Algorithm 1: Constructing percentile bootstrap 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 under sample ${\underset{\_}{x}}^{*}$, denote as ${\widehat{\sigma }}^{*}$ and ${\widehat{\lambda }}^{*}$. step 4Repeat step 3 $N_{boot}$ times, then we can obtain a series of bootstrap MLEs $\left({\widehat{\sigma }}_{**}^{(1)},{\widehat{\sigma }}_{**}^{(2)},\cdots ,{\widehat{\sigma }}_{**}^{(N_{boot})}\right)$ and $({\widehat{\lambda }}_{**}^{(1)},{\widehat{\lambda }}_{**}^{(2)},\cdots ,{\widehat{\lambda }}_{**}^{(N_{boot})})$. step 5Arrange $({\widehat{\sigma }}_{**}^{(1)},{\widehat{\sigma }}_{**}^{(2)},\cdots ,{\widehat{\sigma }}_{**}^{(N_{boot})})$ and $\left({\widehat{\lambda }}_{**}^{\left(1\right)},{\widehat{\lambda }}_{**}^{\left(2\right)},\cdots ,{\widehat{\lambda }}_{**}^{\left(N_{boot}\right)}\right)$ in ascending order, respectively, and obtain $({\widehat{\sigma }}_{**}^{\left[1\right]},{\widehat{\sigma }}_{**}^{\left[2\right]},\cdots ,{\widehat{\sigma }}_{**}^{\left[N_{boot}\right]})$ and $({\widehat{\lambda }}_{**}^{\left[1\right]},{\widehat{\lambda }}_{**}^{\left[2\right]},\cdots ,{\widehat{\lambda }}_{**}^{\left[N_{boot}\right]})$.