Inducement of population sparsity

The pioneering work on parameter orthogonalization by Cox and Reid is presented as an inducement of abstract population-level sparsity. This is taken as a unifying theme for this article, in which sparsity-inducing parameterizations or data transformations are sought. Three recent examples are framed in this light: sparse parameterizations of covariance models, the construction of factorizable transformations for the elimination of nuisance parameters, and inference in high-dimensional regression. Strategies for the problem of exact or approximate sparsity inducement appear to be context-specific and may entail, for instance, solving one or more partial differential equations or specifying a parameterized path through transformation or parameterization space. Open problems are emphasized.