Régression linéaire en petite et en grande dimension

Let $X=(X^{(1)},\cdots, X^{(p)})$ a vector of $p$ explanatory variables and $Y \in \R$ an output. Let $(X_1,Y_1), \cdots , (X_n,Y_n)$, $n$ independent and identically distributed copies of $(X,Y)$. In this thesis, we consider the linear regression in two different settings.\\ \textbf{Explanatory aspect in low dimension ($p < n$):} To study the multiple correlation between the features $(X^{(1)},\cdots, X^{(p)})$ and the output $Y$ it is common to use the coefficient of determination or $R^2$: \begin{equation*} R^2 = \frac{\var[\proj_V(Y)]}{\var[Y]}. \end{equation*} where $V = \Span(\{1,\cdots,X^{(p)}\})$ and $\proj_{V}(Y)$ is the orthogonal projection of $Y$ on $V$. In practice, we can not compute the $R^2$ since it depends on unknown variance quantities: without any other assumptions that moment conditions on $X$ we build a confidence interval for the $R^2$.\\ \textbf{Prediction aspect in high dimension ($p \gg n$)}: We consider an extension of the regular linear regression model called the \textit{quadratic regression model} where we add the order interactions of the explanatory variables: \begin{equation*} Y = \sum_{j=1}^p X^{(j)}\beta_j^\star +\!\! \sum_{1 \leq j \leq k \leq p} \!\!\theta_{\varphi(j,k)}^\star X^{(j)}X^{(k)} + \varepsilon, \end{equation*} where $\varphi: \llbracket 1, p\rrbracket^2 \mapsto \llbracket 1,q\rrbracket $ is a mapping function which assigns to every pair of indices $1 \leq j\leq k \leq p$ ann index $ 1\leq l \leq q = p(p+1)/2$. The \texttt{sprinter} \cite{yu2023reluctant} algorithm is a three-step procedure that performs variable selection and estimation. We study its theoretical properties: in particular, we state fast and slow oracle inequalities for \texttt{sprinter}.