  Conditional distribution functions are important statistical objects for the
analysis of a wide class of problems in econometrics and statistics. We propose
flexible Gaussian representations for conditional distribution functions and
give a concave likelihood formulation for their global estimation. We obtain
solutions that satisfy the monotonicity property of conditional distribution
functions, including under general misspecification and in finite samples. A
Lasso-type penalized version of the corresponding maximum likelihood estimator
is given that expands the scope of our estimation analysis to models with
sparsity. Inference and estimation results for conditional distribution,
quantile and density functions implied by our representations are provided and
illustrated with an empirical example and simulations.
