Abstract
Let X 1 ,...,X n be a random sample drawn from distribution function F(x) with density function f(x) and suppose we want to estimate X(x). It is already shown that kernel estimator of F(x) is better than usual empirical distribution function in the sense of mean integrated squared error. In this paper we derive integrated squared error of kernel estimator and compare the error with that of the empirical distribution function. It is shown that the superiority of kernel estimators is not necessarily true in the sense of integrated squared error.
Similar content being viewed by others
References
Azzalini, A. (1981). A note on the estimation of a distribution function and quantiles by a kernel method, Biometrika, 68, 326–328.
Falk, M. (1983). Relative efficiency and deficiency of kernel type estimators of smooth distribution functions. Statist. Neerlandica, 37, 73–83.
Hall, P. (1984). Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal., 14, 1–16.
Hill, P. D. (1985). Kernel estimation of a distribution function, Comm. Statist. Theory Methods, 14, 605–620.
Jones, M. C. (1990). The performance of kernel density functions in kernel distribution function estimation, Statist. Probab. Lett., 9, 129–132.
Mack, Y. P. (1984). Remarks on some smoothed empirical distribution functions and processes, Bull. Inform. Cybernet., 21, 29–35.
Mammitzsch, V. (1984). On the asymptotically optimal solution within a certain class of kernel type estimatiors, Statist. Decisions, 2, 247–255.
Nadaraya, E. A. (1964). Some new estimates for distribution function, Theory Probab. Appl., 9, 497–500.
Parzen, E. (1962). On the estimation of a probability density and mode, Ann. Math. Statist., 33, 1065–1076.
Reiss, R. D. (1981). Nonparametric estimation of smooth distribution functions, Scand. J. Statist., 8, 116–119.
Rosenblatt, m. (1956). Remarks on some non-parametric estimates of a density function, Ann. Math. Statist., 27, 832–837.
Swanepoel, J. W. H. (1988). Mean integrated squared error properties and optimal kernels when estimating a distribution function, Comm. Statist. Theory Methods, 17, 3785–3799.
Watson, G. S. and Leadbetter, M. R. (1964). Hazard analysis II, Sankhyā Ser. A, 26, 101–116.
Winter, B. B. (1973). Strong uniform consistency of integrals of density estimators, Canad. J. Statist., 1, 247–253.
Winter, B. B. (1979). Convergence rate of perturbed empirical distribution functions, J. Appl. Probab., 16, 163–173.
Yamato, H. (1973). Uniform convergence of an estimator of a distribution function, Bull. Math. Statist., 15, 69–78.
Author information
Authors and Affiliations
About this article
Cite this article
Shirahata, S., Chu, IS. Integrated squared error of kernel-type estimator of distribution function. Ann Inst Stat Math 44, 579–591 (1992). https://doi.org/10.1007/BF00050707
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00050707