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Consider a set of multivariate distributions, $F_1,\dots,F_M$, aiming toexplain the same phenomenon. For instance, each $F_m$ may correspond to adifferent candidate background model for calibration data, or to one of manypossible signal models we aim to validate on experimental data. In thisarticle, we show that tests for a wide class of apparently different models$F_{m}$ can be mapped into a single test for a reference distribution $Q$. As aresult, valid inference for each $F_m$ can be obtained by simulating\underline{only} the distribution of the test statistic under $Q$. Furthermore,$Q$ can be chosen conveniently simple to substantially reduce the computationaltime.

K-2 rotated goodness-of-fit for multivariate data