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To isolate the bias-variance tradeoff of the smoothing step, fix the three neighbor locations and write $Y_{(k)}=m_k+{ϵ}_{(k)}$, with independent zero-mean errors of common variance ${\sigma }^2$. At a target $x$, the local estimate isThe bias and variance of a three-neighbour weighted smoother areConsequently the variance is minimized by equal weights, $w=1/3$, when it equals ${\sigma }^2/3$. It decreases as $w$ increases from zero to $1/3$, and increases thereafter toward ${\sigma }^2$ as $w\to 1$. In particular, variance is not increasing over the entire interval $0<w<1$.

Giving more weight to the closest location reduces the influence of more distant responses and often reduces smoothing bias. If the closest location is the target itself, $m_1=m(x)$, put $\Delta =(m_2+m_3)/2-m(x)$. Then the squared bias is $(1-w{)}^2{\Delta }^2$, andThus a locally flat signal favors equal averaging, while stronger local smoothing bias favors greater nearest-point weight. At a general target the bias is affine in $w$, and its absolute value need not decrease: farther neighbors can cancel one another's approximation errors. Even without that cancellation, a smoothness-based worst-case bias bound decreases with $w$ because the first neighbor is closest. The exact tradeoff is therefore lower variance near balanced weights versus potentially lower bias near the closest-point fit. These formulas describe the elementary smoother; fitted partial residuals, centering and repeated backfitting can introduce dependence, so the variance of the final additive fit requires its full fitting operator rather than this single-step calculation.