RATES OF STRONG POINTWISE CONVERGENCE OF NEAREST NEIGHBOUR DENSITY ESITIMATES

Let X₁,…,X_n be i.i.d,samples drawn from an one-dimenslonal,population with density f.Define f_n(x)=\left(n a_n(x)\right)^-1 \sum_i=1^n K\left(\fracX-X_ia_n(x)\right). We study the strong convergence rate of f_n（x） to f（x）at a predetermined point x₀. Under some properly chosen conditions,for f（x₀） and g_n（x₀）proposed in 3,we have pointwiseby \beginaligned & \limsup _n \rightarrow \infty\left(\fracn\log n\right)^\fracr2 r+1 C_n^-1\left|f_n\left(x_0\right)-f\left(x_0\right)\right| \rightarrow 0 \quad \text a.s. \\ & \limsup _n \rightarrow \infty\left(\fracn\log n\right)^\fracr2 r+1 O_n^-1\left|f_n\left(x_0\right)-g_n\left(x_0\right)\right| \rightarrow 0 \quad \text a.s. \endaligned where C_n is any sequence tending to ∞,and n approaches ∞.If f（x）is only assumed to be continuous at x₀.Then f_n（x₀）may converges to f（x₀）arbitrarily slowly.