## 6 Applications
### First Data: Failure Times of Devices  
The first real data set refers to the failure times of 50 items put under a life test. It is generally known that this data set displays bathtub behavior of the hazard rate function [43]. This data set has been analyzed by many authors. The data are 0.1, 1.0, 1.0, 1.0, 2.0, 3.0, 82.0, 83.0, 84.0, 6.0, 7.0, 11.0, 36.0, 40.0, 45.0, 12.0, 18.0, 18.0, 21.0, 18.0, 0.2, 1.0, 1.0, 18.0, 18.0, 32.0, 45.0, 47.0, 63.0, 63.0, 67.0, 67.0, 50.0, 55.0, 60.0, 67.0, 67.0, 72.0, 85.0, 85.0, 85.0, 75.0, 79.0, 82.0, 84.0, 84.0, 85.0, 86.0, 86.0. Figure 9 displays the density, total time on test transform (TTT), probability-probability (P-P) plot, and quantile-quantile (Q-Q) plot of the PZD for the failure times data.  
Figure 9: _Cont._  
Table 4 represents the values of the parameter estimators for the hypothesized distribution \(PZD\) obtained by the different methods, and Kolmogorov-Smirnov (KS) and the corresponding \(p\)-values. Because \(n=50>30\), it is established by the simulation study that the \(MLE\) methods give the best results.  
As the maximum likelihood parameter estimators for the competing distributions are required to calculate their corresponding NRR test statistic \(Y^{2}\), the MLE values are calculated and given in Table 5.  
To distinguish between the proposed model and its alternatives, we used classical criteria for model selection, and the NRR statistics with the corresponding values are summarized in Table 6.  
From Table 6, we can see that the smallest values of the different criteria for model selection are obtained for the PZD, which confirms that the proposed PZD model describes these data better than all the alternatives.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Method** & \(\beta\) & \(\theta\) & **KS** & \(p\)**-Value** \\ \hline
**MLE** & 0.6763 & 4.6777 & 0.1726 & 0.9496 \\
**WLS** & 0.3646 & 2.3162 & 0.1976 & 0.9284 \\
**MPS** & 0.2346 & 3.4677 & 0.2011 & 0.9202 \\
**CMEs** & 0.3679 & 5.6626 & 0.1938 & 0.9334 \\
**LSE** & 0.5976 & 4.7647 & 0.1784 & 0.9412 \\
**ADE** & 0.6696 & 6.6133 & 0.2134 & 0.9177 \\
**ADEL-R** & 0.8010 & 7.6727 & 0.2212 & 0.9086 \\
**ADEL-T** & 0.9262 & 8.7689 & 0.2276 & 0.9010 \\ \hline \end{tabular}
\end{table}
Table 4: Different parameter estimators for failure times data.  
\begin{table}
\begin{tabular}{c c} \hline
**Distributions** & **Estimates** \\ \hline
**PZD** & \(\hat{\beta}=0.6763,\hat{\theta}=4.6777\) \\
**ZD** & \(\hat{\theta}=1.7826\) \\
**Lomax** & \(\hat{\alpha}=2.3467\), \(\hat{\lambda}=1.0934\) \\
**Xgamma** & \(\hat{\theta}=1.3476\) \\
**Gamma** & \(\hat{\alpha}=1.7266\), \(\hat{\beta}=0.0839\) \\
**Darna** & \(\hat{\alpha}=1.6346\), \(\hat{\theta}=2.7682\) \\
**Power Lindley** & \(\hat{\alpha}=1.6235\), \(\hat{\beta}=2.0145\), \\
**Power Darna** & \(\hat{\alpha}=2.856\), \(\hat{\beta}=3.1624\), \(\hat{\theta}=0.0935\) \\
**EP LD** & \(\hat{\alpha}=8.6321\), \(\hat{\beta}=0.8562\), \(\hat{\theta}=3.4152\) \\ \hline \end{tabular}
\end{table}
Table 5: ML parameter estimates for the alternative distributions based on failure times data.  
Figure 9: The density, TTT, P–P, and Q–Q plots of the PZD for the failure times data.