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

  • K. F. Cheng
  • R. L. Taylor
Article

Abstract

Let (X1,Y1),...(Xn,Yn) 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),...,bk(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 sequencesbk(n)'s.

Key words and phrases

Weight function bandwidth sequences regression function estimates and complete convergence 

AMS Classification Number

Primary 62E15 and 62E40 Secondary 60F15 

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References

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Copyright information

© The Institute of Statistical Mathematics, Tokyo 1980

Authors and Affiliations

  • K. F. Cheng
    • 1
    • 2
  • R. L. Taylor
    • 1
    • 2
  1. 1.The Florida State UniversityUSA
  2. 2.University of South CarolinaUSA

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