varrho_3(\alpha_{1},\lambda) & = \tau^2 e^{-\lambda \tau^2} \frac{\{1-r_3(\lambda)^{\alpha_{1}}\}\big[r_3(\lambda)^{\alpha_{1}}\{\alpha_{1}\ln(r_3(\lambda) + 1\}\big]} {r(\lambda)\{1-r_3(\lambda)^{\alpha_{1}}\}^2} &&\\ & \:\:\:\:+ \tau^2 e^{-\lambda \tau^2} \frac{ \alpha_{1}r_3(\lambda)^{2\alpha_{1}}\ln(r_3(\lambda)} {r(\lambda)\{1-r_3(\lambda)^{\alpha_{1}}\}^2}, &&\\ {} \psi_3(\lambda) & = - \sum_{k=1}^{n_{1}}\frac{t_{k:n}^{4}e^{-\lambda t_{k:n}^{2}}}{\big(1-e^{-\lambda t_{k:n}^{2}}\big)^{2}}, &&\\ {} \kappa_3(\alpha_{1},\lambda_{1}) & = \alpha_{1} \tau^{2} \frac{\{1-r_3(\lambda)^{\alpha_{1}}\}\big[r_3(\lambda)^{\alpha_{1}-2}\tau^{2} e^{-\lambda \tau^{2}} \{\alpha_{1} e^{-\lambda \tau^{2}} - 1\}\big]} {\{1-r_3(\lambda)^{\alpha_{1}}\}^2}&&\\ & \:\:\:\:+ \alpha_{1} \tau^{2} \frac{ \alpha_{1}\tau^{2} e^{-2 \lambda \tau^{2}}r_3(\lambda)^{2\alpha_{1}-2}} {\{1-r_3(\lambda)^{\alpha_{1}}\}^2},&&\\ {} \sigma_3(\alpha_{2},\lambda) & = \alpha_{2} \tau^{2} \frac{\{1-r_3(\lambda)^{\alpha_{2}}\}\big[r_3(\lambda)^{\alpha_{2}-2}\tau^{2} e^{-\lambda \tau^{2}} \{\alpha_{2} e^{-\lambda \tau^{2}} - 1\}\big]} {\{1-r_3(\lambda)^{\alpha_{2}}\}^2}&&\\ & \:\:\:\:+ \alpha_{2} \tau^{2} \frac{\alpha_{2}\tau^{2} e^{-2 \lambda \tau^{2}}r_3(\lambda)^{2\alpha_{2}-2}} {\{1-r_3(\lambda)^{\alpha_{2}}\}^2},&&\\ {} \delta_3(\lambda) & = - \sum_{k=n_{1}+1}^{n_{1}+n_{2}}\left[\frac{t_{k:n}^{4}e^{-\lambda t_{k:n}^{2}}}{\big(1-e^{-\lambda t_{k:n}^{2}}\big)^{2}} \right], \\ {} \xi_3(\alpha_{2},\lambda) & = \tau^{2} e^{-\lambda \tau^{2}} \frac{\{1-r_3(\lambda)^{\alpha_{2}}\}\big[r_3(\lambda)^{\alpha_{2}}\{\alpha_{2}\ln(r_3(\lambda) + 1\}\big]} {r_3(\lambda)\{1-r_3(\lambda)^{\alpha_{2}}\}^2} &&\\ & \:\:\:\:+ \tau^{2} e^{-\lambda \tau^{2}} \frac{ \alpha_{2}r_3(\lambda)^{2\alpha_{2}}\ln(r_3(\lambda)} {r_3(\lambda)\{1-r_3(\lambda)^{\alpha_{2}}\}^2}, &&\\ {} r_3(\lambda) &= \big(1-e^{-\lambda \tau^{2}}\big). \end{aligned} $$
