Publication Date

7-9-1973

Abstract

In this paper, three classes of spline functions are applied to the problem of estimating probability distribution and density functions. In each case, it is shown that the spline function approximating the sample cumulative distribution function converges almost surely to the unknown distribution, F, given that F has at least two continuous derivatives. If F has three or more continuous derivatives, the derivative of the spline converges to the density function. For polynomial splines of sufficiently high degree if F has k continuous derivative, the rate of convergence for the j-th derivative is shown to be 0(hk-j-1/2) where h is the maximum spacing of the approximation points.

Of a particular interest is the fact that one of the types of splines, called splines in tension can be constructed in such a way that it has a non-negative derivative and thus is a distribution function itself.

A discussion of the algorithm and several examples of different applications conclude the work.

Degree Name

Mathematics

Level of Degree

Doctoral

Department Name

Mathematics & Statistics

First Committee Member (Chair)

Herbert Thaddeus Davis III

Second Committee Member

Lambert Herman Koopmans

Third Committee Member

Cleve Barry Moler

Language

English

Document Type

Dissertation

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