  In this paper we propose a general class of covariate-adjusted
response-adaptive (CARA) designs based on a new functional urn model. We prove
strong consistency concerning the functional urn proportion and the proportion
of subjects assigned to the treatment groups, in the whole study and for each
covariate profile, allowing the distribution of the responses conditioned on
covariates to be estimated nonparametrically. In addition, we establish joint
central limit theorems for the above quantities and the sufficient statistics
of features of interest, which allow to construct procedures to make inference
on the conditional response distributions. These results are then applied to
typical situations concerning Gaussian and binary responses.
