UC San Diego

This dissertation addresses a large class of irregular models in economics and statistics -- settings in which the parameters of interest take the form [phi]([theta]₀), where [phi] is a known directionally differentiable function and [theta]₀ is estimated by [theta]n. Chapter 1 provides a tractable framework for conducting inference, Chapter 2 focuses on optimality of estimation, and Chapter 3 applies the developed theory to construct a test whether a Hilbert space valued parameter belongs to a convex set and to derive the uniform weak convergence of the Grenander distribution function -- i.e. the least concave majorant of the empirical distribution function -- under minimal assumptions