Fast multivariate empirical cumulative distribution function with connection to kernel density estimation

This paper revisits the problem of computing empirical cumulative distribution functions (ECDF) efficiently on large, multivariate datasets. Computing an ECDF at one evaluation point requires $\mathcal{O}(N)$ operations a dataset composed $N$ data points. Therefore, direct ECDFs points quadratic $\mathcal{O}(N^2)$ operations, which is prohibitive for large-scale problems. Two fast and exact methods are proposed compared. The first based summation in lexicographical order, with $\mathcal{O}(N{\log}N)$ complexity to lie regular grid. second divide-and-conquer principle, $\mathcal{O}(N\log(N)^{(d-1){\vee}1})$ coincide input two algorithms described detailed general $d$-dimensional case, numerical experiments validate their speed accuracy. Secondly, establishes connection between kernel density estimation (KDE) large class kernels. paves way regression. Numerical tests Laplacian accuracy algorithms. A broad range estimation, survival function regression problems can benefit from methods.