Density estimation using inverse and reciprocal inverse Gaussian kernels

This paper introduces two new nonparametric estimators for probability density functions which have support on the non-negative real line. These kernel estimators are based on some inverse Gaussian (IG) and reciprocal inverse Gaussian (RIG) probability density functions used as kernels. We show that they share the same properties as those of gamma kernel estimators: they are free of boundary bias, always non-negative and achieve the optimal rate of convergence for the mean integrated squared error (MISE). Monte Carlo results concerning finite sample properties are reported for different distributions and sample sizes. Keywords: Boundary biasInverse Gaussian kernelReciprocal inverse Gaussian kernelGamma kernelVariable kernelDensity estimation Acknowledgement We thank M. Akritas and the referee for very constructive criticism. We are grateful to I. Gijbels and T. Bouezmarni for many stimulating discussions. We have received fruitful comments from S. X. Chen, A. Chesher, S. Galluccio, M. Hagmann, O. Renault, K. Thompson, and participants at BNP Paribas seminar and the NP conference. The author gratefully acknowledges financial support from the Belgian Program on Interuniversity Poles of Attraction (PAI nb. P4/01) as well as from the Swiss National Science Foundation through the National Center of Competence: Financial Valuation and Risk Management. Part of this research was done when he was visiting THEMA and IRES. Downloadable at http://www.hec.unige.ch/professeurs/SCAILLET/ Olivier/pages/web/Home/Page/of/Olivier/Scaillet.htm.