ガウス＝マルコフの定理
が最小になる ( β 0 , β 1 , ⋯ , β p ) {\displaystyle (\beta _{0},\beta _{1},\cdots ,\beta _{p})} を最小二乗推定量と呼ぶ。ここで
Y = [ Y 1 Y 2 ⋮ Y n ] , X = [ 1 x 11 x 12 ... x 1 p 1 x 21 x 22 ... x 2 p ⋮ ⋮ ⋮ ⋮ 1 x n 1 x n 2 ... x n p ] , β = [ β 0 β 1 ⋮ β p ] , ε = [ ε 1 ε 2 ⋮ ε n ] {\displaystyle \mathbf {Y} ={\begin{bmatrix}Y_{1}\\Y_{2}\\\vdots \\Y_{n}\end{bmatrix}},\ \mathbf {X} ={\begin{bmatrix}1&x_{11}&x_{12}&\dots &x_{1p}\\1&x_{21}&x_{22}&\dots &x_{2p}\\\vdots &\vdots &\vdots &&\vdots \\1&x_{n1}&x_{n2}&\dots &x_{np}\end{bmatrix}},\ {\boldsymbol {\beta }}={\begin{bmatrix}\beta _{0}\\\beta _{1}\\\vdots \\\beta _{p}\end{bmatrix}},\ {\boldsymbol {\varepsilon }}={\begin{bmatrix}\varepsilon _{1}\\\varepsilon _{2}\\\vdots \\\varepsilon _{n}\end{bmatrix}}}