Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-12-03T00:02:28.406Z Has data issue: false hasContentIssue false
  • English
  • Français

CONSISTENCY OF ASYMMETRIC KERNEL DENSITY ESTIMATORS AND SMOOTHED HISTOGRAMS WITH APPLICATION TO INCOME DATA

Published online by Cambridge University Press:  31 March 2005

Taoufik Bouezmarni  and
Olivier Scaillet
Taoufik Bouezmarni
Affiliation:
Institute of Statistics, Université Catholique de Louvain
Olivier Scaillet
Affiliation:
HEC Genève, Université de Genève and FAME
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider asymmetric kernel density estimators and smoothed histograms when the unknown probability density function f is defined on [0,+∞). Uniform weak consistency on each compact set in [0,+∞) is proved for these estimators when f is continuous on its support. Weak convergence in L1 is also established. We further prove that the asymmetric kernel density estimator and the smoothed histogram converge in probability to infinity at x = 0 when the density is unbounded at x = 0. Monte Carlo results and an empirical study of the shape of a highly skewed income distribution based on a large microdata set are finally provided.We thank O. Linton and the three referees for constructive criticism and M.P. Feser and J. Litchfield for kindly providing the Brazilian data. We are grateful to I. Gijbels, J.M. Rolin, and I. Van Keilegom for their stimulating remarks and to participants at the workshop on statistical modeling (UCL 2002), LAMES (Sao Paulo 2002), L1 Norm conference (Neuchatel 2002), Geneva econometrics seminar, and KUL econometrics seminar for their comments. Part of this research was done when the second author was visiting THEMA and IRES. The first, resp. second, author gratefully acknowledges financial support from the “Projet d'Actions de Recherche Concertées” grant 98/03-217, and from the IAP research network grant P5/24 of the Belgian state, resp. the Swiss National Science Foundation through the National Centre of Competence in Research: Financial Valuation and Risk Management (NCCR-FINRISK).

Type
Research Article
Information
Econometric Theory , Volume 21 , Issue 2 , April 2005 , pp. 390 - 412
Copyright
© 2005 Cambridge University Press

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

Abadir, K. (1999) An introduction to hypergeometric functions for economists. Econometric Reviews 18, 287–330.Google Scholar
Abadir, K. & S. Lawford (2004) Optimal asymmetric kernels. Economics Letters 83, 61–68.Google Scholar
Abadir, K. & P. Paruolo (1997) Two mixed normal densities from cointegration analysis. Econometrica 65, 671–680.Google Scholar
Abadir, K. & M. Rockinger (2003) Density functionals with an option-pricing application. Econometric Theory 19, 778–811.Google Scholar
Aït-Sahalia, Y. (1996a) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527–560.Google Scholar
Aït-Sahalia, Y. (1996b) Continuous-time models of the spot interest rate. Review of Financial Studies 9, 385–426.Google Scholar
Bolancé, C., M. Guillen, & J.P. Nielsen (2003) Kernel density estimation of actuarial loss functions. Insurance: Mathematics and Economics 32, 3–18.Google Scholar
Bouezmarni, T. & J.M. Rolin (2002) Bernstein Estimator for Unbounded Density Function. Mimeo, Institut de Statistique, Université Catholique de Louvain.
Bouezmarni, T. & J.M. Rolin (2003) Consistency of beta kernel density function estimator. Canadian Journal of Statistics 31, 89–98.Google Scholar
Bouezmarni, T. & O. Scaillet (2003) Consistency of Asymmetric Kernel Density Estimators and Smoothed Histograms with Application to Income Data. DP 0306, Institut de Statistique, Université Catholique de Louvain.
Brown, B. & S.X. Chen (1999) Beta-Bernstein smoothing for regression curves with compact support. Scandinavian Journal of Statistics 26, 47–59.Google Scholar
Chen, S.X. (1999) A beta kernel estimator for density functions. Computational Statistics and Data Analysis 31, 131–145.Google Scholar
Chen, S.X. (2000) Probability density functions estimation using gamma kernels. Annals of the Institute of Statistical Mathematics 52, 471–480.Google Scholar
Chen, S.X. (2002) Local linear smoothers using asymmetric kernels. Annals of the Institute of Statistical Mathematics 54, 312–323.Google Scholar
Cowell, F. (2000) Measurement of inequality. In A. Atkinson & F. Bourguignon (eds.), Handbook of Income Distribution, pp. 87–166. North Holland.
Cowell, F., F. Ferreira, & J. Litchfield (1998) Income distribution in Brazil 1981–1990: Parametric and non-parametric approaches. Journal of Income Distribution 8, 63–76.Google Scholar
Devroye, L. (1997) Universal smoothing factor selection in density estimation: Theory and practice. Test 6, 223–320.Google Scholar
Devroye, L. & L. Gyorfi (1985) Nonparametric Density Estimation: The L1 View. Wiley.
Devroye, L. & G. Lugosi (1996) A universally acceptable smoothing factor for kernel density estimates. Annals of Statistics 24, 2499–2512.Google Scholar
Dvoretzky, A., J. Kiefer, & J. Wolfowitz (1956) Asymptotic minimax character of the sample distribution function and of the classical multinomial estimator. Annals of Mathematical Statistics 1, 444–453.Google Scholar
Engle, R. (2000) The econometrics of ultra-high frequency data. Econometrica 68, 1–22.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, vol. 2. Wiley.
Fenton, V. & R. Gallant (1996) Convergence rates of SNP density estimators. Econometrica 64, 719–727.Google Scholar
Fernandes, M. & J. Grammig (2000) Nonparametric specification tests for conditional duration models. Journal of Econometrics (forthcoming).Google Scholar
Foldes, A. & P. Revesz (1974) A general method for density estimation. Scientiarum Mathematicarun Hungarica 7, 90–94.Google Scholar
Gawronski, N. & U. Stadtmüller (1980) On density estimation by means of Poisson's distribution. Scandinavian Journal of Statistics 7, 90–94.Google Scholar
Gawronski, N. & U. Stadtmüller (1981) Smoothing histograms by means of lattice and continuous distributions. Metrika 28, 155–164.Google Scholar
Hagmann, M. & O. Scaillet (2003) Local multiplicative bias correction for asymmetric kernel density estimators. DP 91, FAME, Switzerland.
Härdle, W. & O. Linton (1994) Applied nonparametric methods. In R. Engle & D. McFadden (eds.), Handbook of Econometrics, vol. 4, pp. 2295–2339. North-Holland.
Hall, P. & M. Wand (1988) Minimizing L1 distance in nonparametric density estimation. Journal of Multivariate Analysis 26, 59–88.Google Scholar
Jones, M. (1993) Simple boundary correction for kernel density estimation. Statistics and Computing 3, 135–146.Google Scholar
Jones, M. & P. Foster (1996) A simple nonnegative boundary correction method for kernel density estimation. Statistica Sinica 6, 1005–1013.Google Scholar
Klugman, S., H. Panjer, & G. Willmot (1998) Loss Models: From Data to Decisions. Wiley.
Marron, J.S. & D. Ruppert (1994) Transformations to reduce boundary bias in kernel density estimation. Journal of the Royal Statistical Society, Series B 56, 653–671.Google Scholar
Massart, P. (1990) The tight constant in Dvoretzky-Kiefer-Wolfowitz inequality. Annals of Probability 18, 1269–1283.Google Scholar
Müller, H. (1991) Smooth optimum kernel estimators near endpoints. Biometrika 78, 521–530.Google Scholar
Pagan, A. & A. Ullah (1999) Nonparametric Econometrics. Cambridge University Press.
Renault, O. & O. Scaillet (2003) On the way to recovery: A nonparametric bias free estimation of recovery rate densities. Journal of Banking and Finance (forthcoming).Google Scholar
Rice, J. (1984) Boundary modification for kernel regression. Communications in Statistics—Theory and Methods 13, 893–900.Google Scholar
Scaillet, O. (2004) Density estimation using inverse and reciprocal inverse gaussian kernels. Journal of Nonparametric Statistics 16, 217–226.Google Scholar
Schuster, E. (1985) Incorporating support constraints into nonparametric estimators of densities. Communications in Statistics—Theory and Methods 14, 1123–1136.Google Scholar
Stadtmüller, U. (1983) Asymptotic distribution of smoothed histograms. Metrika 30, 145–158.Google Scholar
Walter, G. & J. Blum (1979) Probability density estimation using delta sequences. Annals of Statistics 7, 328–340.Google Scholar