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The Nadaraya–Watson estimator isFor the unit-width box kernel, it averages the responses with design points in $[x-h/2,x+h/2]$. Define it to be zero if that window contains no observations; the supplied small-denominator bound must refer to a defined estimator.

Write $g=f^X$, $D_n=(nh{)}^{-1}{\sum }_i{1}_{\{|X_i-x|\le h/2\}}$ and $U_n=(nh{)}^{-1}{\sum }_i{Y}_i{1}_{\{|X_i-x|\le h/2\}}$. Set $G_n=U_n-m(x)D_n$. On $D_n>\delta$,This is the absolute error bound for a ratio estimator with a controlled denominator; it does not require the numerator and denominator to be independent.

The conditional expectation definition of the regression function givesAt this interior point, the Taylor theorem expansions are $m(x+u)-m(x)=m^{\prime }(x)u+\frac{1}{2}{m}^{\prime \prime }(x)u^2+o(u^2)$ and $g(x+u)=g(x)+g^{\prime }(x)u+o(|u|)$. Multiplying and using the symmetric interval removes the odd linear term. More precisely,This cancellation underlies the interior absolute-error rate of the Nadaraya-Watson estimator.

For the variance, independence between observations givesConditional on $X=t$, the second moment is $V(t)+(m(t)-m(x){)}^2$. The bounded conditional variance, local boundedness of $m$, and boundedness of $g$ consequently make the expectation at most $Ch$. Thus $\mathrm{Var}(G_n)=O((nh{)}^{-1})$. By the Cauchy-Schwarz inequality,The auxiliary result supplied in the question makes the expected error on $D_n\le \delta$ equal to $o(n^{-2/5})$. Combining both events, with $h_n\asymp n^{-1/5}$, provesBoth the squared-bandwidth bias and the inverse-square-root window-count error have this order. The supplied separate consistency result for the design-density estimator is not needed once the stronger small-denominator expectation bound is used.