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We consider conditional distribution and conditional density functionals in the space of generalized functions. The approach follows Phillips (1985, 1991, 1995) who employed generalized functions to overcome non-differentiability in order to develop expansions. We obtain the limit of the kernel estimators for weakly dependent data, even under non-differentiability of the distribution function; the limit Gaussian process is characterized as a stochastic random functional (random generalized function) on the suitable function space. An alternative simple to compute estimator based on the empirical distribution function is proposed for the generalized random functional. For test statistics based on this estimator, limit properties are established. A Monte Carlo experiment demonstrates good finite sample performance of the statistics for testing logit and probit specification in binary choice models.