Mathematical formalization of the statistical regression problem

Although a rigorous formalization of the regression problem is not necessary in most cases, the theoretical study of the regression problem requires a precise mathematical context than that given in the Regression analysis article.

$(Omega,mathcal\{A\},\; P)$ will denote a probability space and $(Gamma,\; S)$ will be a measure space. $ThetasubseteqGamma$ is a set of coefficients.

Very often, $Gamma\; =\; mathbb\{R\}^n$ and $S=mathcal\{B\}\_n$ with $ninmathbb\{N\}^*$.

The dependent variable "Y" is a random variable, i.e. a measurable function:

: $Y:(Omega,mathcal\{A\})\; ightarrow(Gamma,\; S).$

This variable will be "explained" using other random variables called "factors".

Let $pinmathbb\{N\}^*$. $p$ is called "number of factors".

: $forall\; iin\; \{1,dots,p\},\; X\_i:(Omega,mathcal\{A\})\; ightarrow(Gamma,\; S).$

Let

We finally define $varepsilon:=Y-f(X\_1,dots,X\_p;\; heta)$.