Summary
By representing the location and scale parameters of an absolutely continuous distribution as functionals of the usually unknown probability density function, it is possible to provide estimates of these parameters in terms of estimates of the unknown functionals.
Using the properties of well-known methods of density estimates, it is shown that the proposed estimates possess nice large sample properties and it is indicated that they are also robust against dependence in the sample. The estimates perform well against other estimates of location and scale parameters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmad, I. A. (1980). Integrated mean square properties of density estimation by orthogonal series methods for dependent variables,Ann. Inst. Statist. Math., A, to appear.
Ahmad, I. A. (1979). Strong consistency of density estimation by orthogonal series methods for dependent variables,Ann. Inst. Statist. Math.,31, A, 279–288.
Ahmad, I. A. (1973). Nonparametric estimation of p.d.f.'s for dependent variables with application, Ph. D. dissertation, Florida State University.
Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H. and Tukey, J. W. (1972).Robust Estimates of Location, Princeton University Press, Princeton, N.J.
Bhattacharyya, G. K. and Roussas, G. G. (1969). Estimation of a certain functional of probability density function,Skand Aktuarietidskr.,52, 203–206.
Cencov, N. N. (1962). Estimation of an unknown density function from observations,Dokl. Akad. Nauk SSSR,147, 45–48 (in Russian).
Higgins, J. J. and Tichenor, D. M. (1977). Window estimates of location and scale with applications to the Cauchy distribution,Applied Mathematics and Computations,3, 113–126.
Kronmal, R. and Tarter, M. (1968). The estimation of probability densities and cumulatives by Fourier series methods,J. Amer. Statist. Ass.,63, 925–952.
Murthy, V. K. (1965). Estimation of a probability density,Ann. Math. Statist.,36, 1027–1031.
Parzen, E. (1962). On the 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.
Royden, H. L. (1968).Real Analysis (Second Edition), McMillan, New York.
Schwartz, S. C. (1967). Estimation of a probability density by an orthogonal series,Ann. Math. Statist.,38, 1262–1265.
Szego, G. (1959).Orthogonal Polynomial, Amer. Math. Soc. Coll. Publ. 32, Providence, R.I.
Tapia, R. A. and Thompson, T. R. (1978).Nonparametric Probability Density Estimation, Johns Hopkins Univ. Press, Baltimore, Maryland.
Author information
Authors and Affiliations
About this article
Cite this article
Ahmad, I.A. Nonparametric estimation of the location and scale parameters based on density estimation. Ann Inst Stat Math 34, 39–53 (1982). https://doi.org/10.1007/BF02481006
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02481006