**6. Conclusions**

When deciding between two competing models, practitioners of statistics normally utilize traditional hypothesis testing methods that rely on the assumption that one of the candidate models is properly specified. This approach is problematic because it is unreasonable to assume that one of the proposed models is precisely true. In addition, these methods are only applicable for nested models. To avoid any underlying assumptions and model structure limitations, Riedle, Neath and Cavanaugh [1] propose the use of the bootstrap discrepancy probability (BDCP) to assess the propriety of the fit of two candidate models. However, the bootstrap discrepancy (BD) utilized in this work provides a biased estimator of the Kullback–Leibler discrepancy (KLD).

When hypothesis testing assumptions are met, the BDCP asymptotically approximates the likelihood ratio test *p*-value. Therefore, similarly to *p*-values, the distribution of the BDCP is uniform if the null hypothesis is true. Hence, in settings when the null is true, the BDCP would be of limited value in choosing the appropriate model.

In this paper, we proposed utilizing the *kb* or the *k* corrected BDCP, namely BDCPb and BDCPk, respectively. The BDCPb employs the BDb, a bootstrap corrected estimator of the KLD, while the BDCPk uses the BDk, a BD corrected by adding the number of functionally independent parameters in the candidate model. We showed that for most settings, the BDb serves as an over-corrected estimator of the KLD, but the corresponding BDCPb is less biased than the BDCPk for the estimation of the KLDCP. However, in the case when there is distributional misspecification, we showed that the BDb has negligible bias for the estimation of expected value of the KLD.

Moreover, the estimation of the bootstrap correction *kb* utilizes the same bootstrap samples that were used to calculate the BD; therefore, we argue that the computational requirements of estimating *kb* are not too burdensome. However, if the sample size is moderately large compared to the number of parameters in the model, then we showed that using *k* to correct the bias generally results in comparable values of the KLDCP estimates.

**Author Contributions:** Conceptualization, A.D. and J.C.; Formal analysis, A.D. and J.C.; Methodology, A.D. and J.C.; Supervision, J.C.; Writing—original draft, A.D. and J.C.; Writing—review and editing, A.D. and J.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Data Availability Statement:** The R code used in generating the data for the simulation study is available on request from the corresponding author. The data for the application are not publicly available since the dataset is confidential.

**Conflicts of Interest:** The authors declare no conflict of interest.
