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How non-uniform can a uniform sample be: a histogram approach
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  • Published: September 1986

How non-uniform can a uniform sample be: a histogram approach

  • Jeesen Chen1,
  • Burgess Davis2 &
  • Herman Rubin2 

Probability Theory and Related Fields volume 73, pages 245–254 (1986)Cite this article

  • 62 Accesses

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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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References

  1. Alexander, K.S.: Probability inequalities for empirical processes and a law of the iterated logarithm. Ann. Probab. 12, 1041–1067 (1984)

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  2. Cairoli, R.: Une inegalité pour martingales à indices multiples et ses applications. Séminaire de Strasbourg, 1–27. Berlin Heidelberg New York: Springer 1970

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  3. Chen, J., Rubin, H.: On the consistency property of the data-based histogram density estimators. Purdue mimeograph series 84-11 (1984)

  4. Doob, J.L.: Stochastic processes. New York: John Wiley 1953

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Cincinnati, 45221, Cincinnati, OH, USA

    Jeesen Chen

  2. Department of Statistics, Purdue University, 47907, W. Lafayette, IN, USA

    Burgess Davis & Herman Rubin

Authors
  1. Jeesen Chen
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  2. Burgess Davis
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  3. Herman Rubin
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Additional information

Supported by NSF Grant MCS 8201128

Supported by NSF Grant DMS-8401996

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Cite this article

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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  • Received: 02 February 1985

  • Accepted: 12 January 1986

  • Issue Date: September 1986

  • DOI: https://doi.org/10.1007/BF00339939

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Keywords

  • Uniform Distribution
  • Stochastic Process
  • Probability Measure
  • Probability Theory
  • Statistical Theory
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