Abstract
In many statistical applications, where the dimension of a random vector highly exceeds the number of available measurements, the estimation of covariance matrix poses a challenge. The sample covariance matrix has several undesirable properties in this case, specifically low rank and poor accuracy of estimation of its single elements.
This paper provides an overview of methods that are used for covariance matrix estimation in high-dimensional problems. First, we pay attention to computationally simple methods which usually work element-wise, such as shrinkage, tapering, etc.
Further, more complex approaches are presented, which employ parametric models based on additional assumptions about the properties of the random vector, especially normality, covariance stationarity and Markov property. Parametric models are used to describe the decay of eigenvalues or to model the covariance matrix or its inverse.
Parameters of the corresponding models can be estimated by standard statistical techniques.