  One of the major challenges in multivariate analysis is the estimation of
population covariance matrix from sample covariance matrix (SCM). Most recent
covariance matrix estimators use either shrinkage transformations or asymptotic
results from Random Matrix Theory (RMT). Shrinkage techniques help in pulling
extreme correlation values towards certain target values whereas tools from RMT
help in removing noisy eigenvalues of SCM. Both of these techniques use
different approaches to achieve a similar goal which is to remove noisy
correlations and add structure to SCM to overcome the bias-variance trade-off.
In this paper, we first critically evaluate the pros and cons of these two
techniques and then propose an improved estimator which exploits the advantages
of both by taking an optimally weighted convex combination of covariance
matrices estimated by an improved shrinkage transformation and a RMT based
filter. It is a generalized estimator which can adapt to changing sampling
noise conditions in various datasets by performing hyperparameter optimization.
We show the effectiveness of this estimator on the problem of designing a
financial portfolio with minimum risk. We have chosen this problem because the
complex properties of stock market data provide extreme conditions to test the
robustness of a covariance estimator. Using data from four of the world's
largest stock exchanges, we show that our proposed estimator outperforms
existing estimators in minimizing the out-of-sample risk of the portfolio and
hence predicts population statistics more precisely. Since covariance analysis
is a crucial statistical tool, this estimator can be used in a wide range of
machine learning, signal processing and high dimensional pattern recognition
applications.
