## 3 Methodology
### Bootstrap DEA  
Since this study aims to compare and verify the distribution in management, service quality, network quality, and market efficiencies among the DMUs, Bootstrap DEA is performed following the suggestion by [PERSON] and [PERSON] [41] to overcome the non-parametric restrictions that DEA has in general. While DEA provides the advantage of being able to evaluate efficiency using linear metric without statistical assumptions, this can also work as a disadvantage as the resulting efficiency scores are relative values that change with the change in the number of DMUs. That is, DEA can lead to a false evaluation as it lacks statistical verification for the efficiency score.  
In recognition of these problems, [PERSON] and [PERSON] [41] applied the bootstrap method to the existing DEA model. As a non-parametric statistical technique introduced by [PERSON] [42], bootstrap DEA helps to compare the differences in efficiency among groups. The application of bootstrap on the non-parametric DEA model enables scholars to calculate the confidence interval and standard error for the frontier model, and also opens ways to explain the differences among efficient DMUs when DEA results evaluated multiple DMUs as efficient. In this study, the bootstrap DEA method is applied to eliminate the bias in the efficiency score to provide statistical significance as well as to tie-break the DMUs that are shown as efficient in the DEA analysis results.  
The process to estimate the efficiency of the BCC model using bootstrap DEA follows that outlined by [PERSON] and [PERSON] [41] and [PERSON] and [PERSON] [43]:  
1. Calculate the technical efficiency score \(\hat{\mathcal{O}}_{k}\ (k=1,2,3,\cdots L)\) of individual DMUs through the standard linear programming DEA model
2. Generate a random sample in the size \(L\) from \(\{\hat{\mathcal{O}}_{k}\ (k=1,2,3,\cdots L)\}\) to utilize kernel density estimation to provide \(\{\hat{\mathcal{O}}_{1b}^{*},\hat{\mathcal{O}}_{2b}^{*},\ \hat{\mathcal{O}}_{3b}^{*},\cdots\hat{ \mathcal{O}}_{Lb}^{*}\}\)
3. Calculate \(\{(x_{k},\ y_{kb}^{*}),\ k=1,2,3,\cdots L\}\) as a pseudo data set to generate reference bootstrap technology
4. Calculate the bootstrap efficiency estimation \(\hat{\mathcal{O}}_{kb}^{*}\ (k=1,2,3,\cdots L)\) of each DMU's technical efficiency score, \(\hat{\mathcal{O}}_{k}\ (k=1,2,3,\cdots L)\), by finding the values of a Bootstrap corresponding model
5. Repeatedly calculate \(B\) times (\(B\) is a larger number) in order to obtain the Bootstrap efficiency estimation \(\{\hat{\mathcal{O}}_{kb}^{*}\ (b=1,2,3,\cdots B)\}\). [PERSON] [44] suggested \(B=1000\) to ensure the proper range of the confidence interval, and [PERSON] and [PERSON] [45] suggested \(B=2000\).  
The bootstrap efficiency estimation gained through the five steps above can be calculated using the following equation:  
\[\overline{\theta}_{k}^{*}=\frac{1}{B}\sum_{b=1}^{B}\hat{\theta}_{kb}^{*} \tag{2}\]  
[PERSON] and [PERSON] [41] suggested \(b=1,\ \ldots,\ B\) to be arranged in an incremental order of \(\hat{\theta}_{kb}^{*}-\hat{\theta}_{k}\) values and to eliminate the value of \(\left(\frac{\alpha}{2}\times 100\right)\%\) from values at the far ends. Then, the values at the far ends (the smallest and the largest values) become \(-\hat{b}_{\alpha}^{*}\) and \(-\hat{a}_{\alpha}^{*}\). Thus, the confidence interval of \((1-\alpha)\times 100\%\) for the estimated score can be expressed in the following equation:  
\[\hat{\theta_{k}}+{{a_{k}}^{*}}\leq\theta_{k}\leq\hat{\theta_{k}}+{\hat{b_{k}} ^{*}} \tag{3}\]  
At the same time, the bias of the efficiency estimation \(\hat{\theta_{k}}\) can be found using the bootstrap sample through the following equation:  
\[b\hat{i}\hat{a}_{k}\left(\hat{\theta_{k}}\right)=\overline{\theta}_{k}^{*}- \hat{\theta_{k}}\Therefore, the bootstrap DEA efficiency score is the efficiency score gained through the original efficiency score minus the bias score.