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On estimation of a density and its derivatives

  • K. F. Cheng
Article

Summary

Letf n (p) be a recursive kernel estimate off(p) thepth order derivative of the probability density functionf, based on a random sample of sizen. In this paper, we provide bounds for the moments of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_{L_2 } = \left[ {\smallint [f_n^{(p)} (x) - f^{(p)} (x)]^2 dx} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \) and show that the rate of almost sure convergence of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_{L_2 } \) to zero isO(n−α), α<(r−p)/(2r+1), iff(r),r>p≧0, is a continuousL2(−∞, ∞) function. Similar rate-factor is also obtained for the almost sure convergence of\(\left\| {f_n^{(p)} - f^{(p)} } \right\|_\infty = \mathop {\sup }\limits_x \left| {f_n^{(p)} (x) - f^{(p)} (x)} \right|\) to zero under different conditions onf.

AMS 1970 subject classification

Primary 62G05 Secondary 60F15 

Key words and phrases

Recursive kernel density derivatives almost sure convergence rates 

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References

  1. [1]
    Bhattacharya, P. K. (1967). Estimation of a probability density function and its derivatives,Sankhyá, A,29, 373–382.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Carroll, R. J. (1976). On sequential density estimation,Z. Wahrscheinlichkeitsth.,36, 137–151.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Davies, H. I. (1973). Strong consistency of a sequential estimator of a probability density function,Bull. Math. Statist.,15, 49–54.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Davies, H. I. and Wegman, E. J. (1975). Sequential nonparametric density estimation,IEEE Trans. Inf. Theory, IT-21, 619–628.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Deheuvals, P. (1974). Conditions necessaires et suffisantes de convergence ponctuelle presque sûre et uniforme presque sûre des estimateurs de la densité,C. R. Acad. Sci. Paris, A.278, 1217–1220.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Fryer, M. J. (1977). A review of some non-parametric methods of density estimation,J. Inst. Math. Appl.,20, 335–354.MathSciNetCrossRefGoogle Scholar
  7. [7]
    Lamperti, J. (1966).Probability, W. A. Benjamin, Inc. N.Y.zbMATHGoogle Scholar
  8. [8]
    Lin, P. E. (1975). Rates of convergence in empirical Bayes problems: Continuous case,Ann. Statist.,3, 155–164.MathSciNetCrossRefGoogle Scholar
  9. [9]
    Nadaraya, E. A. (1965). On nonparametric estimates of density functions and regression curves,Theory Prob. Appl.,10, 186–190.CrossRefGoogle Scholar
  10. [10]
    Nadaraya, E. A. (1973). On convergence in theL 2-norm of probability density estimates,Theory Prob. Appl.,18, 808–811.CrossRefGoogle Scholar
  11. [11]
    Parzen, E. (1962). On estimation of a probability density function and mode,Ann. Math. Statist.,33, 1065–1076.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Silverman, B. S. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives,Ann. Statist.,6, 177–184.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Singh, R. S. (1977a). Improvement on some known nonparametric uniformly consistent estimators of derivatives of a density,Ann. Statist.,5, 394–400.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Singh, R. S. (1977b). Applications of estimators of a density and its derivatives to certain statistical problems,J.R. Statist. Soc., B,39, 357–363.MathSciNetzbMATHGoogle Scholar
  15. [15]
    Singh, R. S. (1979). On necessary and sufficient conditions for uniform strong consistency of estimators of a density and its derivatives,J. Multivariate Anal.,9, 157–164.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Singh, R. S. (1981). On the exact asymptotic behavior of estimators of a density and its derivatives,Ann. Statist.,9, 453–456.MathSciNetCrossRefGoogle Scholar
  17. [17]
    Walter, G. G. (1977). Properties of Hermite series estimation of probability density,Ann. Statist.,5, 1258–1264.MathSciNetCrossRefGoogle Scholar
  18. [18]
    Walter, G. G. (1980). Addendum to “Properties of Hermite series estimation of probability density”,Ann. Statist.,8, 454–455.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Wegman, E. J. (1972a). Nonparametric probability density estimation: I. A. summary of available methods,Technometrics,14, 533–546.CrossRefGoogle Scholar
  20. [20]
    Wegman, E. J. (1972b). Nonparametric probability density estimation: II. A comparison of density estimation methods,J. Statist. Comp. Simul.,1, 225–245.CrossRefGoogle Scholar
  21. [21]
    Wegman, E. J. and Davies, H. I. (1979). Remarks on some recursive estimators of a probability density,Ann. Statist.,7, 316–327.MathSciNetCrossRefGoogle Scholar
  22. [22]
    Wolverton, C. T. and Wagner, T. J. (1969). Asymptotically optimal discriminant functions for pattern classification,IEEE Trans. Inf. Theory, IT-15, 258–265.MathSciNetCrossRefGoogle Scholar
  23. [23]
    Yamato, H. (1971). Sequential estimation of a continuous probability density and mode,Bull. Math. Statist.,14, 1–12.MathSciNetCrossRefGoogle Scholar

Copyright information

© The Institute of Statistical Mathematics, Tokyo 1982

Authors and Affiliations

  • K. F. Cheng
    • 1
  1. 1.State University of New York at BuffaloBuffaloUSA

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