CONVERGENCE RATE OF MULTIVARIATE K-NEAREST NEIGHBOR DENSITY ESTIMATES

Let be the collection of m-times continuously differentiable probability densities fon Rd such that 丨Daf（x1）-Daf（x2）丨≤M‖x1-x2‖β for x1,x2∈Rd,[a]=m,where Dadenotes the differential operator defined by Da=（[a]）/（x1a…xdad）.Under rather weak conditionson K（x）,the necessary and sufficient conditions for sup丨n（x）-f（x）丨=0((（logn/n）λ/（d+3λ）,λ=m+β,f∈ are that ∫xaK（xi）dx=0 for 0【[a]≤m.Finally the convergenco rate at apoint is given.

Let F be the collection of m-times continuously differentiable probability densities f on R(d) such that \D(a)f(x1)-D(a)f(x2)\ less-than-or-equal-to M parallel-to x1 - x2 parallel-to beta for x1, x2 is-a-member-of R(d), [a] = m, where D(a) denotes the differential operator defined by [GRAPHICS] Under rather weak conditions on K(x), the necessary and sufficient conditions for sup [GRAPHICS] lambda = m + beta, f is-a-member-of F are that integral x(a)K (xi)dx = 0 for 0 < [a] less-than-or-equal-to m. Finally the convergence rate at a point is given.