= Bias and variance of a three-neighbour weighted smoother
{title2=$\operatorname{Var}(\widehat m_w)=\sigma^2[w^2+(1-w)^2/2]$}
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
Back to article page
