On Bayesian Inference of Covariance Matrix
SAS Institute, Pune.
In this paper we presented that sampling distribution of covariance matrix estimator belongs to the broader class of generalized multivariate Gamma distribution. We discussed different properties multivariate Gamma distribution and showed that posterior mode (or MAP estimator) of the covariance matrix is Bayes estimator by minimizing the posterior expected loss under Kullback-Leibler type loss function. We discussed the conditions where the sampling distribution is degenerate and it turns out that the scenario is very common in financial risk management, especially for very large 'pension fund' or 'mutual fund'. In such cases standard likelihood analysis cannot be done. However we presented simple Bayesian solution to fix the problem using a paralleled Monte Carlo algorithm. We presented an empirical study with data during Lehman crisis of 2008.