To assess goodness of fit, chi-squired test and Kolmogorov-Smirnov test, etc. are traditionally recommended.
However considering necessity of equivalence tests, these tests seems not to be suitable for test of goodness of fit.
Because null hypotheses of these tests are that the data are fitted by some function, according to regular course of hypothesis testing, If the null hypotheses are not rejected, this does NOT mean that accept the null hypotheses, i.e., we can not conclude the data are suitably fitted by some function.
Of cause, these traditional tests have certain significance for quantitative evaluation of goodness of fit. However if one wants to evaluate goodness of fit by hypothesis-testing sense, I think one has to carry out some tests using equivalence tests sense for this. In such case, a null hypothesis is that the data are NOT fitted by some function, more specifically for example, amount of RMS of residuals between the data and fitted function is more than some quantity which indicates criterion of closeness. Is there such a way?
I think that normality tests such as Shapiro-Wilk test have same problem.
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