## 6 Applications
### Second Data: Remission Times  
The second case addressed the lengths of remission times (in months) of 128 bladder cancer patients who are randomly selected, as reported by [44]. The data observations are 0.08, 4.87, 6.94, 8.66, 2.09, 3.48, 13.11, 23.63, 0.20, 13.80, 25.74, 0.50, 2.23, 3.52, 4.98, 6.97, 9.02, 3.88, 5.32, 7.39, 10.34, 13.29, 0.40, 2.26, 9.22, 2.46, 3.64, 5.09, 7.26, 0.51, 2.54, 14.76, 26.31, 0.81, 3.70, 5.17, 7.28, 9.47, 14.24, 25.82, 9.74, 2.62, 3.82, 5.32, 2.69, 4.23, 5.41, 7.62, 7.32, 10.06, 14.77, 32.15, 2.64, 14.83, 34.26, 4.33, 5.49, 7.66, 0.90, 5.34, 7.59, 10.66, 15.96, 2.69, 4.18, 36.66, 12.05, 10.75, 16.62, 43.01, 5.41, 7.63, 11.25, 17.14, 79.05, 17.12, 46.12, 4.40, 5.85, 8.26, 1.26, 2.83, 1.35, 2.87, 5.62, 1.19, 2.75, 4.26, 7.87, 11.64, 4.34, 5.71, 7.93, 11.79, 17.36, 1.40, 3.02, 18.10, 1.46, 11.98, 19.13, 12.02, 2.02, 3.31, 1.76, 3.25, 12.03, 20.28, 2.02, 4.50, 6.25, 8.37, 4.51, 6.54, 8.53, 3.36, 6.76, 8.65, 12.63, 22.69, 12.07, 21.73, 3.57, 5.06, 7.09, 2.07, 3.36, 6.93. Figure 10 represents the density, TTT, P-P, and Q-Q plots of the PZD for the 128 bladder cancer patients' data. Various parameter estimators based on cancer patients' data are summarized in Table 7.  
As the ML parameter estimators for the competing distributions are needed to compute their corresponding NRR test statistic \(Y^{2}\), the MLE values are calculated and given in Table 8. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Distributions** & **Y\({}^{2}\)** & **W** & **A** & **K-S (\(p\)-Value)** & **-NLL** & **AIC** & **CAIC** & **BIC** \\ \hline
**PZD** & 8.4684 & 0.2986 & 0.6844 & 0.1726 (0.9496) & 213.8102 & 431.6204 & 431.8757 & 435.4444 \\
**ZD** & 10.3344 & 0.4144 & 0.7468 & 0.2633 (0.9264) & 236.7156 & 475.4312 & 475.5145 & 477.3432 \\
**Lomax** & 10.9816 & 0.5282 & 0.7677 & 0.4476 (0.9103) & 253.4021 & 510.8042 & 511.0595 & 514.6282 \\
**Xgamma** & 12.6326 & 0.5633 & 0.9436 & 0.3466 (0.9088) & 241.0956 & 484.1312 & 484.2145 & 486.1032 \\
**Gamma** & 11.3392 & 0.4496 & 0.8434 & 0.5233 (0.9126) & 262.3364 & 528.6728 & 528.9281 & 532.4968 \\
**Darma** & 9.2204 & 0.3642 & 0.6939 & 0.1833 (0.9312) & 221.9302 & 447.8604 & 448.1157 & 451.6844 \\
**Power Lindley** & 12.8596 & 0.5698 & 0.9445 & 0.5325 (0.8965) & 268.9236 & 541.8472 & 542.1025 & 545.6710 \\
**Power Darna** & 8.7659 & 0.3245 & 0.6862 & 0.1756 (0.9423) & 221.3546 & 448.7092 & 449.2309 & 454.4450 \\
**EP LD** & 13.0251 & 0.5702 & 0.9512 & 0.5366 (0.8902) & 272.3256 & 550.6512 & 551.1729 & 556.3870 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values of criteria statistics for model selection based on failure times data. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & \(\hat{\beta}\) & \(\mathbf{\theta}\) & **KS** & \(\mathbf{p}\)-Value** \\ \hline MLE & 0.4342 & 5.1267 & 0.0314 & 0.8434 \\ WLS & 0.6764 & 6.3616 & 0.0386 & 0.8383 \\ MPS & 0.6989 & 7.7768 & 0.0497 & 0.8198 \\ CMEs & 0.4213 & 4.9674 & 0.0478 & 0.8266 \\ LSE & 0.3417 & 4.1625 & 0.0412 & 0.8317 \\ ADE & 0.7895 & 8.4616 & 0.0519 & 0.8078 \\ ADEL-R & 0.7618 & 8.7786 & 0.0581 & 0.7984 \\
**ADEL-T** & 0.8216 & 9.7633 & 0.0553 & 0.8010 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Different parameter estimators for cancer patients’ data. \begin{table}
\begin{tabular}{c c} \hline
**Distributions** & **Estimates** \\ \hline
**PZD** & \(\hat{\beta}=0.4342,\hat{\theta}=5.1267\) \\
**ZD** & \(\hat{\theta}=2.3461\) \\
**Lomax** & \(\hat{\alpha}=1.6798,\lambda=0.9436\) \\
**Xgamma** & \(\hat{\theta}=1.9826\) \\
**Gamma** & \(\hat{\alpha}=2.3639,\ \hat{\beta}=0.1364\) \\
**Darma** & \(\hat{\alpha}=0.9671,\ \hat{\theta}=1.3478\) \\
**Power Lindley** & \(\hat{\alpha}=1.8963,\ \hat{\beta}=3.0596\), \\
**Power Darna** & \(\hat{\alpha}=3.0614,\ \hat{\beta}=2.9356,\ \hat{\theta}=0.1796\) \\
**EP LD** & \(\hat{\alpha}=5.6947,\ \hat{\beta}=0.4972,\ \hat{\theta}=1.7693\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: ML parameter estimates for the alternative distributions based on cancer patients’ data.