## 3 Numerical Example
### Criteria  
In this section, we compare the NHPP SRGMs using the two data sets (in Section 3.2) and discuss the goodness of fit of the models. Eight criteria--MSE, PRR, PP, R\({}^{2}\), AIC, SAE, PRV, and RMSPE--are used for the comparison of goodness of fit. The first criterion is the mean squared error (MSE), which measures the distance of the model estimates from the actual data considering the number of observations and number N of parameters in the models. It is defined as follows:  
\[\text{MSE}=\frac{\sum_{i=1}^{n}\left(\text{m}(\text{t}_{i})-\text{y}_{i} \right)^{2}}{\text{n}-\text{N}}, \tag{9}\]  
where y\({}_{i}\) is the total number of failures observed at time t\({}_{i}\) according to the real data and M(t\({}_{i}\)) is the estimated cumulative number of failures at time t\({}_{i}\) for i = 1, 2..., n.  
The second criterion, the predictive ratio risk (PRR) [31], measures the distance of the model estimates from the actual data against the model estimate. It is defined by:  
\[\text{PRR}=\sum_{i=1}^{n}\bigg{(}\frac{\hat{\text{m}}(\text{t}_{i})-\text{y}_{ i}}{\hat{\text{m}}(\text{t}_{i})}\bigg{)}^{2}. \tag{10}\]  
The third criterion, the predictive power (PP) [31], measures the distance of model actual data from the estimates against the actual data. It is defined by:  
\[\text{PP}=\sum_{i=1}^{n}\bigg{(}\frac{\hat{\text{m}}(\text{t}_{i})-\text{y}_{ i}}{\hat{\text{y}}_{i}}\bigg{)}^{2}. \tag{11}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**No.** & **MODEL** & **m(t)** \\ \hline
1 & Goel Okumoto [25] & \(\text{m}(t)=\text{a}\Big{(}1-\text{e}^{-\text{bt}}\Big{)}\) \\
2 & Delayed S-shaped [3] & \(\text{m}(t)=\text{a}\Big{(}1-(1+\text{bt})\text{e}^{-\text{bt}}\Big{)}\) \\
3 & Inflection S-shaped [26] & \(\text{m}(t)=\frac{\text{a}\big{(}1-\text{e}^{-\text{bt}}\big{)}}{1+\beta\text{ e}^{-\text{bt}}}\) \\
4 & Yamada Imperfect [27] & \(\text{m}(t)=\text{a}\Big{(}1-\text{e}^{-\text{bt}}\Big{)}\big{(}1-\frac{\text{a }}{\beta}\big{)}+\text{a}\text{at}\) \\
5 & Pham–Nordmann–Zhang (PNZ) [28] & \(\text{m}(t)=\frac{\text{a}\big{(}1-\text{e}^{-\text{bt}}\big{)}\big{[}1-\frac{ \text{a}}{\beta}\big{)}+\text{a}\text{at}}{1+\beta\text{e}^{-\text{bt}}}\) \\
6 & [PERSON]–[PERSON] (PZ) [7] & \(\text{m}(t)=\frac{\big{(}(c+\text{a})\big{[}1-\text{e}^{-\text{bt}}\big{)} \big{[}-\frac{\text{a}}{\beta\text{e}^{-\text{bt}}}\big{]}\big{(}\text{e}^{- \text{bt}}-\text{e}^{-\text{bt}}\big{)}\big{)}}{1+\beta\text{e}^{-\text{bt}}}\) \\
7 & Dependent Parameter 2 [29] & \(\begin{bmatrix}\gamma t+1\\ \end{bmatrix}e^{-\gamma(t-\text{t}_{0})}+\text{a}(\gamma t+1)\) \\
8 & Testing Coverage [30] & \(\text{m}(t)=\text{N}\bigg{[}1-\bigg{(}\frac{\beta}{\beta+\text{(at)}^{5}}\bigg{)} ^{\text{a}\text{T}}\bigg{]}^{\text{a}\text{T}}\) \\
9 & New model & \(\text{m}(t)=\text{N}\bigg{(}1-\frac{\beta}{\beta+\text{a}(\text{t}-\text{t})^{5} }\bigg{)}^{\text{a}\text{T}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Mean value functions for SRGMs. The fourth criterion is R-square (R\({}^{2}\)) [32], which is used to examine the fitting power of the SRGMs, and it is the correlation index of the regression curve equation, which is expressed as follows:  
\[\mathrm{R}^{2}=1-\frac{\sum_{i=1}^{n}\left(y_{i}-\hat{m}(t_{i})\right)^{2}}{ \sum_{i=1}^{n}(y_{i}-\overline{y})^{2}}. \tag{12}\]  
The fifth criterion is Akaike's information criteria (AIC) [33], which is used to compare for maximization of the likelihood function. It can be considered as an approximate distance from the true probability model:  
\[\mathrm{AIC}=-2\mathrm{logL}+2\mathrm{N}, \tag{13}\]  
where N is the degree of freedom.