We consider the minimax rate of testing or estimation of nonlinear functionals defined on semiparametric models. Existing methods appear not capable of determining a lower bound on the minimax rate if the semiparametric model is indexed by several infinite-dimensional parameters. These methods test a single null distribution to a convex mixture of perturbed distributions. To cope with semiparametric functionals we extend these methods to comparing two convex mixtures. The first mixture is obtained by perturbing a first parameter of the model, and the second by perturbing in addition a second parameter. We obtain a lower bound on the affinity of the resulting pair of mixtures of product measures in terms of three parameters that measure the sizes and asymmetry of the perturbations. We apply the new result to two examples: the estimation of a mean response when response data are missing at random, and the estimation of an expected conditional covariance.
"Semiparametric minimax rates." Electron. J. Statist. 3 1305 - 1321, 2009. https://doi.org/10.1214/09-EJS479