Distribution-Free Distribution Regression

'Distribution regression' refers to the situation where a response Y depends on a covariate P where P is a probability distribution. The model is Y f(P) µ where f is an unknown regression function and µ is a random error. Typically, we do not observe P directly, but rather, we observe a sample from P. In this paper we develop theory and methods for distribution-free versions of distribution regression. This means that we do not make distributional assumptions about the error term µ and covariate P. We prove that when the effective dimension is small enough (as measured by the doubling dimension), then the excess prediction risk converges to zero with a polynomial rate.