## 3 Monte Carlo Simulation  
In this section, we carry out an extensive Monte Carlo simulation study to compare the performances of the maximum likelihood (ML) estimates of \(R_{s,k}\). The comparisons are done based on point and interval estimations. The simulation study is carried out on 5000 simulated samples from exponentiated moment-based exponential distributed stress and strength variables for different sample sizes \(n=15\), 20, 25, 30, 35, 40, 45, and 50 and for the combinations of the following parameters:  
\[(\alpha_{1},\alpha_{2})\ =\ (2.25,\ 1.5),\ (2,\ 1.5),\ (1.75,\ 1.5),\ (1.5,\ 1.5),\ (1.5,\ 1.75),\ (1.5,\ 2),\ (1.5,\ 2.25). \tag{24}\]  
We follow [PERSON] and [PERSON] [11] for the simulation procedure. We insert the estimates of \(\beta\), \(\alpha_{1}\), \(\alpha_{2}\) into the expression of \(\
u\) increases. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Components** & **Sample Size** & \multicolumn{6}{c}{**Parameters**} \\ (s, \(k\)) & **(\(n\),\(m\))** & & & & **(\(\alpha_{1}\),\(\alpha_{2}\))** & & \\ \hline  & & (2.25,1.5) & (2.0,1.5) & (1.75,1.5) & (1.5,1.5) & (1.5,1.75) & (1.5,2.0) & (1.5,2.25) \\ \hline  & (15,15) & 0.0223 & 0.0148 & 0.0069 & \(-\)0.0072 & \(-\)0.0218 & \(-\)0.0354 & \(-\)0.0510 \\  & (20,20) & 0.0217 & 0.0147 & 0.0066 & \(-\)0.0057 & \(-\)0.0199 & \(-\)0.0335 & \(-\)0.0475 \\  & (25,25) & 0.0216 & 0.0147 & 0.0062 & \(-\)0.0051 & \(-\)0.0188 & \(-\)0.0308 & \(-\)0.0446 \\ (1, 3) & (30,30) & 0.0215 & 0.0146 & 0.0057 & \(-\)0.0032 & \(-\)0.0172 & \(-\)0.0291 & \(-\)0.0436 \\  & (35,35) & 0.0215 & 0.0133 & 0.0052 & \(-\)0.0037 & \(-\)0.0152 & \(-\)0.0291 & \(-\)0.0419 \\  & (40,40) & 0.0214 & 0.0132 & 0.0049 & \(-\)0.0028 & \(-\)0.0152 & \(-\)0.0273 & \(-\)0.0412 \\  & (45,45) & 0.0212 & 0.0125 & 0.0043 & \(-\)0.0019 & \(-\)0.0151 & \(-\)0.0272 & \(-\)0.0409 \\  & (50,50) & 0.0211 & 0.0123 & 0.0041 & \(-\)0.0016 & \(-\)0.0144 & \(-\)0.0264 & \(-\)0.0396 \\  & (15,15) & 0.0344 & 0.0240 & 0.0125 & \(-\)0.0062 & \(-\)0.0251 & \(-\)0.0409 & \(-\)0.0581 \\  & (20,20) & 0.0341 & 0.0236 & 0.0121 & \(-\)0.0052 & \(-\)0.0237 & \(-\)0.0401 & \(-\)0.0559 \\  & (25,25) & 0.0339 & 0.0233 & 0.0118 & \(-\)0.0051 & \(-\)0.0231 & \(-\)0.0376 & \(-\)0.0533 \\ (2, 4) & (30,30) & 0.0338 & 0.0231 & 0.0115 & \(-\)0.0027 & \(-\)0.0214 & \(-\)0.0360 & \(-\)0.0528 \\  & (35,35) & 0.0337 & 0.0227 & 0.0111 & \(-\)0.0037 & \(-\)0.0192 & \(-\)0.0364 & \(-\)0.0512 \\  & (40,40) & 0.0336 & 0.0225 & 0.0107 & \(-\)0.0027 & \(-\)0.0194 & \(-\)0.0344 & \(-\)0.0508 \\  & (45,45) & 0.0332 & 0.0223 & 0.0102 & \(-\)0.0016 & \(-\)0.0194 & \(-\)0.0349 & \(-\)0.0507 \\  & (50,50) & 0.0331 & 0.0218 & 0.0101 & \(-\)0.0013 & \(-\)0.0187 & \(-\)0.0337 & \(-\)0.0493 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Bias of the simulated estimates of reliability \(R_{s,k}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Components** & **Sample Size** & \multicolumn{6}{c}{**Parameters**} \\ (s, \(k\)) & **(\(n\),\(m\))** & & & **(\(\alpha_{1}\),\(\alpha_{2}\))** & & \\ \hline  & & (2.25,1.5) & (2.0,1.5) & (1.75,1.5) & (1.5,1.5) & (1.5,1.75) & (1.5,2.0) & (1.5,2.25) \\ \hline  & (15,15) & 0.0036 & 0.0041 & 0.0048 &  
other life forms and abiotic components in most forests means that other elements, such as wildlife or soilnitutrients, are often the focal point. Thus, forest ecology is a highly diverse and important branch of ecological study. A forest ecosystem is a natural woodland unit consisting of all plants, animals, and microorganisms in that area functioning together with all of the non-living physical factors of the environment. Characteristics and methodological approaches of forest ecology data most often followcertain distributions, namely canonical distributions in ecology. We used the forest data and resource assessment of forestry in Vaniatu, 1990-2000, from the project of the University of Wellington, Victoria, Australia, the Department of Meteorology, and the Department of Forests. The data are available from Corrigan (2009).