| Abstrakt: |
Let μ 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 μ is known, take the worst partition of the square into k≦n rectangles R, each with its short side at least δ times as long as the long side, and let Z= n∑|μ( R)−μ( R)|. We prove distribution inequalities for Z implying the right half of c( n,k) ≦ EZ ≦ C( n,k, 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. [ABSTRACT FROM AUTHOR] |
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