Motivated by applications in genetic fields, we propose to estimate the heritability in high dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know neither their proportion nor their positions. The estimator that we consider is strongly inspired by the one proposed by [16], and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is ? n-consistent when both the number of observations n and the number of random effects N tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. SomeMonte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

In this paper, we present a maximum likelihood method for estimating the parameters of a univariate Hawkes process with self-excitation or inhibition. Our work generalizes techniques and results that were restricted to the self-exciting scenario. The proposed estimator is implemented for the classical exponential kernel and we show that, in the inhibition context, our procedure provides more accurate estimations than current alternative approaches.

Summary Motivated by applications in neuroanatomy, we propose a novel methodology to estimate heritability, which corresponds to the proportion of phenotypic variance that can be explained by genetic factors. Since the phenotypic variations may be due to only a small fraction of the available genetic information, we propose an estimator of heritability that can be used in sparse linear mixed models. Since the real genetic architecture is in general unknown in practice, our method enables the user to determine whether the genetic effects are very sparse: in that case, we propose a variable selection approach to recover the support of these genetic effects before estimating heritability. Otherwise, we use a classical maximum likelihood approach. We apply our method, implemented in the R package EstHer that is available on the Comprehensive R Archive Network, on neuroanatomical data from the project IMAGEN.