Nonparametric density estimation by spline projection kernels

Su-Yun Chen, Purdue University

Abstract

Nonparametric density estimation is considered. Suppose we have a set of observed data assumed to be from an unknown distribution. Density estimation is the construction of an estimate for the density function from the observed data. There is a vast of literature in density estimation, a summary of some methods is in Chapter 1. However, there are not many spline methods available. Boneva, Kendall and Stefanov (1971) first proposed the histospline for density estimation, which is basically the derivative of the cubic spline interpolant to the sample c.d.f. $F\sb{n}$($x$) at some points. Lii and Rosenblat (1975) studied the asymptotic behavior concerning the bias and variance for the estimator. Wahba (1975) modified the estimator on the boundaries to get the optimal square error convergence rate. Here we introduce the spline projection kernel for density estimation. A kernel function ${\rm I\!K}(x,y)$ is derived from the $L\sb2$ projection onto certain spline spaces. The projection can be written as ${\rm(I\!P} f)(x)$ = $\int{\rm I\!K}(x,y)f(y)dy$. An estimate based on a sample $X\sb1, dots ,X\sb{n}$ from $f(x)$ is given by ${1\over n}\sum\sbsp{j=1}{n}{\rm I\!K}(x,X\sb{j}).$ The integrated mean square error (IMSE) is derived to measure the discrepancy of the estimator from $f(x)$. The IMSE is minimized with respect to the displacement of knots. It is shown the optimal ISME is$$0\left(n \sp{-{2d\over 2d+1}}\right),$$where d is the order of the spline space. Examples and comparison with histospline are done.

Degree

Ph.D.

Advisors

Studden, Purdue University.

Subject Area

Statistics

COinS