Parametrizations, fixed and random effects

Abstract

We consider the problem of estimating the random element $s$ of a finite-dimensional vector space $S$ from the continuous data corrupted by noise with unknown variance ${\sigma }_w^2$. It is assumed that the mean $E\left(s\right)$ (the fixed effect) of $s$ belongs to a known vector subspace $F$ of $S$, and that the likelihood of the centered component $s-E\left(s\right)$ (the random effect) belongs to an unknown supplementary space $E$ of $F$ relative to $S$. Furthermore, the likelihood is assumed to be proportional to $exp\{-q\left(s\right)/2{\sigma }_s^2\}$, where ${\sigma }_s^2$ is some unknown positive parameter. We introduce the notion of bases separating the fixed and random effects and define comparison criteria between two separating bases using the partition functions and the maximum likelihood method. We illustrate our results for climate change detection using the set $S$ of cubic splines. We show the influence of the choice of separating basis on the estimation of the linear tendency of the temperature and the signal-to-noise ratio ${\sigma }_w^2/{\sigma }_s^2$.