On the uniform complete convergence of estimates for multivariate density functions and regression curves |
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Authors: | K. F. Cheng R. L. Taylor |
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Affiliation: | 1. The Florida State University, Florida, USA 2. University of South Carolina, South Carolina, USA
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Abstract: | Let (X 1,Y 1),...(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 $$hat f_n (x) = frac{1}{{nb_1 (n) cdots b_k (n)}}sumlimits_{i = 1}^n W left( {frac{{x_1 - X_{i1} }}{{b_1 (n)}}, cdots ,frac{{x_k - X_{ik} }}{{b_k (n)}}} right)$$ are considered whereW(x) is a bounded, nonnegative weight function andb 1 (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 $$hat h_n (x) = frac{1}{{nb_1 (n) cdots b_k (n)}}sumlimits_{i = 1}^n {Y_i W} left( {frac{{x_1 - X_{i1} }}{{b_1 (n)}}, cdots ,frac{{x_k - X_{ik} }}{{b_k (n)}}} right)$$ 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. |
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