Cook's Distance

Definition

Single Observation

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $C_i=\frac{1}{p}(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_{(i)}{)}^{⊺}\mathrm{Cov}(\hat{\mathbf{\beta }}{)}^{-1}(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_{(i)})=\frac{1}{p{\sigma }^2}\frac{e_i^2{h}_{ii}}{(1-h_{ii}{)}^2}$ where $p-1$ is the number of explanatory variables, and $h_{ii}$ is the Leverage (Statistics)

$C_i$ is the Cook’s distance. The more influential the data point, the larger the Cook’s distance

Set of Observations

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $C_K=\frac{1}{p}(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_{(K)}{)}^{⊺}\mathrm{Cov}(\hat{\mathbf{\beta }}{)}^{-1}(\hat{\mathbf{\beta }}-{\hat{\mathbf{\beta }}}_{(K)})=\frac{1}{p{\sigma }^2}{\mathbf{e}}_K^{⊺}(I-{\mathbf{H}}_K{)}^{-1}{\mathbf{H}}_K(I-{\mathbf{H}}_K{)}^{-1}{\mathbf{e}}_K$ where $p-1$ is the number of explanatory variables, and $\mathbf{H}$ is the matrix of leverages