Uniform Consistency of the Highly Adaptive Lasso Estimator of Infinite Dimensional Parameters

الأصل · EN

Consider the case that we observe n independent and identically distributed copies of a random variable with a probability distribution known to be an element of a specified statistical model. We are interested in estimating an infinite dimensional target parameter that minimizes the expectation of a specified loss function. In generallyₑfficientₜMLE we defined an estimator that minimizes the empirical risk over all multivariate real valued cadlag functions with variation norm bounded by some constant M in the parameter space, and selects M with cross-validation. We referred to this estimator as the Highly-Adaptive-Lasso estimator due to the fact that the constrained can be formulated as a bound M on the sum of the coefficients a linear combination of a very large number of basis functions. Specifically, in the case that the target parameter is a conditional mean, then it can be implemented with the standard LASSO regression estimator. In generallyₑfficientₜMLE we proved that the HAL-estimator is consistent w.r.t. the (quadratic) loss-based dissimilarity at a rate faster than n⁻¹/² (i.e., faster than n⁻¹/⁴ w.r.t. a norm), even when the parameter space is completely nonparametric. The only assumption required for this rate is that the true parameter function has a finite variation norm. The loss-based dissimilarity is often equivalent with the square of an L²(P₀)-type norm. In this article, we establish that under some weak continuity condition, the HAL-estimator is also uniformly consistent.