Abstract
In many real problems, such as in physical, biological, socio-economic sciences, medicine and certain natural sciences, one is faced with probabilistic models. To completely specify these models one must know the form of either probability distribution or density function. In practice one does not know the exact forms, rather one has to approximate these functions from given data. A great number of researchers have investigated different aspects of this problem. In this paper we concentrate on two broad classes of density estimators, namely kernel and spline function based estimators. The approach adopted is interval analysis, oriented so that the end computations can be easily and economically performed on modern computers. In order to understand clearly and simply, the ideas involved are unified by using the Hilbert spaces.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bartlett, M.S. (1963). Statistical estimation of density functions. Sankhyā (A), 25, 245–254.
Bochner, S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press.
Boneva, L.I., Kendall, D.G. and Stefanov, I. (1971) Spline transformations: Three new diagnostic aids for the statistical data-analyst (with Discussion). J.R. Statist. Soc. B, 33, 1–70.
Daniels, H.E. (1962). The estimation of spectral densities. J.R. Statist. Soc. B, 24, 185–198.
Greville, T.N.E. ed. (1969). Theory and Applications of Spline Functions. Academic Press, New York.
Liusternik, L. and Sobolev, V. (1961). Elements of Functional Analysis. F. Ungar Publ. Co. New York.
Nadaraya, E.A. (1965). On nonparametric estimates of density functions and regression curves. Teoriya Veroyatnost., 10, 199–203.
Nadaraya, E.A. (1970). Remarks on nonparametric estimates of density functions and regression curves. Teoriya Veroyatnost., 15, 139–142.
Parzen, E. (1962). On estimation of a probability density and mode. Ann. Math. Statist., 33, 1065–1076.
Parzen, E. (1967). Time Series Analysis Papers. Holden Day, San Francisco.
Rosenblatt, M. (1956). Remarks on some nonparametric estimates of density function. Ann. Math. Statist., 27, 832–837.
Schönberg, I.J. ed. (1969). Approximations with Special Emphasis on Spline Functions. Academic Press, New York.
Van Ryzin, J. (1969). On strong consistency of density estimates. Ann. Math. Statist., 40, 1765–1772.
Wegman, E.J. (1969). Nonparametric probability density estimation. Inst. Statist. Memo Series No: 638, University of N. Carolina.
Whittle, P. (1958). On the smoothing of probability density functions. J.R. Statist. Soc. B, 20, 334–343.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1975 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ahmad, R. (1975). A distribution-free interval mathematical analysis of probability density functions. In: Nickel, K. (eds) Interval Mathematics. IMath 1975. Lecture Notes in Computer Science, vol 29. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-07170-9_9
Download citation
DOI: https://doi.org/10.1007/3-540-07170-9_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-07170-9
Online ISBN: 978-3-540-37504-3
eBook Packages: Springer Book Archive