## 3 Interval Estimation
### Bootstrap Confidence Interval
The data set we selected was obtained from [[http://cdec.water.ca.gov/cgi-progs/queryMonthly?SHA](http://cdec.water.ca.gov/cgi-progs/queryMonthly?SHA)]([http://cdec.water.ca.gov/cgi-progs/queryMonthly?SHA](http://cdec.water.ca.gov/cgi-progs/queryMonthly?SHA)) (accessed on 10 March 2023) and represents the water capacity of the Shasta reservoir in California, USA. We assumed that the water levels do not lead to excessive dryness if the reservoir capacities in August for at least 2 out of the next 5 years are not lower than the capacity observed in December of that year. Our purpose was to infer whether the area was excessively dry. Therefore, we can build a model to estimate the MSSR, where the capacities in August can be regarded as the strength variables and the capacity in December can be regarded as the stress variable. From 1975 to 2016, the water capacities in August and December were as follows:

\[X=\left( \begin{array}{cccc}0.287785&0.126977&0.768563&0.703119&0.729986\\ 0.811159&0.829569&0.726164&0.423813&0.715158\\ 0.363359&0.463726&0.371904&0.291172&0.414087\\ 0.538082&0.744881&0.722613&0.561238&0.813964\\ 0.668612&0.524947&0.605979&0.715850&0.529518\\ 0.742025&0.468782&0.345075&0.425334&0.767070\\ 0.613911&0.461618&0.294834&0.392917&0.688100\\ \end{array} \right),\quad Y=\left( \begin{array}{cccc}0.667157\\ 0.767135\\ 0.640395\\ 0.650691\\ 0.709025\\ 0.824860\\ 0.679829\\ \end{array} \right).\]

Before proceeding with the estimation, we first performed Kolmogorov-Smirnov (K-S) tests to ensure that \(X\) and \(Y\) did follow the Chen distribution. In addition, three distributions were used to compare with the Chen distribution in the K-S tests. Their PDFs are as follows:

\[\text{[PERSON]: }f(x) =\frac{\theta}{\lambda}\left(\frac{x}{\lambda}\right)^{\theta-1} \!\!e^{-(\frac{x}{\lambda})^{\theta}},\quad x>0,\quad\theta,\,\lambda>0,\] \[\text{Burr XII: }f(x) =\theta\lambda x^{\theta-1}(1+x^{\theta})^{-\lambda-1},\quad x>0, \quad\theta,\,\lambda>0,\] \[\text{Generalized Rayleigh: }f(x) =2\theta\lambda^{2}xe^{-(\lambda x)^{2}}(1-e^{-(\lambda x)^{2}})^{ \theta-1},\quad x>0,\quad\theta,\,\lambda>0.\]

The results of the K-S tests are shown in Table 6. It can be seen that both \(X\) and \(Y\) followed the Chen distribution at the 0.05 significance level. In particular, the Chen distribution demonstrated better performance in the K-S tests for data set \(X\). In the previous sections, the estimations of \(R_{s,k}\) were discussed under the assumption that the stress and strength variables followed the Chen distribution with the same second shape parameter. Therefore, before estimating using the real data, we needed to check whether their second shape parameters were the same. Let \(X_{1},\ X_{2},\...,\ X_{k}\) follow \(Chen(\theta_{1},\ \lambda_{1})\) independently, and let \(Y\) (independent of \(X_{1},X_{2},...,X_{k}\)) follow \(Chen(\theta_{2},\ \lambda_{2})\). Then, the following hypothesis testing question is posed as

\[H_{0}:\lambda_{1}=\lambda_{2}\quad\text{versus}\quad H_{1}:\lambda_{1}\
eq \lambda_{2}.\]

In the complete sample, the corresponding likelihood function is given by \[L(\theta_{1},\lambda_{1},\theta_{2},\lambda_{2})= \prod_{i=1}^{n}\left(\prod_{j=1}^{k}f_{X}(x_{ij})\right)f_{Y}(y_{i})\] \[= \theta_{1}{}^{nk}\lambda_{1}{}^{nk}\theta_{2}{}^{n}\lambda_{2}{}^ {n}\prod_{i=1}^{n}\prod_{j=1}^{k}x_{ij}{}^{\lambda_{1}-1}\text{exp}\{x_{ij}{}^{ \lambda_{1}}+\theta_{1}(1-e^{x_{ij}{}^{\lambda_{1}}})\}\] \[\times\prod_{i=1}^{n}y_{i}{}^{\lambda_{2}-1}\text{exp}\{y_{i}{}^{ \lambda_{2}}+\theta_{2}(1-e^{y_{i}{}^{\lambda_{2}}})\}.\]

Then, the likelihood ratio statistic is constructed as follows:

\[r=\frac{\max_{\theta_{1},\theta_{2}>0,\lambda_{1}=\lambda_{2}>0}L(\theta_{1}, \lambda_{1},\theta_{2},\lambda_{2})}{\max_{\theta_{1},\theta_{2},\lambda_{1} \lambda_{2}>0}L(\theta_{1},\lambda_{1},\theta_{2},\lambda_{2})}.\]

For large \(n\), when \(H_{0}\) holds, one has \(\Lambda=-2\ln r\sim\chi^{2}(1)\) asymptotically.