Abstract
Rosenblatt and Parzen proposed a well-known estimator f n for an unknown density function f, and later Schuster suggested a modification % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] n to rectify certain drawbacks of f n . This paper gives the asymptotically optimum bandwidth and kernel for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] n under the standard measure of IMSE when f is discontinuous at one or both endpoints of its support. We also consider an alternative definition of the IMSE under which the optimum bandwidths and kernels for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaybyaeqale% qabaGaaiOxaaqdbaGaamOzaaaaaaa!3851!\[\mathop f\limits^\^ \] n and f n are derived. The latter supplement van Eeden's results.
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References
Apostol, T. M. (1957). Mathematical Analysis, Addison-Wesley, Reading, Massachusetts.
Bickel, P. J. and Rosenblatt, M. (1973). On some global measures of the deviations of density function estimates, Ann. Statist., 1, 1071–1095 (correction: ibid, (1975) 3, 1370).
Cline, D. B. H. and Hart, J. D. (1990). Kernel estimates of densities with discontinuities or discontinuous derivatives, Statistics, 22, 69–84.
Epanechnikov, V. A. (1969). Nonparametric estimates of a multivariate probability density, Theory Probab. Appl., 14, 153–158.
Ghosh, B. K. and Huang, W.-M. (1991). The power and optimal kernel of the Bickel-Rosenblatt test for goodness of fit, Ann. Statist., 19, 999–1009.
Johnson, N. L. and Kotz, S. (1970). Distributions in Statistics: Continuous Univariate Distributions, Houghton-Mifflin, Boston.
Lehmann, E. L. (1983). Theory of Point Estimation, Wiley, New York.
Parzen, E. (1962). On estimation of a probability density function and mode, Ann. Math. Statist., 33, 1065–1076.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function, Ann. Math. Statist., 27, 832–837.
Rosenblatt, M. (1971). Curve estimates, Ann. Math. Statist., 42, 1815–1842.
Schuster, E. F. (1985). Incorporating support constraints into nonparametric estimates of densities, Comm. Statist. Theory Methods, 14, 1123–1136.
Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Chapman, New York.
vanEeden, C. (1985). Mean integrated squared error of kernel estimators when the density and its derivative are not necessarily continuous, Ann. Inst. Statist. Math., 37, 461–472.
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Ghosh, B.K., Huang, WM. Optimum bandwidths and kernels for estimating certain discontinuous densities. Ann Inst Stat Math 44, 563–577 (1992). https://doi.org/10.1007/BF00050706
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DOI: https://doi.org/10.1007/BF00050706