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Kernel Estimation When Density May Not Exist

  • Zinde-Walsh, Victoria

Nonparametric kernel estimation of density and conditional mean is widely used, but many of the pointwise and global asymptotic results for the estimators are not available unless the density is continuous and appropriately smooth; in kernel estimation for discrete-continuous cases smoothness is required for the continuous variables. Nonsmooth density and mass points in distributions arise in various situations that are examined in empirical studies; some examples and explanations are discussed in the paper. Generally, any distribution function consists of absolutely continuous, discrete, and singular components, but only a few special cases of nonparametric estimation involving singularity have been examined in the literature, and asymptotic theory under the general setup has not been developed. In this paper the asymptotic process for the kernel estimator is examined by means of the generalized functions and generalized random processes approach; it provides a unified theory because density and its derivatives can be defined as generalized functions for any distribution, including cases with singular components. The limit process for the kernel estimator of density is fully characterized in terms of a generalized Gaussian process. Asymptotic results for the Nadaraya–Watson conditional mean estimator are also provided.

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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 24 (2008)
Issue (Month): 03 (June)
Pages: 696-725

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Handle: RePEc:cup:etheor:v:24:y:2008:i:03:p:696-725_08
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  1. Peter C.B. Phillips, 1993. "Robust Nonstationary Regression," Cowles Foundation Discussion Papers 1064, Cowles Foundation for Research in Economics, Yale University.
  2. Zinde-Walsh, Victoria, 2002. "Asymptotic Theory For Some High Breakdown Point Estimators," Econometric Theory, Cambridge University Press, vol. 18(05), pages 1172-1196, October.
  3. Lu, Zhan-Qian, 1999. "Nonparametric Regression with Singular Design," Journal of Multivariate Analysis, Elsevier, vol. 70(2), pages 177-201, August.
  4. Victoria Zinde-Walsh & Peter C.B. Phillips, 2003. "Fractional Brownian Motion as a Differentiable Generalized Gaussian Process," Cowles Foundation Discussion Papers 1391, Cowles Foundation for Research in Economics, Yale University.
  5. Li, Qi & Racine, Jeff, 2003. "Nonparametric estimation of distributions with categorical and continuous data," Journal of Multivariate Analysis, Elsevier, vol. 86(2), pages 266-292, August.
  6. repec:cup:cbooks:9780521355643 is not listed on IDEAS
  7. Susanne M. Schennach, 2004. "Estimation of Nonlinear Models with Measurement Error," Econometrica, Econometric Society, vol. 72(1), pages 33-75, 01.
  8. Frigyesi, Attila & Hössjer, Ola, 1998. "A test for singularity," Statistics & Probability Letters, Elsevier, vol. 40(3), pages 215-226, October.
  9. repec:cup:cbooks:9780521586115 is not listed on IDEAS
  10. Green, David A & Riddell, W Craig, 1997. "Qualifying for Unemployment Insurance: An Empirical Analysis," Economic Journal, Royal Economic Society, vol. 107(440), pages 67-84, January.
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