Sushma Kumari

The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to  give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.

Wishart matrices are one of the fundamental matrix models in multivariate statistics. The classical Wishart ensemble has been generalized to [Formula: see text]-Laguerre ensemble for the values of [Formula: see text]. Many properties such as eigenvalue densities, moments and inverse moments of [Formula: see text]-Laguerre matrices are important in various fields of mathematics and physics. The moments and inverse moments of Wishart matrices have been studied rigorously and the explicit formulas were given in [P. Graczyk, G. Letac and H. Massam, Ann. Statist. 31 (2003) 287–309; S. Matsumoto, General moments of the inverse real Wishart distribution and orthogonal Weingarten functions, J. Theoret. Probab. 25 (2012) 798–822]. We give a necessary and sufficient condition for the existence of finite inverse moments of the [Formula: see text]-Laguerre matrix. Moreover, we extend our result for inverse compound Wishart matrices for the values of [Formula: see text] and [Formula: see text].