Analytic characterization of the critical-regime limiting distribution

Determine whether the technique of Groeneboom, Lalley, and Temme can be applied to obtain an analytic expression for the limiting distribution \(\mathbb{D}_{\beta,\alpha}\) of the additive monotone least squares estimator when \(\beta=1/3\), for which no closed-form expression is currently available.

Background

When the coordinate-resolution exponent satisfies $\beta=1/3$, the normalized estimator converges to the left derivative at zero of the greatest convex minorant of a two-sided Gaussian random walk with quadratic drift. Unlike the Gaussian and Chernoff regimes, the paper does not provide a closed-form description of this random variable.

The authors explicitly identify determining the distribution analytically, potentially using the technique of Groeneboom, Lalley, and Temme, as an unresolved question.