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OPTIMAL CONVERGENCE PROPERTIES OF KERNEL DENSITY

ESTIMATORS WITHOUT DIFFERENTIABILITY CONDITIONS

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R. J. KARUNAMUNI AND K. L. MEHRA

*Department of Statistics and Applied Probability, University of Alberta,*

Edmonton, Alberta, Canada T6G 2G1
(Received June 14, 1989; revised February 15, 1990)

**Abstract.**
Let *X*_{1}, *X*_{2},...., *X*_{n} be independent
observations from an (unknown) absolutely continuous univariate
distribution with density *f* and let
^{^}*f*(*x*) = (*nh*)^{-1} \sum ^{n}_{i=1} *K*[(*x*-*X*_{i})/*h*] be a kernel estimator of
*f*(*x*) at the point *x*, - \infty < *x* < \infty, with *h* = *h*_{n} (*h*_{n} \to 0
and *nh*_{n} \to \infty, as *n* \to \infty) the bandwidth and *K* a kernel
function of order *r*. ``Optimal'' rates of convergence to zero for the
bias and mean square error of such estimators have been studied and
established by several authors under varying conditions on *K* and
*f*. These conditions, however, have invariably included the
assumption of existence of the *r*-th order derivative for *f* at the
point *x*. It is shown in this paper that these rates of convergence
remain valid without any differentiability assumptions on *f* at *x*.
Instead some simple regularity conditions are imposed on the density
*f* at the point of interest. Our methods are based on certain results
in the theory of semi-groups of linear operators and the notions and
relations of calculus of ``finite differences''.

*Key words and phrases*:
Kernel density estimation, bias,
mean square error, finite differences, semi-groups, linear
operators.

**Source**
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