Annals of the Institute of Statistical Mathematics

, Volume 32, Issue 2, pp 187–199

# 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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Nadaraya, E. A. (1970). Remarks on nonparametric estimates for density function and regression curve,Theory Prob. Appl.,15, 134–137.
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Taylor, R. L. and Cheng, K. F. (1978). On the uniform complete convergence of density function estimates,Ann. Inst. Statist. Math.,30, A, 397–406.
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Watson, G. S. (1964). Smooth regression analysis,Sankhyã, A,26, 359–372.

© 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