Mathematical Formalization Of The Statistical Regression Problem

(\Omega,\mathcal{A}, P)

will denote a Probability Space and

(\Gamma, S)

will be a Measure Space .

\Theta\subseteq\Gamma

is a set of coefficients.

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".

orall i\in \{1,\cdots,p\}, X_i:(\Omega,\mathcal{A})
ightarrow(\Gamma, S)

.

Let

f:\left\{
\begin{matrix}
\Gamma^p	imes\Theta&
ightarrow&\Gamma\
(X_1,\cdots,X_p;	heta)&\mapsto&f(X_1,\cdots,X_p,	heta)
\end{matrix}

ight.

.

We finally define

arepsilon:=Y-f(X_1,\cdots,X_p;	heta)

	CATEGORIES ABOUT MATHEMATICAL FORMALIZATION OF THE STATISTICAL REGRESSION PROBLEM

	regression analysis

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