Statistical methods for estimating the dimension of multivariate data

Many dimension reduction methods are based on estimating a kernel matrix whose range space is or contains a lower dimensional projection of the data. The rank of the kernel matrix is the dimension of the subspace the data live in. Methods for estimating the rank of the kernel matrix have been proposed only for some special cases. In this dissertation, we develop sequential asymptotically weighted chi-squared and chi-squared tests for estimating the rank of a symmetric kernel matrix that is the asymptotic mean of a random matrix with an asymptotic normal distribution. Our methods can be extended to estimating the rank of general, i.e. not necessarily symmetric, kernel matrices.; A Wald-type test to assess individual or simultaneous contribution of the original variables to the linear combinations comprising the lower dimensional projection is developed. We apply this methodology to assessing variable contribution to principal components.; Estimating the number of factors in factor analysis has been a major problem since factor analysis was introduced. We propose a solution combining ideas from multivariate multiple regression and permutation tests. Only the standard assumptions of factor analysis are needed.; Power simulation studies show the proposed methodologies work well in various situations.