## 3 Criteria for Model Comparisons  
Theoretically, once the analytical expression for mean value function m(t) is derived, then the parameters in m(t) can be estimated using parameter estimation methods (MLE: the maximum likelihood estimation method, LSE: the least square estimation method); however, in practice, accurate estimates may not be obtained by the MLE, particularly under certain conditions where the mean value function m(t) is too complex. The model parameters to be estimated in the mean value function m(t) can then be obtained using a MATLAB program that is based on the LSE method. Six common criteria; the mean squared error (MSE), Akaike's information criterion (AIC), the predictive ratio risk (PRR), the predictive power (PP), the sum of absolute errors (SAE), and R-square (R\({}^{2}\)) will be used for the goodness-of-fit estimation of the proposed model, and to compare the proposed model with other existing models, as listed in Table 1. These criteria are described as follows.  
The MSE is  
\[\text{MSE}=\frac{\sum_{i=0}^{n}\left(\hat{\text{m}}(\text{t}_{i})-\text{y}_{i} \right)^{2}}{\text{n}-\text{m}}. \tag{10}\]  
AIC [12] is  
\[\text{AIC}=-2\log\text{L}+2\text{m}. \tag{11}\]  
The PRR [13] is  
\[\text{PRR}=\sum_{i=0}^{n}\left(\frac{\hat{\text{m}}(\text{t}_{i})-\text{y}_{i }}{\hat{\text{m}}(\text{t}_{i})}\right)^{2}. \tag{12}\]  
The PP [13] is  
\[\text{PP}=\sum_{i=0}^{n}\left(\frac{\hat{\text{m}}(\text{t}_{i})-\text{y}_{i }}{\text{y}_{i}}\right)^{2}. \tag{13}\]  
The SAE [8] is  
\[\text{SAE}=\sum_{i=0}^{n}|\hat{\text{m}}(\text{t}_{i})-\text{y}_{i}|. \tag{14}\]  
The correlation index of the regression curve equation (R\({}^{2}\)) [9] is  
\[\text{R}^{2}=1-\frac{\sum_{i=0}^{n}\left(\hat{\text{m}}(\text{t}_{i})-\text{ y}_{i}\right)^{2}}{\sum_{i=0}^{n}\left(\text{y}_{i}-\overline{\text{y}_{i}} \right)^{2}}. \tag{15}\]  
Here, \(\hat{\text{m}}(\text{t}_{i})\) is the estimated cumulative number of failures at t\({}_{i}\) for i \(=1,2,\cdots,n\); y\({}_{i}\) is the total number of failures observed at time t\({}_{i}\); n is the actual data which includes the total number of observations; and, m is the number of unknown parameters in the model.  
The MSE measures the distance of a model estimate from the actual data that includes the total number of observations and the number of unknown parameters in the model. AIC is measured to compare the capability of each model in terms of maximizing the likelihood function (L), while considering the degrees of freedom. The PRR measures the distance of the model estimates from the actual data against the model estimate. The PP measures the distance of the model estimates from the actual data. The SAE measures the absolute distance of the model. For five of these criteria, i.e., MSE, AIC, PRR, PP, and SAE, the smaller the value is, the closer the model fits relative to other models run on the same dataset. On the other hand, R\({}^{2}\) should be close to 1.  
We use (8) below to obtain the confidence interval [13] of the proposed NHPP software reliability model. The confidence interval is described as follows;  
\[\hat{\mathbf{m}}(\mathbf{t})\pm Z_{\alpha/2}\sqrt{\hat{\mathbf{m}}(\mathbf{t})}, \tag{16}\]  
where, \(Z_{\alpha/2}\) is \(100(1-\alpha)\), the percentile of the standard normal distribution.  
Table 1 summarizes the different mean value functions of the proposed new model and several existing NHPP models. Note that models 9 and 10 consider environmental uncertainty.