Andrews-Pregibon Statistic

Definition

Single Observation

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $A{P}_i=(1-h_{ii})\frac{1-e_i^2}{(1-h_{ii}){\mathbf{e}}^{⊺}\mathbf{e}}=1-h_{ii}^{\ast }$ where $h_{ii}^{\ast }={\mathbf{x}}_i^{\ast ⊺}({\mathbf{X}}^{\ast ⊺}{\mathbf{X}}^{\ast }{)}^{-1}{\mathbf{x}}_i^{\ast }$ is the Leverage (Statistics) of the matrix ${\mathbf{X}}^{\ast }$

$A{P}_i$ is the Andrews-Pregibon statistic. The more influential the data point, the smaller the Andrews-Pregibon statistic

Set of Observations

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $A{P}_K=\det [I-{\mathbf{H}}_K^{\ast }]$ where ${\mathbf{H}}_K^{\ast }={\mathbf{X}}_K^{\ast }({\mathbf{X}}^{\ast ⊺}{\mathbf{X}}^{\ast }{)}^{-1}{\mathbf{X}}_K^{\ast ⊺}$ is the matrix of leverages