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# 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.

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

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

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

$Y:\left(Omega,mathcal\left\{A\right\}\right)rightarrow\left(Gamma, S\right).$

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

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

$forall iin \left\{1,dots,p\right\}, X_i:\left(Omega,mathcal\left\{A\right\}\right)rightarrow\left(Gamma, S\right).$

Let $f:left\left\{ begin\left\{matrix\right\} Gamma^ptimesTheta&rightarrow&Gamma \left(X_1,dots,X_p;theta\right)&mapsto&f\left(X_1,dots,X_p,theta\right) end\left\{matrix\right\} right.$

We finally define $varepsilon:=Y-f\left(X_1,dots,X_p;theta\right)$.

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