Interior first-order bias cancellation for local constant regression ($\mathrm{Bias}=o(h)$)

Let $x$ be an interior target for fixed-design nonparametric regression with equally spaced sites, regression function $m$ differentiable at $x$, and independent mean-zero errors of common variance ${\sigma }^2<\infty$. Use a nonnegative symmetric unit-integral compactly supported piecewise smooth regression kernel $K$, with $h\to 0$ and $nh\to \infty$. For the Nadaraya–Watson estimator,The noise variance factor is essential; $R(K)/(nh)$ alone assumes unit variance.

For the bias, expand $m(x_i)=m(x)+m^{\prime }(x)(x_i-x)+o(|x_i-x|)$ uniformly in the shrinking window. Symmetry cancels the limiting first moment and the grid error is $O(1/n)=o(h)$. For the variance, independent errors give ${\sigma }^2{\sum }_i{W}_i(x{)}^2$; the two kernel Riemann sums yield ${\sum }_i{W}_i^2=R(K)/(nh)+o((nh{)}^{-1})$. Hence the squared bias of an estimator is $o(h^2)$. This is a pointwise differentiability argument; it does not supply a uniform remainder over an arbitrary class of functions.