This thesis deals with the statistical inference of large dimensional data. The random matrix theory allows to take into account this framework, since most asymptotic results apply to large-dimensional random matrices. A large number of these results concerns the population covariance matrix. First, we are interested in estimating the number of factors/spikes in large dimension. To construct our estimator, we use the fact that the eigenvalue behavior of the sample covariance matrix differs depending on whether they correspond to spikes or not. The estimator is based on differences between consecutive ordered eigenvalues. We establish the consistency of the estimator in the case where all the spikes are different, and compare it to two existing methods through simulation experiments. The estimator depends on a threshold which should satisfy some conditions. Furthermore, we extend our result of consistency to the equality case and improve our estimator by using a dimension-adapted threshold. Secondly, we consider the maximum likelihood estimator in a strict factor model with homoscedastic variance. Using a central limit theorem for linear spectral statistics, we correct the estimator of the common variance in high-dimensional setting by evaluating its bias and establishing its limiting law. We present a corrected version of the goodness-of-fit test for a factor model. Finally, we propose a test for the equality of two spikes.
