Abstract
In this paper, we show that the multiplicative bias correction (MBC) techniques can be applied for generalized Birnbaum-Saunders (GBS) kernel density estimators. First, some properties of the MBC-GBS kernel density estimators (bias, variance and mean integrated squared error) are shown. Second, the choice of bandwidth is investigated by adopting the popular cross-validation technique. Finally, the performances of the MBC estimators based on GBS kernels are illustrated by a simulation study, followed by a real application for nonnegative heavy tailed (HT) data. In general, in terms of integrated squared bias (ISB) and integrated squared error (ISE), the proposed estimators outperform the standard GBS kernel estimators.
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References
Barros, M., Paula, G. A., & Leiva, V. (2009). An R implementation for generalized Birnbaum–Saunders distributions. Computational Statistics and Data Analysis, 53, 1511–1528.
Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics and Data Analysis, 31, 131–145.
Chen, S. X. (2000). Gamma kernel estimators for density functions. Annals of the Institute of Statistical Mathematics, 52, 471–480.
Fang, K. T., Kotz, S., & Ng, W. K. (1990). Symmetric multivariate and related distributions. London: Chapman & Hall.
Hagmann, M., & Scaillet, O. (2007). Local multiplicative bias correction for asymmetric kernel density estimators. Journal of Econometrics, 141, 213–249.
Hirukawa, M. (2010). Nonparametric multiplicative bias correction for kernel-type density estimation on the unit interval. Computational Statistics and Data Analysis, 54, 473–495.
Hirukawa, M., & Sakudo, M. (2014). Nonnegative bias reduction methods for density estimation using asymmetric kernels. Computational Statistics and Data Analysis, 75, 112–123.
Jin, X., & Kawczak, J. (2003). Birnbaum–Saunders and lognormal kernel estimators for modelling durations in high frequency financial data. Annals of Economics and Finance, 4, 103–124.
Jones, M. C., & Foster, P. J. (1993). Generalized jackknifing and higher order kernels. Journal of Nonparametric Statistics, 3, 81–94.
Jones, M. C., Linton, O., & Nielsen, J. P. (1995). A simple bias reduction method for density estimation. Biometrika, 82, 327–338.
Kokonendji, C. C., & Senga Kiessé, T. (2011). Discrete associated kernels method and extensions. Statistical Methodology, 8, 497–516.
Leiva, V., Vilca, F., Balakrishnan, N., & Sanhueza, A. (2010). A skewed sinh-normal distribution and its properties and application to air pollution. Communications in Statistics-Theory and Methods, 39, 426–443.
Marchant, C., Bertin, K., Leiva, V., & Saulo, H. (2013). Generalized Birnbaum–Saunders kernel density estimators and an analysis of financial data. Computational Statistics and Data Analysis, 63, 1–15.
Nadarajah, S. (2008). A truncated inverted beta distribution with application to air pollution data. Stochastic Environmental Research and Risk Assessment, 22, 285–289.
Parzen, E. (1962). On estimation of a probability density function and mode. Annals of Mathematical Statistics, 33, 1065–1076.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Annals of Mathematical Statistics, 27, 832–837.
Saulo, H., Leiva, V., Ziegelmann, F. A., & Marchant, C. (2013). A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data. Stochastic Environmental Research and Risk Assessment, 7, 1479–1491.
Scaillet, O. (2004). Density estimation using inverse and reciprocal inverse Gaussian kernels. Journal of Nonparametric Statistics, 16, 217–226.
Silverman, B. W. (1986). Density estimation for statistics and data analysis. New York: Chapman and Hall.
Terrell, G. R., & Scott, D. W. (1980). On improving convergence rates for nonnegative kernel density estimators. Annals of Statistics, 8, 1160–1163.
Ziane, Y., Adjabi, S., & Zougab, N. (2015). Adaptive Bayesian bandwidth selection in asymmetric kernel density estimation for nonnegative heavy-tailed data. Journal of Applied Statistics, 42(8), 1645–1658.
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Zougab, N., Adjabi, S. Multiplicative bias correction for generalized Birnbaum-Saunders kernel density estimators and application to nonnegative heavy tailed data. J. Korean Stat. Soc. 45, 51–63 (2016). https://doi.org/10.1016/j.jkss.2015.07.001
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DOI: https://doi.org/10.1016/j.jkss.2015.07.001