Strong noise estimation in cubic splines

Abstract

The data $(y_i,x_i)\in$ $\textbf{R}\times[a,b]$, $i=1,\ldots,n$ satisfy $y_i=s(x_i)+e_i$ where $s$ belongs to the set of cubic splines. The unknown noises $(e_i)$ are such that $var(e_I)=1$ for some $I\in \{1, \ldots, n\}$ and $var(e_i)=\sigma^2$ for $i\neq I$. We suppose that the most important noise is $e_I$, i.e. the ratio $r_I=\frac{1}{\sigma^2}$ is larger than one. If the ratio $r_I$ is large, then we show, for all smoothing parameter, that the penalized least squares estimator of the $B$-spline basis recovers exactly the position $I$ and the sign of the most important noise $e_I$.