The uniform convergence of the nadaraya‐watson regression function estimate

If ( X 1 , Y 1 ), …, ( X n ,Y n ) is a sequence of independent identically distributed R d × R ‐valued random vectors then Nadaraya (1964) and Watson (1964) proposed to estimate the regression function m(x) = ϵ {Y 1 |X 1 = x{ by\documentclass{article}\pagestyle{empty}\begin{document}$$ m_n \left( x \right) = \sum\limits_{i = 1}^n {Y_i K\left( {\left( {x - X_i } \right)/h_n } \right)} /\sum\limits_{i = 1}^n {K\left( {\left( {x - X_i } \right)/h_n } \right)}, $$\end{document}where K is a known density and { h n } is a sequence of positive numbers satisfying certain properties. In this paper a variety of conditions are given for the strong convergence to 0 of ess X sup| m n ( X )‐ m ( X )| (here X is independent of the data and distributed as X 1 ). The theorems are valid for all distributions of X 1 and for all sequences { h n } satisfying h n → 0 and nh   d n /log n→0.