Consistent and Efficient Density Estimation

Abstract

In density estimation, suggestions to use an FFT to efficiently compute densities have been put forward. At first, this would seem quite sensible however, unlike problems where one samples at regularly spaced intervals, such is not the case for a random sample. Instead one would have to use the random sample to reallocate the data onto a grid followed by applying the FFT. This is the procedure that Silverman (1986) proposes for density estimation on the unit interval and borrowing that idea, is the procedure proposed by Healy and Kim (1996) for the case of the sphere. There is however a potential setback that neither authors address. Indeed, imposing an FFT invokes an aliasing effect and therefore one has to consider how big this is likely to be. It is not so obvious that the effect of aliasing is completely benign with respect to the L ²-rate of convergence achievable with the ordinary implementation scheme and therefore it is the purpose of this paper to quantify this aliasing effect. In particular, sufficient conditions that guarantee the preservation of the L ²-rate of convergence with an FFT implementation are established. The corresponding analysis for the sphere has been given in Hendriks(2003).

Financial assistance for carrying out this research was supported in part by grants from NATO (CRG951309) and NSERC (OGP46204)