Minimum variance quadratic unbiased estimators as a tool to identify compound normal distributions

We derive the minimum variance quadratic unbiased estimator (MIVQUE) of the variance of the components of a random vector having a compound normal distribution (CND). We show that the MIVQUE converges in probability to a random variable whose distribution is essentially the mixing distribution characterising the CND. This fact is very important, because the MIVQUE allows us to make out the signature of a particular CND, and notably allows us to check if an hypothesis of normality for multivariate observations y1,…,yM is plausible.