Bias and variance of a three-neighbour weighted smoother ($\mathrm{Var}({\hat{m}}_w)={\sigma }^2[w^2+(1-w{)}^2/2]$)

With fixed neighbor positions, independent equal-variance errors and weights $w,(1-w)/2,(1-w)/2$, the displayed variance is minimized at $w=1/3$, with value ${\sigma }^2/3$. Its bias is $w{m}_1+(1-w)(m_2+m_3)/2-m(x)$. Larger nearest-point weight often reduces smoothing bias, but neither the bias nor the variance is universally monotone over $0<w<1$. If the nearest point is the target, the squared bias is $(1-w{)}^2[(m_2+m_3)/2-m(x){]}^2$.

// Target: probability-and-statistics.bigb