Elliptical multivariate analysis (Q1118935)

We investigate the interpretation and the asymptotic properties of the well-known statistics \({\hat \theta}\) derived in multivariate analysis, when the parent distribution of the observations is not necessarily normal but belongs to the wider class of elliptical distributions (including the normal). Our conclusion is that those statistics keep their geometric interpretation and consistency and that for most of those statistics the asymptotic distribution of \(\sqrt{T}({\hat \theta}- \theta)\) is \(N(0,(1+\kappa)var_ N)\), where \(var_ N\) is the covariance matrix under the normality assumption and where \(\kappa\) is the joint kurtosis parameter. For statistics of the \(\chi^ 2\)-type used in asymptotic test criteria, we find similar expressions. It follows that all multivariate software may be utilized in the elliptical context under the provision that \(\kappa\) is estimated in addition.