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Histogram-kernel error and its application for bin width selection in histograms

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Abstract

Histogram and kernel estimators are usually regarded as the two main classical data-based nonparametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. Just as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott’s optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.

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References

  1. Beer, C.F., Swanepoel, J.W.H. Simple and effective number-of-bins circumference selectors for a histogram. Statistics and Computing, 9: 27–35 (1999)

    Article  Google Scholar 

  2. Bowman, A.W. An alternative method of cross-validation for the smoothing of density estimates. Biometrika, 71: 353–360 (1984)

    Article  MathSciNet  Google Scholar 

  3. Cencov, N.N. Estimation of an unknown distribution density from observations. Soviet Math., 3: 1159–1562 (1962)

    Google Scholar 

  4. Daly, J.E. The construction of optimal histogram. Commun. Statist. Theory Meth., 17(9): 2921–2931 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Devroye, L. The double kernel method in density estimation”, Annales de L’Institut Henri Poincare, 25: 533–580 (1989)

    MathSciNet  MATH  Google Scholar 

  6. Faraway, J.J., Jhun, M. Bootstrap choice of bandwidth for density estimation. Journal of Statistical Planning and Inference, 85: 1119–1122 (1990)

    MathSciNet  Google Scholar 

  7. Freedman, D., Diaconis, P. On the histogram as a density estimation: L 2 theory. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 57: 453–476 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. He, K., Meeden, G. Selecting the number of bins in a histogram: A decision theoretical approach. Journal of Statistical Planning and Inference, 61: 49–59 (1997)

    Article  MATH  Google Scholar 

  9. Parzen, E. Nonparametric statistical data modeling (with discussion). Journal of the American Statistical Association, 74: 105–131 (1979)

    MathSciNet  MATH  Google Scholar 

  10. Rosenblatt, M. Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27: 832–837 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  11. Rudemo, M. Empirical choice of histogram and kernel density estimators. Scandinavian Journal of Statistics, 9: 65–78 (1982)

    MathSciNet  MATH  Google Scholar 

  12. Sain, S.R., Scott, D.W. On locally adaptive density estimation. Journal of the American Statistical Association, 91: 1525–1534 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Scott, D.W. On Optimal and Data-Based Histograms. Biometrika, 66: 605–610 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  14. Scott, D.W., Terrell, G.R. Biased and unbiased cross-validation in density estimation. Journal of the American Statistical Association, 82: 1131–1146 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Scott, D.W. Multivariate density estimation-Theory, practice and visualization. John Wiley & Sons, New York, 1992

    Book  MATH  Google Scholar 

  16. Simonoff, J.S., Udina, F. Measuring the stability of histogram appearance when the anchor position is changed. Computational Statistics and Data Analysis, 23: 335–353 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sturges, H.A. The choice of a class interval. Journal of the American Statistical Association, 21: 65–66 (1926)

    Article  Google Scholar 

  18. Taylor, C.C. Bootstrap choice of the smoothing parameter in kernel density estimation. Biometrika, 76: 705–712 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Terrell, G.R. The maximal smoothing principle in density estimation. Journal of the American Statistical Association, 85: 470–477 (1990)

    MathSciNet  Google Scholar 

  20. Terrell, G.R., Scott, D.W. Over-smoothed nonparamtric density estimates. Journal of the American Statistical Association, 80: 209–214 (1985)

    MathSciNet  Google Scholar 

  21. Wang, X.X. Nonparametric density estimation: Histogram and binned kernel density estimation theories and their applications. Master Thesis, Graduate University of Chinese Academy of Sciences, 2007

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Authors and Affiliations

  1. Department of Mathematics, Graduate University of Chinese Academy of Sciences, Beijing, 100049, China

    Xiu-xiang Wang

  2. College of Management, Graduate University of Chinese Academy of Sciences, Beijing, 100190, China

    Jian-fang Zhang

Authors
  1. Xiu-xiang Wang
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  2. Jian-fang Zhang
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Correspondence to Xiu-xiang Wang.

Additional information

Supported by the National Natural Science Foundation of China (No. 70371018, 70572074)

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Wang, Xx., Zhang, Jf. Histogram-kernel error and its application for bin width selection in histograms. Acta Math. Appl. Sin. Engl. Ser. 28, 607–624 (2012). https://doi.org/10.1007/s10255-007-7081-y

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  • DOI: https://doi.org/10.1007/s10255-007-7081-y

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