Efficient Non-parametric Estimation of Probability Density Functions
Ömer Eğeci̇oğlu · 1995
Accurate and fast estimation of probability density functions is crucial for satisfactory computational performance in many scientific problems. When the type of density is known a priori, then the problem becomes statistical estimation of parameters from the observed values. In the nonparametric case, usual estimators make use of kernel functions. If X j ; j = 1; 2; : : : ; n is a sequence of i.i.d. random variables with estimated probability density function fn , in the kernel method the computation of the values fn (X 1 ); fn (X 2 ); : : : ; fn (Xn ) requires O(n 2 ) operations, since each kernel needs to be evaluated at every X j . We propose a sequence of special weight functions for the nonparametric estimation of f which requires almost linear time: if m is a slowly growing function that increases without bound with n, our method requires only O(m 2 n) arithmetic operations. We derive conditions for convergence under a number of metrics, which turn out to be similar to those...