Continuity of the argmax distribution in multidimensional maximum score estimation

Establish continuity of the cumulative distribution function F_{hat{s}}(t) = P[hat{s} ≤ t] of hat{s} = argmax_{s ∈ R^d} G(s) for the multidimensional (d > 1) maximum score estimator in the semiparametric binary response model y = 1{w + x'θ_0 ≥ u} with Median(u | w, x) = 0, where the limiting Gaussian process G has mean μ(s) = − s' E[ f_{u|w,x}(0 | −x'θ_0, x) f_{w|x}(−x'θ_0 | x) x x' ] s and covariance kernel C(s, t) = E[ f_{w|x}(−x'θ_0 | x) C_BM(x' s, x' t) ], with C_BM the covariance kernel of a two-sided standard Brownian motion.

Background

The maximum score estimator, introduced by Manski (1975), is defined as any maximizer of the non-smooth score Σ_i (2y_i−1) 1{w_i + x_i'θ ≥ 0}. Under cube-root asymptotics, the estimator’s rescaled deviation converges to the argmax of a Gaussian process whose mean is quadratic and whose covariance kernel is a functional of a two-sided Brownian motion, as established by Abrevaya and Huang (2005).

In the univariate case (d = 1), continuity of the argmax distribution follows from known Chernoff-type results. However, for d > 1 and non-bilinear covariance kernels, the literature had not established continuity of the argmax distribution function. The paper frames this as an open question in the maximum score setting and then provides sufficient conditions to answer it affirmatively, supporting inference methods that rely on continuity (e.g., bootstrap-based procedures).