Eecient Non-parametric Estimation of Probability Density Functions

Accurate and fast estimation of probability density functions is crucial for satisfactory computational performance in many scientiic problems. When the type of density is known a priori, then the problem becomes statistical estimation of parameters from the observed values. In the non-parametric 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 f n , in the kernel method the computation of the values f n (X 1); f n (X 2); : : : ; f n (X n) 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 non-parametric 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 required for the convergence of kernel based methods. We also discuss experiments on diierent distributions and compare the eeciency and the accuracy of our computations with kernel based estimators for various values of n and m.