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The problem of quickest detection of a change in the distribution of streaming data is considered. It is assumed that the pre-change distribution is known, while the only information about the post-change is through a (small) set of labeled data. This post-change data is used in a data-driven minimax robust framework, where an uncertainty set for the post-change distribution is constructed. The robust change detection problem is studied in an asymptotic setting where the mean time to false alarm goes to infinity. It is shown that the least favorable distribution (LFD) is an exponentially tilted version of the pre-change density and can be obtained efficiently. A Cumulative Sum (CuSum) test based on the LFD, which is referred to as the distributionally robust (DR) CuSum test, is then shown to be asymptotically robust. The results are extended to the case with multiple post-change uncertainty sets and validated using synthetic and real data examples.