Résumés
Abstract
Standard kernel density estimation methods are very often used in practice to estimate density functions. It works well in numerous cases. However, it is known not to work so well with skewed, multimodal and heavy-tailed distributions. Such features are usual with income distributions, defined over the positive support. In this paper, we show that a preliminary logarithmic transformation of the data, combined with standard kernel density estimation methods, can provide a much better fit of the density estimation.
Parties annexes
Bibliography
- Abadir, K. M., and A. Cornea (2014). “Link of moments before and after transformations, with an application to resampling from fat-tailed distributions.’’ Paper presented at the 2nd Conference of the International Society for Nonparametric Statistics, Cadiz, Spain.
- Abadir, K. M., and S. Lawford (2004). “Optimal asymmetric kernels.’’ Economics Letters, 83: 61-68.
- Ahamada, I., and E. Flachaire (2011). Non-Parametric Econometrics. Oxford University Press.
- Bouezmarni, T., and J. V. K. Rombouts (2010). “Nonparametric density estimation for multivariate bounded data.’’ Journal of Statistical Planning and Inference, 140: 139-152.
- Bouezmarni, T., and O. Scaillet (2005). “Consistency of asymmetric kernel density estimators and smoothed histograms with application to income data.’’ Econometric Theory, 21: 390-412.
- Bowman, A. (1984). “An Alternative Method of Cross-validation for the Smoothing of Kernel Density Estimates.’’ Biometrika, 71: 353-360.
- Bowman, A. W., and A. Azzalini (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations. Oxford University Press.
- Buch-Larsen, T., J. P. Nielsen, M. Guillén, and C. Bolanccé (2005). “Kernel density estimation for heavy-tailed distribution using the champernowne transformation.’’ Statistics, 39: 503-518.
- Charpentier, A., and A. Oulidi (2010). “Beta Kernel Quantile Estimators of Heavy-tailed Loss Distributions.’’ Statistics and Computing, 20: 35-55.
- Chen, S. X. (1999). “Beta Kernel Estimators for Density Functions.’’ Computational Statistics & Data Analysis. 31: 131-145.
- Chen, S. X. (2000), “Probability Density Function Estimation Using Gamma Kernels.’’ Annals of the Institute of Statistical Mathematics, 52: 471-480.
- Cowell, F. A., and E. Flachaire (2007). “Income distribution and inequality measurement: The problem of extreme values.’’ Journal of Econometrics, 141: 1044-1072.
- Cowell, F. A., and E. Flachaire (2015). “Statistical methods for distributional analysis.’’ in A. B. Atkinson and F. Bourguignon (Eds.), Handbook of Income Distribution, Volume 2A, Chapter 6. New York: Elsevier Science B. V.
- Davidson, R. (2012). “Statistical inference in the presence of heavy tails.’’ The Econometrics Journal, 15: C31-C53.
- Davidson, R., and E. Flachaire (2007). “Asymptotic and bootstrap inference for inequality and poverty measures.’’ Journal of Econometrics, 141: 141 - 166.
- Departmentof Social Security (1996). “Households Below Average Income: Methodological Review Report of a Working Group.’’ Corporate Document Services, London.
- Department of Social Security (1998). “Households Below Average Income 1979-1996/7.’’ Corporate Document Services, London.
- Devroye, L., and L. Györfi (1985). Nonparametric Density Estimation: The L1 View. Wiley Series in Probability and Mathematical Statistics.
- Hagmann, M., and O. Scaillet (2007). “Local multiplicative bias correction for asymmetric kernel density estimators.’’ Journal of Econometrics, 141: 213-249.
- Härdle, W. (1989). Applied Nonparametric Regression. Econometric Society Monograph, Cambridge.
- Kleiber, C., and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken. N.J.: John Wiley.
- Kuruwita, C. N., K. B. Kulasekera. and W. J. Padgett (2010). “Density estimation using asymmetric kernels and bayes bandwidths with censored data.’’ Journal of Statistical Planning and Inference, 140: 1765-1774.
- Li, Q., and J. S. Racine (2006). Nonparametric Econometrics. Princeton University Press.
- Markovich, N. (2007). Nonparametric Analysis of Univariate Heavy-Tailed Data. Wiley Series in Probability and Statistics.
- Marron, J. S., and D. Ruppert (1994). “Transformations to reduce boundary bias in kernel density estimation.’’ Journal of the Royal Statistical Society, Series B 56: 653-671.
- Marron, J. S., and H. P. Schmitz (1992). “Simultaneous density estimation of several income distributions.’’ Econometric Theory, 8: 476-488.
- Pagan, A., and A. Ullah (1999). Nonparametric Econometrics. Cambridge: Cambridge University Press.
- Pearson, R. K. (2005). Mining Imperfect Data: Dealing with Contamination and Incomplete Records. Society for Industrial and Applied Mathematics.
- Portnoy, S., and R. Koenker (1989). “Adaptive L estimation of linear models.’’ Annals of Statistics. 17: 362-381.
- Rudemo, M. (1982). “Empirical choice of histograms and kernel density estimators.’’ Scandinavian Journal of Statistics, 9: 65-78.
- Scaillet, O. (2004). “Density estimation using inverse and reciprocal inverse Gaussian kernels.’’ Journal of Nonparametric Statistics, 16: 217-226.
- Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. New-York: John Wiley & Sons.
- Sheather, S. J., and M. C. Jones (1991). “A reliable data-based bandwidth selection method for kernel density estimation.’’ Journal of the Royal Statistical Society, 53: 683-690.
- Silverman, B. (1986). Density Estimation for Statistics and Data Analysis, London: Chapman & Hall.
- Simono, J. S. (1996), Smoothing Methods in Statistics. New-York: Springer Verlag.
- Stone, M. (1974). “Cross-validatory choice and assessment of statistical predictions (with discussion).’’ Journal of the Royal Statistical Society, B 36: 111-147.
- Wand, M. P., and M. C. Jones (1995). Kernel Smoothing. London: Chapman & Hall.