Leverage (Statistics)

Consider a Multiple Linear Regression model $\mathbf{y}=\mathbf{X}\mathbf{\beta }+\mathbf{ϵ}$ $h_{ii}={\mathbf{x}}_i^{⊺}({\mathbf{X}}^{⊺}\mathbf{X}{)}^{-1}{\mathbf{x}}_i$

$h_{ii}$ is the $i$-th leverage, the distance between ${\mathbf{x}}_i$ and $\bar{\mathbf{x}}$

A measure of how far away the independent variable value of an observation is from those of the other observations It is used to detect outlier

Facts

$\sum _{i=1}^n{h}_{ii}=p$, where $p-1$ is the number of explanatory variables $\frac{1}{n}\le h_{ii}\le 1$ $h_{ij}^2\le h_{ii}{h}_{jj}$