It follows from (1) and (3) that, based on a given type-II censored sample **x** drawn from the GB distribution, the joint PDF of the papulation parameters *β* and *λ* is given by:

$$ L(\beta,\,\lambda|{\mathbf{x}}) \ \propto\ {\beta}^{r}{\lambda}^{r}\, {e^{-2\,\beta\, T_{1}+T_{2}}}, $$

(6)

where

$$\begin{array}{*{20}l} {}T_{1}\!&=\!{(n\,-\,r)\, x^{\lambda}_{r}\,+\,\sum_{j=1}^{r} x^{\lambda}_{j} },\\ {}T_{2}\!&=\!(n\,-\,r)\!\ln\! \left(3\,-\,2\,{e^{-\beta\,x^{\lambda}_{r}}} \right)\,+\,{\lambda}\!\sum_{j=1}^{r}{{\ln(x_{j})}}\,+\,\!\sum_{j=1}^{r} \!\ln\left(\!1\,-\,{e^{-\beta\, x^{\lambda}_{j}}}\!\right)\!. \end{array} $$

### When *λ* is known

In this case, for fixed *λ*, say *λ*=*λ*^{(0)}, let *θ*=1/*β* and \(y_{i}=x_{i}^{\lambda ^{(0)}}\), *i*=1, 2, ⋯ *r*. Then, *y*_{1},⋯,*y*_{r} is a type-II random sample from *B**i**l**a**l*(*θ*) distribution. Abd-Elrahman and Niazi [7] established the existence and uniqueness theorem for the maximum likelihood estimate (MLE) of the parameter *θ*, say \(\hat \theta _{M}\). The MLE for the parameter *β* is then by \(\hat \beta _{M}\left (\lambda ^{(0)}\right)=1/\hat \theta _{M}\). Clearly, \(\hat \beta _{M}\left (\lambda ^{(0)}\right)\) exists and it is unique.

Now, we provide an iterative technique for finding \(\hat \beta _{M}\left (\lambda ^{(0)}\right)\) as follows. Let,

$$ {}\begin{aligned} W_{1}&={\frac{\beta\,{x^{{\lambda}^{(0)}}_{r}}{e^{-\beta\,{x^{{\lambda}^{(0)}}_{r}} }}}{3-2\,{e^{-\beta\,{x^{{\lambda}^{(0)}}_{r}}}}}},\qquad W_{2j}={\frac{\beta\,{x^{{\lambda}^{(0)}}_{j}}{e^{-\beta\,{x^{{\lambda}^{(0)}}_{j}} }}}{1-{e^{-\beta\,{x^{{\lambda}^{(0)}}_{j}}}}}},\\ j&=1, \, 2,\, \cdots,\, r. \end{aligned} $$

(7)

In view of (6) and (7), the likelihood equation of *β* is then given by:

$$\begin{array}{@{}rcl@{}} \frac{\partial\,{\ln L(\beta,\,\lambda^{(0)}|{\mathbf{x}})}}{\partial\,{\beta}}&\,=\,&\frac{r+\,2\, (n\,-\,r) \,W_{1}+\sum_{j=1}^{r}W_{2j}}{\beta}\\&&-{2\left((n\,-\,r)\, {x^{{\lambda}^{(0)}}_{r}}\,+\,\sum_{j=1}^{r}{x^{{\lambda}^{(0)}}_{j}}\right)}. \end{array} $$

For *ν*=0,1,2,⋯, we calculate \(\hat \beta _{M}({\lambda }^{(0)})\) by using the following formula:

$$ {}\begin{aligned} \hat\beta^{(\nu+1)}_{M}&\left(\lambda^{(0)}\right)\\ &=\left.\frac{r+2\, \left(n\,-\,r \right) W_{1}+\sum_{j=1}^{r} W_{2j}}{2\,\left(\left(n\,-\,r \right) {x^{\lambda}_{r}}+\sum_{j=1}^{r}{x^{\lambda}_{j}}\right)} \right|_{\beta=\hat\beta^{(\nu)}_{M}(\lambda^{(0)}),\,\lambda=\lambda^{(0)}}, \end{aligned} $$

(8)

iteratively until some level of accuracy is reached.

###
**Remark 1**

Note that, all of the functions *W*_{1} and *W*_{2j}, *j*=1,2, ⋯, *r*, which appear in (8), need to have some initial value for *β*, say \(\hat \beta ^{(0)}\). This initial value can be obtained based on the available type-II censored sample as if it is complete, see Ng et al. [10]. We use the moment estimator of *β* as a starting point in the iterations (8). That is, in view of (3), \(\hat \beta ^{(0)}\) is given by

$$ \hat{\beta}^{(0)}= \frac{5\,r\,}{6\,{\sum_{i=1}^{r} x_{i}^{\lambda^{(0)}}}}. $$

(9)

### When *β* is known

When *β* is assumed to be known, say *β*^{(0)}, it follows from (6) that the likelihood equation of *λ* is given by

$$ {}\begin{aligned} \frac{\partial\,{\ln L(\beta^{(0)},\,\lambda|{\mathbf{x}})}}{\partial\,{\lambda}}\!&=\! {\frac{r}{\lambda}}\,-\,2\, (n\,-\,r) \ln (x_{r}) \left(\beta^{(0)}{x^{\lambda}_{r}}\,-\,W_{1}\right)\\ &\quad+\sum_{j=1}^{r}\ln (x_{j}) \left(1\,-\,2\!\,\beta^{(0)}{x^{\lambda}_{j}}\,+\,W_{2j} \right), \end{aligned} $$

(10)

where *W*_{1} and *W*_{2j}, *j* = 1,2,⋯,*r*, are as given by (7) after replacing *β*, *λ*^{(0)} by *β*^{(0)} and *λ*, respectively. In order to established the existence and uniqueness of the MLE for *λ*, the following theorem is needed.

###
**Theorem 1**

For a given fixed value of the parameter *β* = *β*^{(0)}, the MLE for the parameter *λ*, \(\hat \lambda _{M}\left (\beta ^{(0)}\right)\), exists and it is unique.

###
*Proof*

See Appendix. □

The MLE \(\hat \lambda _{M}\left (\beta ^{(0)}\right)\) can be iteratively obtained by using Newton’s method, i.e.,

$$ \begin{aligned} \hat\lambda^{(\nu+1)}_{M}\left(\beta^{(0)}\right)&= \hat\lambda^{(\nu)}_{M}\left(\beta^{(0)}\right)\\ &\quad-\left.\left\{ \frac{\lambda\,{\mathcal{G}}_{1} (\beta^{(0)},\,\lambda|{ \mathbf{x}})} {\lambda\,{\mathcal{G}}_{2} (\beta^{(0)},\,\lambda|{\mathbf{x}})+{\mathcal{G}}_{1} \left(\beta^{(0)},\,\lambda|{\mathbf{x}}\right)} \right\} \right|_{\lambda=\hat\lambda^{(\nu)}_{M}\left(\beta^{(0)}\right)}\, {,} \end{aligned} $$

(11)

for *ν*=0,1,2,⋯, where \({\mathcal {G}}_{1}(\cdot,\,\lambda |{\mathbf {x}})\) is as given by (10) and \({\mathcal {G}}_{2}(\cdot,\,\lambda |{\mathbf {x}})\) is the second derivative of ln*L*(·, *λ*|**x**) with respect to (w.r.t.) *λ*, which is given in the “Appendix” section.

###
**Remark 2**

An initial value for *λ*, \(\hat \lambda ^{(0)}_{M}\), can be obtained as follows: (1) Calculate the sample coefficient of variation (CV) based on a given type-II censored sample data as if it is complete. (2) Equating the sample CV with its corresponding CV from the population would results in an equation of *λ* only. (3) \(\hat \lambda ^{(0)}_{M}\) would be the solution of this equation, which provides a good starting point for (11). This technique have been used by, e.g., Kundu and Howlader [11] and Abd-Elrahman [1].

Here, the population CV of the GB distribution is given by

$$ {}\begin{aligned} {\mathcal{C}}(\lambda)&= \sqrt {{\frac{ \left({3}^{m_{2}}-{2}^{m_{2}} \right) \Gamma \left(m_{2} \right) }{ \left({3}^{m_{1}}-{2}^{m_{1}} \right)^{2} \left(\Gamma \left(m_{1} \right) \right)^{2}}}-1},\\ m_{1}&=1+\frac{1}{\lambda},\quad m_{2}=1+\frac{2}{\lambda}. \end{aligned} $$

(12)

### When both *β* and *λ* are unknown

In this case, first an initial value for *λ*, \(\hat \lambda ^{(0)}\), can be obtained as described in “When *β* is known” section. Once \(\hat \lambda ^{(0)}\) is obtained, an initial value for the parameter *β*, \(\hat \beta ^{(0)}\), can be calculated as the right hand side of (9) after replacing *λ*^{(0)} by \(\hat \lambda ^{(0)}\).

Based on the initials \(\hat \beta ^{(0)}\) and \(\hat \lambda ^{(0)}\), an updated value for *β*, \(\hat \beta ^{(1)}\), can be obtained by using (8). Similarly, based on the pair (\(\hat \beta ^{(1)},\hat \lambda ^{(0)}\)), an updated value for *λ*, \(\hat \lambda ^{(1)}\), can be obtained by using (11), and so on. As a stopping rule, the iterations will be terminated after some value *s*<1000 with a level of accuracy, *ε*≤1.2×10^{−7}, which is defined as

$$\epsilon\,=\, \left\vert\frac{\hat\beta^{(s+1)}-\hat\beta^{(s)}} {\hat\beta^{(s)}}\right\vert +\left\vert\frac{\hat\lambda^{(s+1)}-\hat\lambda^{(s)}}{\hat\lambda^{(s)}}\right\vert. $$

Hence, the limiting pair of estimates \(\left (\hat \beta ^{(s)}, \hat \lambda ^{(s)}\right)\) exists and it is unique, which would maximizes the likelihood function (6) w.r.t., the unknown population parameters *β* and *λ*. That is, \(\hat \beta _{M}\,=\,\hat \beta ^{(s)}\) and \(\hat \lambda _{M}\,=\,\hat \lambda ^{(s)}\).

Substituting the values of *β* and *λ* in (4) by their MLEs, the MLE for reliability function *s*(*t*) at some value of *t* = *t*_{0} can then be obtained.

### Fisher information matrix (FIM)

In this section, by using the *missing information principle*, the Fisher information matrix (FIM) about the underlying population parameters based on type-II censoring is provided. Suppose that, **x** = (*x*_{1}, *x*_{2}, …,*x*_{r})^{′} and **Y** = (*X*_{r+1}, *X*_{r+2}, …, *X*_{n})^{′} denote the ordered observed censored and the unobserved ordered data, respectively. The vector **Y** can be thought of as the missing data. Combine **x** and **Y** to form the complete data set **W**. It is easy to show that the amount of information about the unknown parameters *β* and *λ*, which is provided by **W** is given by:

$$\begin{array}{@{}rcl@{}} I_{\mathbf{W}}\left({\beta}, \, \lambda\right) \,=\,\left[\begin{array}{cc} \frac{\,c_{1}}{\beta^{2}}&{\frac{\,c_{2}\,-\,c_{1}\,\ln \left(\beta \right)}{\beta\,\lambda}} \\ {\frac {\,c_{2}\,-\,c_{1}\,\ln \left(\beta \right)}{\beta\,\lambda}}&{\frac {\,c_{3}\,+\,\ln \left(\beta \right) \left\{ c_{1}\,\ln \left(\beta \right)\! -\! c_{4} \right\}}{{\lambda}^{2}}}\end{array} \right] \end{array} $$

(13)

with *c*_{1}=1.92468,*c*_{2}=0.05606,*c*_{3}=1.79061, and *c*_{4}=0.11211.

For *s* = *r* + 1,*r* + 2,…,*n*, the conditional distribution of each *X*_{s}∈**Y** given *X*_{s} > *x*_{r} follows the truncated underlying distribution with left truncation at *x*_{r}, see Ng et al. [10]. Therefore, in view of (1) and (3), the PDF of *X*_{s}∈**Y** given *X*_{s} > *x*_{r} is given by

$$ \begin{aligned} f (x|X_{s}>x_{r};\,\beta,\:\lambda) \!&=\!\frac{6\,\beta\,e^{-2\,\beta\, \left(x^{\lambda}\,-\,x^{\lambda}_{r}\right)}\, \left(1\,-\,e^{-\beta\,x^{\lambda}}\right)} {\left(3\,-\,2\,e^{-\beta\,{x^{\lambda}_{r}}}\right) },\\ &\quad x>x_{r}, \: (\beta,\:\lambda>0). \end{aligned} $$

(14)

Hence, the expected ordered unobserved (missing) information matrix *I*_{Y}(*β*, *λ*), which is related to the vector **Y**, is then given by

$$ \begin{aligned} I_{\mathbf{Y}|\mathbf{x}}(\beta,\,\lambda)\,=\,-(n\,-\,r)\,{{{\mathrm{I}\!\mathrm{E}}}} \left[ \begin{array}{cc} \frac{\partial^{2} \,\ln [f(x|X_{s\,}\!>\!x_{r};\,\beta,\,\lambda)]}{\partial\,\beta^{2}} &\ \frac{\partial^{2} \,\ln [f(x|X_{s\,}\!>\!x_{r};\,\beta,\,\lambda)]}{\partial\,\beta\,\partial\,\lambda} \\ \frac{\partial^{2} \,\ln [f(x|X_{s\,}\!>\!x_{r};\,\beta,\,\lambda)]}{\partial\,\lambda\,\partial\,\beta} &\frac{\partial^{2} \,\ln [f(x|X_{s\,}\!>\!x_{r};\,\beta,\,\lambda)]}{\partial\,\lambda^{2}} \end{array} \right]. \end{aligned} $$

(15)

In order to evaluate of the expectations involved in (15), calculations for the following expressions are required.

1) Part 1

$$ {}I^{(k)}(y)=\int_{y}^{\infty}{\left\{\ln (t)\right\}}^{k}\, G_{1}(t)\,{\mathrm{d}\!\,} t,\qquad y>0,\quad k=0,1,2, $$

(16)

where

$${G_{1}(t)=\frac{t\,{e^{-2\,t}}\, \left[ {t\,{e^{-t}\,+\,\left(1-\,{e^{-t}} \right)\,\left(2-3\,{e^{-t}} \right)} } \right] }{1-{e^{-t}}}}. $$

Denote \( I_{0} = {{{\lim }_{y\,\to \,0^{+}}}} I^{(0)}(y) = 0.32078\), \( I_{1} = {{{\lim }_{y\,\to \,0^{+}}}} I^{(1)}(y) = 0.00934\) and \( I_{2} ={{{\lim }_{y\,\to \,0^{+}}}} I^{(2)}(y) = 0.13177\). Then, (16) can be rewritten as

$$ {}I^{(k)}(y)=I_{k}- \int_{\,0}^{y}{\left\{\ln (t)\right\}}^{k}\, G_{1}(t)\,{\mathrm{d}\!\,} t,\qquad y>0,\quad k=0,1,2. $$

(17)

The integrals involved in (17) can be calculated by using a simple numerical integration tool, e.g., Simpson’s rule.

2) Part 2

$$\begin{array}{@{}rcl@{}} I^{(3)}(y)\!&=&\!\int_{y}^{\infty} {\frac{t^{2}\,e^{-3\,t}}{1-{e^{-t}}}} \, \mathrm{d}\, t \! =\! I_{3 }- \int_{\,0}^{y} {\frac{t^{2}\,e^{-3\,t}}{1-{e^{-t}}}} \, \mathrm{d}\, t,\quad y>0, \\ &=&I_{3 }-\,\sum_{j=0}^{\infty}\left\{\int_{\,0}^{y} { {t^{2}\,e^{-(j+3)\,t}}} \, \mathrm{d}\, t\right\},\\ &=&{e^{-3\,y}}\sum_{j=0}^{\infty }{\frac{\left(1+ \left(1+ \left(3+j \right) y \right)^{2} \right) {e^{-j\,y}}}{ \left(3+j \right)^{3}}}, \end{array} $$

(18)

where \(I_{3 }\,=\,{\lim }_{y\,\to \, 0^{+}} I^{(3)}(y)\,=\,-\frac {9}{4}\,+\,2\, \sum _{i=1}^{\infty }\,i^{-3}\,=\,0.154114\,\).

Now, in view of (17) and (18), it is easy to show that the elements *I*_{i j} of *I*_{Y|x}(*β*, *λ*) after division by (*n* − *r*), *i*, *j*=1,2, are given by

$$\begin{array}{*{20}l} {}I_{11} &= \frac{1}{\beta^{2}}\left\{1 +6\,\left({\frac{e^{-y}I^{(3)}(y)}{3 - 2\,e^{-y}}} {- \frac{y^{2} e^{-y}}{{\left(3 - 2\, e^{-y} \right)}^{2}}}\right) \right\}, \, y\,=\,\beta\,x^{\lambda}_{r}, \end{array} $$

(19)

$$\begin{array}{*{20}l} {}I_{12} &= -\,\frac{6}{\beta\,\lambda} \,\left\{\!\frac{t_{1}(x_{r}) + \left[I^{(0)}{(y)} - \ln \left(\beta \right)I^{(1)}{(y)}\right] \,e^{2\,y}}{\left(3 - 2\,{e^{-y}} \right) }\!\right\} \,=\,I_{21}, \end{array} $$

(20)

$$\begin{array}{*{20}l} I_{22} &= \frac{1}{\lambda^{2}}\begin{array}{l} \left\{{1 + \frac{6 \left[e^{2\,y}\left[{\left(\ln (\beta) \right)}^{2} I^{(0)}{(y)} - 2 \ln(\beta)\,I^{(1)}(y) + I^{(2)}(y)\right] - t_{2}(x_{r})\right]}{ \left(3 - 2\,e^{-y}\right)}}\right\}, \end{array} \end{array} $$

(21)

where

$${}t_{1}(x_{r})\,=\,{\frac {{\beta\,x^{\lambda}_{r}}\ln\! \left(x^{\lambda}_{r} \right)\! \left[\!\left(\! 1\,-\,{e^{-\beta\,{x^{\lambda}_{r}}}} \!\right)\! \left(\! 3\,-\,2\,{e^{- \beta\,{x^{\lambda}_{r}}}} \!\right)\! +\!\beta\, {x^{\lambda}_{r}}{e^{-\beta\,{x^{\lambda}_{r}}}} \right] }{ \left(3\,-\,2\,{e^{-\beta\,{x^{\lambda}_{r}}}} \right)}} $$

and

$${}t_{2}(x_{r})\,=\,{\frac {\beta\,{x^{\lambda}_{r}}\! \left(\ln\! \left({x^{\lambda}_{r}} \right) \right)^{2} \!\left[ \!\beta\,{x^{\lambda}_{r}} {e^{-\beta\,{x^{\lambda}_{r}}}}\,+\, \left(\! \!1\,-\,{e^{-\beta\,{x^{\lambda}_{r }}}}\! \right) \!\left(\! 3\,-\,2\,{e^{-\beta\,{x^{\lambda}_{r}}}}\! \right) \!\right] }{ \left(3\,-\,2\,{e^{-\beta\,{x^{\lambda}_{r}}}} \right)}}. $$

Note that the elements *I*_{i j}, *i*,*j* = 1,2, constitute the Fisher information related to each *X*_{s}, *s* = *r*+1,*r*+2,⋯,*n*, where *X*_{s} is distributed as in (14). Therefore, in view of (19–21), the elements of the FIM about the parameters *β* and *λ* related to the complete data set **W** can be obtained as \(n\, {\lim }_{y\,\to \, 0^{+}}\, I_{i\,j},\,i,\,j=1,2\), which give as the same results as in (13).

Therefore, the FIM gains about the two unknown parameters *β* and *λ* from a given type-II censored sample, (*x*_{1},*x*_{2},⋯*x*_{r})^{′}, is then given by

$$I_{\mathbf{x}}(\beta,\,\lambda)= I_{\mathbf{W}}(\beta,\,\lambda) - I_{{\mathbf{Y}}|\mathbf{x}}(\beta,\,\lambda).$$

### Asymptotic variances and covariance

Once *I*_{x}(*β*, *λ*) is calculated, at \(\beta \,=\,\hat \beta _{M}\) and \(\lambda \,=\,\hat \lambda _{M}\), the asymptotic variance-covariance matrix of the MLEs of the two unknown parameters *β* and *λ* is then given by

$${}{\mathbf{Var-Cov}}\left(\hat\beta_{M},\,\hat\lambda_{M}\right)= {I^{-1}_{\mathbf{x}}\left(\hat\beta_{M},\,\hat\lambda_{M}\right)}= \left[ \begin{array}{cc} {\hat\sigma_{1}^{2}}&\hat\sigma_{12}\\\noalign{\medskip}\hat\sigma_{21}&{\hat\sigma_{2}^{2}}\end{array} \right]. $$

Again, once \({I^{-1}_{\mathbf {x}}\left (\hat \beta _{M},\,\hat \lambda _{M}\right)}\) is obtained, the asymptotic variance of the reliability function *s*(*t*_{0}) can then be calculated as the lower bound of the Cram\(\acute {\mathrm {e}}\)r-Rao inequality of the variance of any unbiased estimator for *s*(*t*_{0}). That is,

$$ {} \begin{aligned} \text{Var}[ \widehat{s(t_{\,0})}]&= 36\,{t^{2\,\hat\lambda_{M}}_{0}}{e^{-4\,\hat\beta_{M}{t^{\hat\lambda_{M}}_{0}}}}\left[{\hat\sigma_{2}^{2}} {{\hat\beta_{M}^{2}}} \left[ \ln ({t_{\,0}}) \right]^{2}\right.\\ &\quad\left.+\hat\beta_{M}\,\ln({t_{\,0}})\, {\hat\sigma_{12}}\,+\, {{\hat\sigma_{1}^{2}}} \right] {\left[1\,-\,{e^ {-\hat\beta_{M} t^{\hat\lambda_{M}}_{0}}}\right]}^{2}. \end{aligned} $$

(22)

Consequently, the asymptotic (1 − *α*) 100 *%* confidence intervals, ACIs, for \(\hat {\beta }_{M}\), \(\hat {\lambda }_{M}\), and \(\widehat {s(t_{\,0})}_{M}\) are given by

$$ \begin{aligned} {}&\left[\hat{\beta}_{M}\,\mp\, Z_{\frac{\alpha}{2}}\,{\hat\sigma_{1}}\right],\, \left[\hat{\lambda}_{M}\,\mp\, Z_{\frac{\alpha}{2}}\,{\hat\sigma_{2}}\right] \, \text{and}\\ &\qquad\left[\widehat{s(t_{\,0})}_{M}\,\mp\, Z_{\frac{\alpha}{2}}\,\sqrt{\text{Var}[\widehat{s(t_{\,0})}]}\right], \end{aligned} $$

(23)

respectively, where \(Z_{\frac {\alpha }{2}}\) is the percentile \((1\,-\,{\frac {\alpha }{2}})\) of the standard normal distribution.