Minimizers of U-processes and their domains of attraction
Dietmar Ferger
TL;DR
The paper develops a unified asymptotic theory for minimizers of U-processes in M-estimation, establishing necessary and sufficient conditions under which the scaled minimizer $(m_n-m)/a_n$ converges in distribution to a Smirnov-class limit. By linking the limit through $H(x)=\Phi_\sigma(\delta(x))$ and deriving the functional equations $\delta(x)=\sqrt{k}\;\delta(\alpha_k x)$, it classifies all possible limit laws into four Smirnov classes and characterizes their domains of attraction via the local behavior of $V(t)=D^+U(t)$ around the minimizer $m$. The results extend classical square-root asymptotics to a broader set of normalizing sequences, extend to general minimizers, and illustrate with a broad array of statistics examples, including quantiles, $L_p$-estimators, Hodges–Lehmann, Theil–Sen, and $U$-statistics. Overall, the framework unifies and broadens M-estimator asymptotics, clarifying when non-normal or non-square-root limits arise and how to determine the appropriate normalization from the underlying kernel $h$ and distributional structure.
Abstract
In this paper, we study the minimizers of U-processes and their domains of attraction. U-processes arise in various statistical contexts, particularly in M-estimation, where estimators are defined as minimizers of certain objective functions. Our main results establish necessary and sufficient conditions for the distributional convergence of these minimizers, identifying a broad class of normalizing sequences that go beyond the standard square-root asymptotics with normal limits. We show that the limit distribution belongs to exactly one of the four classes introduced by Smirnov. These results do not only extend Smirnov's theory but also generalize existing asymptotic theories for M-estimators, including classical results by Huber and extensions to higher-degree U-statistics. Furthermore, we analyze the domain of attraction for each class, providing alternative characterizations that determine which types of statistical estimators fall into a given asymptotic regime.