Nonparametric Sequential Goodness-of-Fit Testing under Heterogeneity

We consider sequential goodness-of-fit testing when the data generating distribution is an unknown mixture of several reference distributions, with heterogeneity arising through variation in the mixing proportions. In many contemporary applications, this reflects latent or structurally distinct subpopulations in the data that occur at different rates while their underlying distributions remain unchanged. Such changes in composition can substantially alter the overall data distribution and cause standard sequential procedures to spuriously reject. We develop a nonparametric sequential testing framework for detecting distributional departures in the presence of such heterogeneity. Our framework is based on a composite null that allows the data generating distribution to arise from the same collection of reference distributions but with possibly different mixing weights. Motivated by the principle of testing by betting, we propose SeqGoF, which uses kernel mean embeddings to quantify departures from this composite null and constructs a sequential betting strategy against it, without imposing a parametric model for the alternative. We show that SeqGoF has anytime-valid level-$alpha$ control under arbitrary stopping and almost-sure power one against every fixed alternative outside the null, and we study its first-order stopping-time asymptotics. Numerical experiments and an application to hard-drive SMART telemetry demonstrate the strong finite-sample performance of SeqGoF.