Nonstandard estimation for the von Mises Fisher distribution

Abstract

The main focus of this thesis is to provide a theoretical analysis of certain statistical models incorporating constraints of various kinds. The models are motivated with data examples using spherical distributions fitted to asset allocation data related to financial portfolios. We fit models consisting of several von Mises Fisher distributions of di↵erent dimensions to a sample of such financial data. Our analysis of the properties of the corresponding hypothesis test statistics under this model combines Silvey’s approach towards constraints on the parameter space with Cherno↵’s innovations to find the asymptotic distributions of the likelihood ratio (deviance) statistics. Silvapulle and Sen detail situations when some constraints are imposed on either the parameter space or on the underlying distribution, where the information matrix is positive definite. But, in some examples, the expectation of the second derivative need not be a positive definite matrix. In such cases, methodology developed by Maller and others can be applied as it does not require the information matrix to be positive definite. Consequently, we apply this methodology to hypothesis tests for the equality of concentration parameters in the spherical subcomponent model. Properties of the tests are examined by simulations and a real data application is given.