An Oracle Approach to Goodness-of-Fit Testing via Linear Combination of Degenerate V-Statistics

Here, we present a novel model-free approach for combining test statistics of the V-statistic type, both with and without estimated parameters, and derive the limiting distribution of the resulting combined statistics. We investigate their empirical power across a wide range of data-generating scenarios and conclude that the proposed method can serve as a compromise solution when several statistics exhibit opposing behavior against different alternatives. Moreover, it can act as a simple and effective alternative to tuning-parameter selection within families of test statistics indexed by such parameters.