Leave-one-out residual identity for a linear smoother
= Leave-one-out residual identity for a linear smoother
If a linear smoother has fitted vector $\widehat Y=HY$, its residual after fitting without observation $i$ is
$$
Y_i-\widehat Y_{-i,i}=\frac{Y_i-\widehat Y_i}{1-H_{ii}}.
$$
The identity follows from a <block matrix inverse> or a rank-one inverse update.