A likelihood index of goodness of fit using directional approximation

In goodness of fit tests, the test statistics and the p-value which is calculated by using test statistics are utilized for acceptance of the null hypothesis. That is, the p-value has a critical meaning for many researchers. However, a small change in the p-value or not well-chosen of significance level leads to the rejection or acceptance of the null hypothesis. In this study, it is aimed to develop a likelihood index that can be used as a criterion for the degree of reliability of the goodness-of-fit tests to avoid these deficiencies. The proposed likelihood index was developed based on the directional approximation and gives the degree to which the sample fits well to the distribution being tested. In addition, the distribution of this index value was examined, and a free chi-square goodness-of-fit test was proposed. Several simulation studies have been conducted to comprehensively test the accuracy, validity, and performance of the likelihood index and the free chi-square goodness-of-fit test. The proposed test method has demonstrated superiority in many aspects compared to the other selected test methods. Furthermore, the proposed test is distribution-free, degree of freedom-free, binning-free compared to other tests in the literature, it is easy to calculate and use. Finally, the proposed method has been applied to real data sets and its performance has been demonstrated.