On the uniform complete convergence of estimates for multivariate density functions and regression curves

Abstract

Let (X₁,Y₁),...(X_n,Y_n) be a random sample from the (k+1)-dimensional multivariate density functionf^*(x,y). Estimates of thek-dimensional density functionf(x)=∫f^*(x,y)dy of the form

$$\hat f_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n W \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$

are considered whereW(x) is a bounded, nonnegative weight function andb₁(n),...,b_k(n) and bandwidth sequences depending on the sample size and tending to 0 asn→∞. For the regression function

$$m(x) = E(Y|X = x) = \frac{{h(x)}}{{f(x)}}$$

whereh(x)=∫y(f)^*(x, y)dy , estimates of the form

$$\hat h_n (x) = \frac{1}{{nb_1 (n) \cdots b_k (n)}}\sum\limits_{i = 1}^n {Y_i W} \left( {\frac{{x_1 - X_{i1} }}{{b_1 (n)}}, \cdots ,\frac{{x_k - X_{ik} }}{{b_k (n)}}} \right)$$

are considered. In particular, unform consistency of the estimates is obtained by showing that$||\hat f_n (x) - f(x)||_\infty$ and$||\hat m_n (x) - m(x)||_\infty$ converge completely to zero for a large class of “good” weight functions and under mild conditions on the bandwidth sequencesb_k(n)'s.