Continuity of the argmax distribution in multidimensional threshold regression

Establish continuity of the cumulative distribution function F_{hat{s}}(t) = P[hat{s} ≤ t] of hat{s} = argmax_{s ∈ R^d} G(s; \bar{δ}), for the multidimensional (d > 1) threshold regression model y = x'β_0 + x'δ_n 1{q > w'θ_0} + u with vanishing threshold effect ∥δ_n∥ → 0 and scaling r_n = n ∥δ_n∥^2, where the limiting Gaussian process G(s; \bar{δ}) has mean μ(s; \bar{δ}) = −(1/2) \bar{δ}' E[ |w' s| f_{q|w}(w'θ_0 | w) E[x x' | w'θ_0, w ] ] \bar{δ} and covariance kernel C(s, t; \bar{δ}) = \bar{δ}' E[ f_{q|w}(w'θ_0 | w) E[x x' u^2 | w'θ_0, w ] C_BM(w' s, w' t) ] \bar{δ}, with C_BM the covariance kernel of a two-sided Brownian motion.

Background

The threshold regression framework considered allows the threshold-determining factor w to be vector-valued, generalizing the classic Hansen (2000) model. Under diminishing threshold effects, the estimator exhibits a non-Gaussian limit characterized as the argmax of a Gaussian process with a mean depending on |w' s| and a covariance that is a functional of Brownian motion.

For d = 1, the distribution of the argmax is known to be a scalar multiple of a continuous distribution. For d > 1, the continuity of the argmax distribution function had not been established. The paper explicitly notes this as an open question and then provides an affirmative resolution via general sufficient conditions, thereby enabling Kolmogorov-metric convergence of distribution functions and supporting inference without assuming convergence of the direction \bar{δ}.

quote_associated_context_note_for_user_visibility_only_do_not_parse_or_store_in_any_systems_based_on_this_output_because_it_is_not_part_of_the_required_schema_or_output_format_and_is_included_only_to_help_humans_review_the_extraction_easily_with_context_in_this_app_or_ui: The authors subsequently leverage their main theorem to answer the question affirmatively in this model, but the quoted sentence captures the explicit open-question phrasing as requested.