Summary
Let μn be the empirical probability measure associated with n i.i.d. random vectors each having a uniform distribution in the unit square S of the plane. After μn is known, take the worst partition of the square into k≦n rectangles R i, each with its short side at least δ times as long as the long side, and let Z= n∑|μn(R j)−μ(R j)|. We prove distribution inequalities for Z implying the right half of c p,δ(n,k)p/2 ≦ EZ p ≦ C p,δ(n,k p/2, p > 0. (The left half follows easily by considering non-random partitions.) Similar results are obtained in other dimensions, and for population distributions other than uniform, and our results are related to data based histogram density estimation.
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Supported by NSF Grant MCS 8201128
Supported by NSF Grant DMS-8401996
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Chen, J., Davis, B. & Rubin, H. How non-uniform can a uniform sample be: a histogram approach. Probab. Th. Rel. Fields 73, 245–254 (1986). https://doi.org/10.1007/BF00339939
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DOI: https://doi.org/10.1007/BF00339939