statistical regression

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})rightarrow(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})rightarrow(Gamma, S).

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

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

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