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On the uniform complete convergence of density function estimates

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Abstract

Letf be a uniformly continuous density function. LetW be a non-negative weight function which is continuous on its compact support [a, b] and ∫ ba W(x)dx=1. The complete convergence of

$$\mathop {\sup }\limits_{ - \infty< s< \infty } \left| {\frac{1}{{nb\left( n \right)}}\sum\limits_{k - 1}^n {W\left( {\frac{{s - X_k }}{{b\left( n \right)}}} \right)} - f\left( s \right)} \right|$$

to zero is obtained under varying conditions on the bandwidthsb(n), support or moments off, and smoothness ofW. For example, one choice of the weight function for these results is Epanechnikov's optimal function andnb 2(n)>n δ for some δ>0. The uniform complete convergence of the mode estimate is also considered.

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References

  1. Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates,Ann. Statist.,1, 1071–1095.

    Article  MathSciNet  Google Scholar 

  2. Billingsley, P. (1968).Convergence of Probability Measures, Wiley, New York.

    MATH  Google Scholar 

  3. Chow, Y. S. (1966). Some convergence theorems for independent random variables,Ann. Math. Statist.,35, 1482–1493.

    Article  MathSciNet  Google Scholar 

  4. Deheuvels, P. (1974). Conditions nécessaires et suffisantes de convergence ponctuelle presque sûre et uniforme presque sûre des estimateurs de la densité,C. R. Acad. Sci., A,278, 1217–1220.

    MathSciNet  MATH  Google Scholar 

  5. Devroye, L. P. and Wagner, T. J. (1977). The strong uniform consistency of nearest neighbor density estimates,Ann. Statist.,5, 536–540.

    Article  MathSciNet  Google Scholar 

  6. Epanechnikov, V. A. (1969). Nonparametric estimates of multivariate probability density,Theory Prob. Appl.,14, 153–158.

    Article  MathSciNet  Google Scholar 

  7. Nadaraya, E. A. (1965). On nonparametric estimates of density functions and regression curves,Theory Prob. Appl.,10, 186–190.

    Article  Google Scholar 

  8. Parzen, E. (1962). On estimation of a probability density and mode,Ann. Math. Statist.,33, 1065–1076.

    Article  MathSciNet  Google Scholar 

  9. Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function,Ann. Math. Statist.,27, 832–835.

    Article  MathSciNet  Google Scholar 

  10. Rosenblatt, M. (1971). Curve estimates,Ann. Math. Statist.,42, 1815–1842.

    Article  MathSciNet  Google Scholar 

  11. Schuster, E. F. (1969). Estimation of probability density function and its derivatives,Ann. Math. Statist.,40, 1187–1195.

    Article  MathSciNet  Google Scholar 

  12. Silverman, B. W. (1978). Weak and strong uniform consistency of the kernel estimate of a density and its derivatives,Ann. Statist.,6, 177–184.

    Article  MathSciNet  Google Scholar 

  13. Stout, W. (1968). Some results on the complete and almost sure convergence of linear combinations of independent random variables and martingale differences,Ann. Math. Statist.,39, 1549–1562.

    Article  MathSciNet  Google Scholar 

  14. Taylor, R. L. and Cheng, K. F. (1977). On the uniform complete convergence of density function estimates, FSU Statistics Report M446-ONR Technical Report No. 129, Florida State University.

  15. Woodroofe, M. (1967). On the maximum deviation of the sample density,Ann. Math. Statist.,38, 475–481.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. University of Soth Carolina, South Carolina, USA

    R. L. Taylor & K. F. Cheng

  2. The Florida State University, Florida, USA

    R. L. Taylor & K. F. Cheng

Authors
  1. R. L. Taylor
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  2. K. F. Cheng
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Additional information

While visiting at Florida State University.

Research supported by National Institute of Environmental Health Sciences under Grant 5 T32 ES0711-03.

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Taylor, R.L., Cheng, K.F. On the uniform complete convergence of density function estimates. Ann Inst Stat Math 30, 397–406 (1978). https://doi.org/10.1007/BF02480229

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  • DOI: https://doi.org/10.1007/BF02480229

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