Exact mean integrated squared error and bandwidth selection for kernel distribution function estimators

References

• Abdous, B. 1993. Note on the minimum mean integrated squared error of kernel estimates of a distribution function and its derivatives. Communications in Statistics—Theory and Methods 22 (2):603–9. doi:10.1080/03610929308831040.
   Web of Science ®Google Scholar
• Aldershof, B., J. S. Marron, B. U. Park, and M. P. Wand. 1995. Facts about the Gaussian probability density function. Applicable Analysis 59 (1–4):289–306. doi:10.1080/00036819508840406.
   Google Scholar
• Altman, N., and C. Léger. 1995. Bandwidth selection for kernel distribution function estimation. Journal of Statistical Planning and Inference 46 (2):195–214. doi:10.1016/0378-3758(94)00102-2.
   Web of Science ®Google Scholar
• Azzalini, A. 1981. A note on the estimation of a distribution function and quantiles by a kernel method. Biometrika 68 (1):326–8. doi:10.1093/biomet/68.1.326.
   Web of Science ®Google Scholar
• Bartlett, M. S. 1963. Statistical estimation of density functions. Sankhyā 25 (3):245–54. URL: www.jstor.org/stable/25049271.
   Google Scholar
• Bowman, A., P. Hall, and T. Prvan. 1998. Bandwidth selection for the smoothing of distribution functions. Biometrika 85 (4):799–808. doi:10.1093/biomet/85.4.799.
   Web of Science ®Google Scholar
• Chacón, J. E., P. Monfort, and C. Tenreiro. 2014. Fourier methods for smooth distribution function estimation. Statistics and Probability Letters 84:223–30. doi:10.1016/j.spl.2013.10.010.
   Web of Science ®Google Scholar
• Chernozhukov, V., I. Fernández-Val, and A. Galichon. 2009. Improving point and interval estimators of monotone functions by rearrangement. Biometrika 96 (3):559–75. doi:10.1093/biomet/asp030.
   Web of Science ®Google Scholar
• Falk, M. 1983. Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Statistica Neerlandica 37 (2):73–83. doi:10.1111/j.1467-9574.1983.tb00802.x.
   Google Scholar
• Falk, M. 1984. Relative deficiency of kernel type estimators of quantiles. The Annals of Statistics 12 (1):261–8. doi:10.1214/aos/1176346405.
   Web of Science ®Google Scholar
• Fraley, C., and A. E. Raftery. 2002. Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association 97 (458):611–31. doi:10.1198/016214502760047131.
   Web of Science ®Google Scholar
• Fryer, M. J. 1976. Some errors associated with the non-parametric estimation of density functions. IMA Journal of Applied Mathematics 18 (3):371–80. doi:10.1093/imamat/18.3.371.
   Google Scholar
• Glad, I. K., N. L. Hjort, and N. G. Ushakov. 2003. Correction of density estimators that are not densities. Scandinavian Journal of Statistics 30 (2):415–27. doi:10.1111/1467-9469.00339.
   Web of Science ®Google Scholar
• Glad, I. K., N. L. Hjort, and N. G. Ushakov. 2007. Density estimation using the sinc kernel. Preprint statistics 2/2007. Trondheim, Norway: Norwegian University of Science and Technology. URL: www.math.ntnu.no/preprint/statistics/2007/S2-2007.pdf.
   Google Scholar
• Granovsky, B. L., and H.-G. Müller. 1991. Optimizing kernel methods: A unifying variational principle. International Statistical Review 59 (3):373–88. doi:10.2307/1403693.
   Web of Science ®Google Scholar
• Hansen, B. E. 2005. Exact mean integrated squared error of higher order kernel estimators. Econometric Theory 21 (6):1031–57. doi:10.1017/S0266466605050528.
   Web of Science ®Google Scholar
• Jones, M. C. 1990. The performance of kernel density functions in kernel distribution function estimation. Statistics & Probability Letters 9 (2):129–32. doi:10.1016/0167-7152(92)90006-q.
   Web of Science ®Google Scholar
• Mammitzsch, V. 1984. On the asymptotically optimal solution within a certain class of kernel type estimators. Statistics & Decisions 2 (3–4):247–55. doi:10.1524/strm.1984.2.34.247.
   Google Scholar
• Marron, J. S., and D. Nolan. 1988. Canonical kernels for density estimation. Statistics & Probability Letters 7 (3):195–9. doi:10.1016/0167-7152(88)90050-8.
   Web of Science ®Google Scholar
• Marron, J. S., and M. P. Wand. 1992. Exact mean integrated squared error. The Annals of Statistics 20 (2):712–36. doi:10.1214/aos/1176348653.
   Web of Science ®Google Scholar
• McLachlan, G., and D. Peel. 2000. Finite mixture models. New York: John Wiley & Sons.
   Google Scholar
• Nadaraya, E. A. 1964. Some new estimates for distribution functions. Theory of Probability & Its Applications 9 (3):497–500. doi:10.1137/1109069.
   Google Scholar
• Olver, F. W. J., Lozier, D. W., Boisvert, R. F., and Clark, C. W., eds. 2010. NIST handbook of mathematical functions. Cambridge, UK: Cambridge University Press.
   Google Scholar
• Polansky, A. M., and E. R. Baker. 2000. Multistage plug-in bandwidth selection for kernel distribution function estimates. Journal of Statistical Computation and Simulation 65 (1–4):63–80. doi:10.1080/00949650008811990.
   Web of Science ®Google Scholar
• Rao, B. L. S. P. 1983. Nonparametric functional estimation. Cambridge, MA: Academic Press.
   Google Scholar
• Reiss, R. 1981. Nonparametric estimation of smooth distribution functions. Scandinavian Journal of Statistics 8 (2):116–9. www.jstor.org/stable/4615820.
   Web of Science ®Google Scholar
• Roeder, K., and L. Wasserman. 1997. Practical Bayesian density estimation using mixtures of normals. Journal of the American Statistical Association 92 (439):894–902. doi:10.1080/01621459.1997.10474044.
   Web of Science ®Google Scholar
• Silverman, B. W. 1986. Density estimation. Boca Raton, FL: Chapman and Hall.
   Google Scholar
• Swanepoel, J. W. H. 1988. Mean integrated squared error properties and optimal kernels when estimating a distribution function. Communications in Statistics—Theory and Methods 17 (11):3785–99. doi:10.1080/03610928808829835.
   Web of Science ®Google Scholar
• Tenreiro, C. 2006. Asymptotic behaviour of multistage plug-in bandwidth selections for kernel distribution function estimators. Journal of Nonparametric Statistics 18 (1):101–16. doi:10.1080/10485250600578334.
   Web of Science ®Google Scholar
• Wand, M. P., and W. R. Schucany. 1990. Gaussian-based kernels. Canadian Journal of Statistics 18 (3):197–204. doi:10.2307/3315450.
   Web of Science ®Google Scholar
• Watson, G. S., and M. R. Leadbetter. 1964. Hazard analysis II. Sankhyā 26 (1):101–16. URL: www.jstor.org/stable/25049316.
   Google Scholar
• Yamato, H. 1973. Uniform convergence of an estimator of a distribution function. Bulletin of mathematical statistics. 15 (3–4):69–78. URL: ci.nii.ac.jp/Naid/120001036895.
   Google Scholar