Kybernetika 53 no. 2, 198-219, 2017

Density estimation via best $L^2$-approximation on classes of step functions

Dietmar Ferger and John VenzDOI: 10.14736/kyb-2017-2-0198

Abstract:

We establish consistent estimators of jump positions and jump altitudes of a multi-level step function that is the best $L^2$-approximation of a probability density function $f$. If $f$ itself is a step-function the number of jumps may be unknown.

Keywords:

density estimation, argmin-theorem, step functions, martingale inequalities, multivariate cadlag stochastic processes

Classification:

62F10, 62G07, 60G44

paper.pdf

References:

  1. P. Billingsley: Convergence of Probability Measures. Second Edition. John Wiley and Sons, Inc., New York 1999.   DOI:10.1002/9780470316962
  2. Z. Birnbaum and A. Marshall: Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32 (1961), 687-703.   DOI:10.1214/aoms/1177704964
  3. C. K. Chu and P. E. Cheng: Estimation of jump points and jump values of a density function. Statist. Sinica 6 (1996), 79-95.   CrossRef
  4. R. .M. Dudley: Uniform Central Limit Theorems. Cambridge University Press, New York 1999.   DOI:10.1017/cbo9780511665622
  5. D. Ferger: Minimum distance estimation in normed linear spaces with Donsker-classes. Math. Methods Statist. 19 (2010), 246-266.   DOI:10.3103/s1066530710030038
  6. D. Ferger: Arginf-sets of multivariate cadlag processes and their convergence in hyperspace topologies. Theory Stoch. Process. 20 (2015), 36, 13-41.   CrossRef
  7. P. Gaenssler and W. Stute: Wahrscheinlichkeitstheorie. Springer-Verlag, Berlin, Heidelberg, New York 1977.   CrossRef
  8. Y. Kanazawa: An optimal variable cell histogram. Comm. Statist. Theory Methods 17 (1988), 1401-1422.   DOI:10.1080/03610928808829688
  9. Y. Kanazawa: An optimal variable cell histogram based on the sample spacings. Ann. Statist. 20 (1992), 291-304.   DOI:10.1214/aos/1176348523
  10. H. L. Koul: Weighted Empirical Processes in Dynamic Nonlinear Models. Second Edition. Springer-Verlag, New York 2002.   DOI:10.1007/978-1-4613-0055-7
  11. P. Massart: The tight constant in the Dvoretzky-Kiefer-Wolfowitz inequality. Ann. Probab. 18 (1990), 1269-1283.   DOI:10.1214/aop/1176990746
  12. R. T. Rockefellar and R. J.-B. Wets: Variational Analysis. Springer-Verlag, Berlin, Heidelberg 1998.   CrossRef
  13. G. R. Shorack and J. A. Wellner: Empirical Processes With Applications to Statistics. John Wiley and Sons, New York, Chichester, Brisbane, Toronto, Singapore 1986.   CrossRef