Algorithm 2: Constructing bootstrap-t confidence intervals

step 1Set the simulation number $N_{boot}$ times ahead. step 2Compute the MLEs of $\sigma$ and $\lambda$ under the original censored sample $\underset{\_}{x}=(x_1,x_2,\cdots ,x_m)$, denoted as $\widehat{\sigma }$ and $\widehat{\lambda }$. (If we carry out a simulation study, we should first generate an adaptive progressive type II censored sample $\underset{\_}{x}=(x_1,x_2,\cdots ,x_m)$ from EHL($\lambda$,$\sigma$) with $T,n,m,R$ as the original sample.) step 3Generate a bootstrap sample ${\underset{\_}{x}}^{*}$ using $\widehat{\sigma }$,$\widehat{\lambda }$ and the same censoring pattern $(n,m,T,R)$. Then, calculate the bootstrap MLEs ${\widehat{\sigma }}^{*}$ and ${\widehat{\lambda }}^{*}$ and their variances $Var({\widehat{\sigma }}^{*})$ and $Var({\widehat{\lambda }}^{*})$. step 4Calculate the t-statistics $\widetilde{S_1}=\frac{{\widehat{\sigma }}^{*}-\widehat{\sigma }}{\sqrt{Var\left({\widehat{\sigma }}^{*}\right)}}$ for ${\widehat{\sigma }}^{*}$ and $\widetilde{S_2}=\frac{{\widehat{\lambda }}^{*}-\widehat{\lambda }}{\sqrt{Var\left({\widehat{\lambda }}^{*}\right)}}$ for ${\widehat{\lambda }}^{*}$. step 5Repeat steps 2–3 $N_{boot}$ times to acquire a series of bootstrap t-statistics $\left({\widetilde{S_1}}_{**}^{\left(1\right)},{\widetilde{S_1}}_{**}^{\left(2\right)},\cdots ,{\widetilde{S_1}}_{**}^{\left(N_{boot}\right)}\right)$ and $\left({\widetilde{S_2}}_{**}^{\left(1\right)},{\widetilde{S_2}}_{**}^{\left(2\right)},\cdots ,{\widetilde{S_2}}_{**}^{\left(N_{boot}\right)}\right)$. step 6Arrange $\left({\widetilde{S_1}}_{**}^{\left(1\right)},{\widetilde{S_1}}_{**}^{\left(2\right)},\cdots ,{\widetilde{S_1}}_{**}^{\left(N_{boot}\right)}\right)$ and $\left({\widetilde{S_2}}_{**}^{\left(1\right)},{\widetilde{S_2}}_{**}^{\left(2\right)},\cdots ,{\widetilde{S_2}}_{**}^{\left(N_{boot}\right)}\right)$ in ascending order respectively and obtain $\left({\widetilde{S_1}}_{**}^{\left[1\right]},{\widetilde{S_1}}_{**}^{\left[2\right]},\cdots {\widetilde{S_1}}_{**}^{\left[N_{boot}\right]}\right)$ and $\left({\widetilde{S_2}}_{**}^{\left[1\right]},{\widetilde{S_2}}_{**}^{\left[2\right]},\cdots ,{\widetilde{S_2}}_{**}^{\left[N_{boot}\right]}\right)$.