Keywords: [ histograms ] [ distribution testing ] [ probability distributions ] [ binning ] [ lower bounds ] [ sub-linear algorithms ]

Abstract:
We investigate the problem of testing whether a discrete probability distribution over an ordered domain is a histogram on a specified number of bins. One of the most common tools for the succinct approximation of data, $k$-histograms over $[n]$, are probability distributions that are piecewise constant over a set of $k$ intervals. Given samples from an unknown distribution $\mathbf p$ on $[n]$, we want to distinguish between the cases that $\mathbf p$ is a $k$-histogram versus far from any $k$-histogram, in total variation distance. Our main result is a sample near-optimal and computationally efficient algorithm for this testing problem, and a nearly-matching (within logarithmic factors) sample complexity lower bound, showing that the testing problem has sample complexity $\widetilde \Theta (\sqrt{nk} / \epsilon + k / \epsilon^2 + \sqrt{n} / \epsilon^2)$.

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