ror.org/002rjbv21
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.10.1016/j.econlet.2003.07.017 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Abadir, K. M., and S. Lawford (2004). “Optimal asymmetric kernels.’’ Economics Letters, 83: 61-68.10.1016/j.econlet.2003.07.017 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Ahamada, I., and E. Flachaire (2011). Non-Parametric Econometrics. Oxford University Press.10.1017/S0266466605050218 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Bouezmarni, T., and J. V. K. Rombouts (2010). “Nonparametric density estimation for multivariate bounded data.’’ Journal of Statistical Planning and Inference, 140: 139-152.10.1016/j.jspi.2009.07.013 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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. 10.1017/S0266466605050218 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Bowman, A. (1984). “An Alternative Method of Cross-validation for the Smoothing of Kernel Density Estimates.’’ Biometrika, 71: 353-360.10.1093/biomet/71.2.353 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Bowman, A. W., and A. Azzalini (1997). Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations. Oxford University Press.10.1016/S0167-9473(99)00010-9 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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: 503-518.10.1080/02331880500439782 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Charpentier, A., and A. Oulidi (2010). “Beta Kernel Quantile Estimators of Heavy-tailed Loss Distributions.’’ Statistics and Computing, 20: 35-55.10.1007/s11222-009-9114-2 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Chen, S. X. (1999). “Beta Kernel Estimators for Density Functions.’’ Computational Statistics & Data Analysis. 31: 131-145.10.1016/S0167-9473(99)00010-9 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Chen, S. X. (2000), “Probability Density Function Estimation Using Gamma Kernels.’’ Annals of the Institute of Statistical Mathematics, 52: 471-480.10.1023/A:1004165218295 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Cowell, F. A., and E. Flachaire (2007). “Income distribution and inequality measurement: The problem of extreme values.’’ Journal of Econometrics, 141: 1044-1072.10.1016/j.jeconom.2007.01.001 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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.10.1016/B978-0-444-59428-0.00007-2 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Davidson, R. (2012). “Statistical inference in the presence of heavy tails.’’ The Econometrics Journal, 15: C31-C53.10.1111/j.1368-423X.2010.00340.x Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Davidson, R., and E. Flachaire (2007). “Asymptotic and bootstrap inference for inequality and poverty measures.’’ Journal of Econometrics, 141: 141 - 166.10.1016/j.jeconom.2007.01.009 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Departmentof Social Security (1996). “Households Below Average Income: Methodological Review Report of a Working Group.’’ Corporate Document Services, London.10.1002/9780470723609 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Department of Social Security (1998). “Households Below Average Income 1979-1996/7.’’ Corporate Document Services, London.10.1017/S0266466600013177 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Devroye, L., and L. Györfi (1985). Nonparametric Density Estimation: The L1 View. Wiley Series in Probability and Mathematical Statistics.10.1017/CBO9780511612503 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Hagmann, M., and O. Scaillet (2007). “Local multiplicative bias correction for asymmetric kernel density estimators.’’ Journal of Econometrics, 141: 213-249.10.1016/j.jeconom.2007.01.018 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Härdle, W. (1989). Applied Nonparametric Regression. Econometric Society Monograph, Cambridge.10.1214/aos/1176347022 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Kleiber, C., and S. Kotz (2003). Statistical Size Distributions in Economics and Actuarial Sciences. Hoboken. N.J.: John Wiley.10.1002/0471457175 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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: 1765-1774.10.1016/j.jspi.2010.01.001 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Li, Q., and J. S. Racine (2006). Nonparametric Econometrics. Princeton University Press.10.1007/978-1-4899-3324-9 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Markovich, N. (2007). Nonparametric Analysis of Univariate Heavy-Tailed Data. Wiley Series in Probability and Statistics.10.1002/9780470723609 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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: 653-671.10.1007/978-1-4899-4493-1 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Marron, J. S., and H. P. Schmitz (1992). “Simultaneous density estimation of several income distributions.’’ Econometric Theory, 8: 476-488.10.1017/S0266466600013177 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Pagan, A., and A. Ullah (1999). Nonparametric Econometrics. Cambridge: Cambridge University Press.10.1017/CBO9780511612503 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Pearson, R. K. (2005). Mining Imperfect Data: Dealing with Contamination and Incomplete Records. Society for Industrial and Applied Mathematics.10.1137/1.9780898717884 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Portnoy, S., and R. Koenker (1989). “Adaptive L estimation of linear models.’’ Annals of Statistics. 17: 362-381.10.1214/aos/1176347022 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Rudemo, M. (1982). “Empirical choice of histograms and kernel density estimators.’’ Scandinavian Journal of Statistics, 9: 65-78.Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Scaillet, O. (2004). “Density estimation using inverse and reciprocal inverse Gaussian kernels.’’ Journal of Nonparametric Statistics, 16: 217-226.10.1080/10485250310001624819 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Scott, D. W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. New-York: John Wiley & Sons.10.1002/9780470316849 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- 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.Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Silverman, B. (1986). Density Estimation for Statistics and Data Analysis, London: Chapman & Hall.10.1007/978-1-4899-3324-9 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Simono, J. S. (1996), Smoothing Methods in Statistics. New-York: Springer Verlag.10.1007/978-1-4612-4026-6 Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Stone, M. (1974). “Cross-validatory choice and assessment of statistical predictions (with discussion).’’ Journal of the Royal Statistical Society, B 36: 111-147.10.1111/j.2517-6161.1991.tb01857.x Google Scholar Rechercher cette référence bibliographique sur Google Scholar
- Wand, M. P., and M. C. Jones (1995). Kernel Smoothing. London: Chapman & Hall.10.1007/978-1-4899-4493-1 Google Scholar Rechercher cette référence bibliographique sur Google Scholar