## 6 Real Data Analysis  
A real data set was employed in this section to illustrate the utility of the Burr XII distribution and its corresponding estimators. The data set comprises 128 observations recording the remission time (in years) of bladder cancer patients, as studied in [23]. An effective and accurate estimator of the reliability functions can greatly assist doctors in understanding the remission time of bladder patients globally and in managing the therapeutic interventions for these patients. Before estimating the reliability of this data set, we initially calculated the Bayesian information criterion (BIC) for this data set for various commonly used lifetime distributions, including the Burr XII, Gamma, Weibull, Lindley, log-logistic, and linear-exponential distributions, to determine whether it is plausible to fit this data set with the Burr XII distribution. The PDFs of the considered candidate distributions are structured as follows:  
\[\text{Gamma: }f(x;\lambda,\omega)=\frac{\omega^{\lambda}}{\Gamma(\lambda)}x^{ \lambda-1}e^{-\omega x},x>0,\lambda>0,\omega>0,\]  
\[\text{Weibull: }f(x;\lambda,\omega)=\frac{\lambda}{\omega}\Big{(}\frac{x}{ \omega}\Big{)}^{\lambda-1}e^{-\left(\frac{x}{\omega}\right)^{\lambda}},x>0, \lambda>0,\omega>0,\]  
\[\text{Lindley: }f(x;\eta)=\frac{\eta^{2}}{1+\eta}(1+x)e^{-\eta x},x>0,\eta>0,\]  
\[\text{Log-logistic: }f(x;\lambda,\omega)=\frac{\lambda\omega x^{\omega-1}}{ \left(1+\lambda x^{\omega}\right)^{2}},x>0,\lambda>0,\omega>0,\]  
\[\text{Linear-exponential: }f(x;\lambda,\omega)=\left(\lambda+\omega x\right)e^{- \left(\lambda x+\frac{1}{2}\omega x^{2}\right)},\lambda+\omega x>0.\]  
The goodness-of-fit results are shown in Table 6.  
The results in Table 6 show that the Burr XII distribution should be selected over the other candidate distributions due to its smaller BIC values. Figure 6: The MLE and UMVUE of the Burr XII distribution reliability function. In order to enhance the persuasive power of this study, a Kolmogorov-Smirnov test was conducted as a validation supplement. Let \(F_{0}\), \(\hat{F}\), and \(F_{n}\) denote the underlying distribution of the data, the estimated Burr XII distribution, and the empirical distribution, respectively. The test statistic for the null hypothesis \(H_{0}:F_{0}=\hat{F}\) versus the alternative hypothesis \(H_{1}:F_{0}\
eq\hat{F}\) is shown below:  
\[D_{n}=\underset{-\infty<x<\infty}{sup}\big{|}\hat{F}(x)-F_{n}(x)\big{|}.\]  
The Kolmogorov-Smirnov distance between the estimated Burr XII distribution and the empirical distribution was calculated as \(D_{n}=0.035015<D_{n}(0.05)\). This suggests that under a significance level of 0.05, there are no significant discrepancies between the estimated Burr XII distribution and the empirical distribution of the data. Moreover, the p-value of the test was calculated as \(p=0.9976>0.05\). A high p-value indicates a high probability of obtaining the observed results, thus we cannot reject the null hypothesis, which verifies the correctness of choosing the Burr XII distribution as the best-fitting distribution. In order to visualize this data set more intuitively and compare it with the fitted Burr XII distribution, the histogram of the remission time data and the corresponding Burr XII distribution are shown in Figure 7. This suggests that both the histogram of the real data and the PDF of the underlying distribution exhibit similar patterns. Table 6 indicates that for the Burr XII distribution, the MLE of \(\alpha\) is \(\hat{\alpha}=1.428458\). In order to verify that the real data are consistent with the case of a fixed \(\alpha\) and unknown \(\beta\) as studied in this paper, we carried out a likelihood ratio test for the integrity of our studies using the Burr XII distribution. Let the null and alternative hypotheses be \(H_{0}:\alpha=1.428458\) and \(H_{1}:\alpha\
eq 1.428458\), respectively. The test statistic is