Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-25T05:47:14.674Z Has data issue: false hasContentIssue false
  • English
  • Français

KERNEL ESTIMATION WHEN DENSITY MAY NOT EXIST

Published online by Cambridge University Press:  26 February 2008

Victoria Zinde-Walsh
Victoria Zinde-Walsh*
Affiliation:
McGill University and CIREQ
*
Address correspondence to Victoria Zinde-Walsh, Department of Economics, McGill University, 855 Sherbrooke Street West, Montreal, Quebec, CanadaH3A 2T7; e-mail: victoria.zinde-walsh@mcgill.ca.
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Econometric Theory , Volume 24 , Issue 3 , June 2008 , pp. 696 - 725
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Ahmad, I.A. ’ Cerrito, P.B. (1994) Nonparametric estimation of joint discrete-continuous probability densities with applications. Journal of Statistical Planning and Inference 41, 349–364.CrossRefGoogle Scholar
Bierens, H.J. (1987) Kernel estimation of regression functions. In Bewley, T.F. (ed.), Advances in Econometrics, vol. 1, pp. 99–144. Cambridge University Press.Google Scholar
Devroye, L.P. ’ Györfi, L. (1985) Nonparametric Density Estimation. Wiley.Google Scholar
Encyclopaedia of Mathematics: An Updated and Annotated Translation of the Soviet Matematicheskaya Encyclopedia (1988). Kluwer.Google Scholar
Fan, J. ’ Gijbels, Z. (1995) Local Polynomial Modelling and Its Applications. Chapman and Hall.Google Scholar
Frigyesi, A. (2004) Topics in multifractal measures, nonparametrics and biostatistics. Doctoral Thesis, Centre for Mathematical Sciences, Lund University.Google Scholar
Frigyesi, A. ’ Hössjer, O. (1998) A test for singularity. Statistics and Probability Letters 40, 215–226.Google Scholar
Gel'fand, I.M. ’ Shilov, G.E. (1964) Generalized Functions, vol. 1: Properties and Operations. Academic Press.Google Scholar
Gel'fand, I.M. ’ Shilov, G.E. (1964) Generalized Functions, vol. 2: Spaces of Test Functions and Generalized Functions. Academic Press.Google Scholar
Gel'fand, I.M. ’ Vilenkin, N.Y. (1964) Generalized Functions, vol. 4: Applications of Harmonic Analysis. Academic Press.Google Scholar
Green, D.A. ’ Riddell, W.C. (1997) Qualifying for unemployment insurance: An empirical analysis. Economic Journal 107, 67–84.CrossRefGoogle Scholar
Halperin, I. (1952) Introduction to the Theory of Distributions (Based on Lectures by L. Schwartz). University of Toronto Press.Google Scholar
Härdle, W., Kerkyacharian, G., Picard, D., ’ Tsybakov, A. (1998) Wavelets, Approximations and Statistical Applications. Springer-Verlag.Google Scholar
Li, Q. ’ Racine, J. (2003) Nonparametric estimation of distributions with categorical and continuous data. Journal of Multivariate Analysis 86, 266–292.CrossRefGoogle Scholar
Lu, Z.-Q. (1999) Nonparametric regression with singular design. Journal of Multivariate Analysis 70, 177–201.CrossRefGoogle Scholar
Müller, H.-G. (1992) Change-points in nonparametric regression analysis. Annals of Statistics 20, 737–761.Google Scholar
Pagan, A. ’ Ullah, A. (1999) Nonparametric Econometrics. Cambridge University Press.CrossRefGoogle Scholar
Phillips, P.C.B. (1991) A shortcut to LAD estimator asymptotics. Econometric Theory 7, 450–463.CrossRefGoogle Scholar
Phillips, P.C.B. (1995) Robust non-stationary regression. Econometric Theory 11, 912–951.Google Scholar
Schennach, S. (2004) Estimation of nonlinear models with measurement error. Econometrica 72, 33–75.Google Scholar
Schwartz, L. (1950) Théorie des Distributions, vols. 1 and 2. Hermann.Google Scholar
Sobolev, S. (1992) Cubature Formulas and Modern Analysis. Gordon and Breach Science Publishers.Google Scholar
Zinde-Walsh, V. (2002) Asymptotic theory for some high breakdown point estimators. Econometric Theory 18, 1172–1196.Google Scholar
Zinde-Walsh, V. ’ Phillips, P.C.B. (2003) Fractional Brownian motion as a differentiable generalized Gaussian process. In Athreya, K., Majumdar, M., Puri, M., ’ Waymire, W. (eds.), Probability, Statistics and Their Applications: Papers in Honor of Rabi Bhattacharya, Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol. 41, pp. 285–292.CrossRefGoogle Scholar