Eecient Online Non-parametric Density Estimation 1 Overview and Motivation

Non-parametric density estimation has broad applications in computational nance especially in cases where high frequency data are available. However, the technique is often intractable , given the run times necessary to evaluate a density. We present a new and eecient algorithm based on multipole techniques. Given the n kernels that estimate the density, current methods take O(n) time to directly sum the kernels to perform a single density query. The cumulative O(n 2) running time for n queries makes it very costly, if not impractical, to compute the density for large n. Our new Multipole-accelerated Online Density Estimation (MODE) algorithm is general in that it can be applied to any kernel (in arbitrary dimensions) that admits a Taylor series expansion. The running time reduces to O(log n) or even constant time, depending on the kernel chosen, and hence, the cumulative running time is reduced to O(n log n) or O(n), respectively. Our results show that the MODE algorithm provides dramatic advantages over the direct approach to density evaluation. For example, we show using a modest computing platform that on-line density updates and queries for one million points and two dimensions take 8 days to compute using the direct approach versus 40 seconds with the MODE approach. The usual approach in nance to estimating a relation between n variables is to make distribu-tional assumptions about the data generating process or to directly impose parametric restrictions on the functional relation. Recently, however, there is considerable interest in an alternative approach: non-parametric density estimation. This approach \lets the data speak for itself." Rather than imposing assumptions, the non-parametric technique allows us to directly approximate the d-dimensional density that describes how these variables interact. Non-parametric density estimation techniques have been used by A t-Sahalia and Lo 1] to estimate the state-price density for options, by Boudoukh, Richardson, Stanton, and Whitelaw 7] and Harvey 15] to design hedging 1