  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 Pirinen et al. (2013), 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 $\sqrt{n}$-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.
Some Monte-Carlo experiments are also conducted in order to show the finite
sample performances of our estimator.
