Linear Subspace Spanned by Principal Points of a Mixture of Spherically Symmetric Distributions

For each positive integer k, a set of k-principal points of a distribution is the set of k points that optimally represent the distribution in terms of mean squared distance. However, explicit form of k-principal points is often difficult to obtain. Hence a theorem established by Tarpey et al. (1995) has been influential in the literature, which states that when the distribution is elliptically symmetric, any set of k-principal points is in the linear subspace spanned by some principal eigenvectors of the covariance matrix. This theorem is called a “principal subspace theorem”. Recently, Yamamoto and Shinozaki (2000b) derived a principal subspace theorem for 2-principal points of a location mixture of spherically symmetric distributions. In their article, the ratio of mixture was set to be equal. This article derives a further result by considering a location mixture with unequal mixture ratio.