Conference Agenda

We introduce the notion of generalized $\psi$-estimators as unique points of sign change of some appropriate functions.This notion is a generalization of usual $\psi$-estimators (also called $Z$-estimators). We give necessary as well as sufficient conditions for the (unique) existence of generalized $\psi$-estimators. Our results are well-applicable in statistical estimation theory, for example, in case of empirical quantiles, empirical expectiles, some (usual) $\psi$-estimators in robust statistics, and some maximum likelihood estimators as well.

Furthermore, we give axiomatic characterisations of generalized $\psi$-estimators and (usual) $\psi$-estimators, respectively. The key properties of estimators that come into play in the characterisation theorems are the symmetry, the (strong) internality and the asymptotic idempotency. In the proofs, a separation theorem for Abelian subsemigroups plays a crucial role.

The talk is based on our papers [1], [2] and [3].

Mátyás Barczy has been supported by the project TKP2021-NVA-09. Project no. TKP2021-NVA-09 has been implemented with the support provided by the Ministry of Innovation and Technology of Hungary from the National Research, Development and Innovation Fund, financed under the TKP2021-NVA funding scheme.

References

[1] Barczy, M., Páles Zs.: Existence and uniqueness of weighted generalized $\psi$-estimators, ArXiv 2211.06026.

[2] Barczy, M., Páles Zs.: Basic properties of generalized $\psi$-estimators.To appear in Publicationes Mathematicae Debrecen. Available also at Arxiv 2401.16127.

[3] Barczy, M., Páles Zs.: Axiomatic characterisation of generalized $\psi$-estimators, ArXiv 2409.16240.

Let $\theta_n = (\tau_n, \alpha_n) \in \mathbb{R}^{2}$, be an estimator consisting of two parts: $\tau_{n}$ is an infimising point of a process $X_n$ in the Skorokhod space and $\alpha_{n}$ is a real-valued random variable. For this estimator we give conditions under which weak convergence of $(X_n)_{n \in \mathbb{N}}$ and $(\alpha_n)_{n \in \mathbb{N}}$ leads to a convergence result for $(\theta_n)_{n \in \mathbb{N}}$, which may coincide with distributional convergence. Usually, arginf continuous mapping theorems are used for distributional convergence of $(\tau_n)_{n \in \mathbb{N}}$. For this purpose, it is assumed that the processes $X_n$ weakly converge to a process $X$, $(\tau_n)_{n \in \mathbb{N}}$ is stochastically bounded and the set of infimising points of $X$ is almost surely a singleton. However, we consider the case, in which this uniqueness is not satisfied. In fact, in this case we obtain weak convergence of $(\tau_n)_{n \in \mathbb{N}}$ in the topological space $(\mathbb{R}, \mathcal{O})$, where $\mathcal{O}$ is the right- or left-order topology. Further, we have to deal with the fact that Slutsky's theorem is not applicable to connect the convergence of $(\tau_n)_{n \in \mathbb{N}}$ and $(\alpha_n)_{n \in \mathbb{N}}$. The result is generalised for $\theta_n \in \mathbb{R}^{d+l}$ with $d \in \mathbb{N}$ and $l \in \mathbb{N}$. Then, we use this in regression when estimating the best least-squares approximation of an unknown regression function and constructing confidence regions.

In two landmark papers, Akaike introduced the AIC and FPE, demonstrating their significant usefulness for prediction. In subsequent seminal works, Shibata developed a notion of asymptotic efficiency and showed that both AIC and FPE are optimal, setting the stage for decades-long developments and research in this area and beyond. Conceptually, the theory of efficiency is universal in the sense that it (formally) only relies on second-order properties of the underlying process $(X_t)_{t \in \mathbb{Z}}$, but, so far, almost all (efficiency) results require the much stronger assumption of a linear process with independent innovations. In this work, we establish sharp oracle inequalities subject only to a very general notion of weak dependence, establishing a universal property of the AIC and FPE. A direct corollary of our inequalities is asymptotic efficiency of these criteria. Our framework contains many prominent dynamical systems such as random walks on the regular group, functionals of iterated random systems, functionals of (augmented) Garch models of any order, functionals of (Banach space valued) linear processes, possibly infinite memory Markov chains, dynamical systems arising from SDEs, and many more.

We provide close bounds for the Clopper-Pearson binomial confidence bounds, similar to, but necessarily a bit more complicated than, the well-known Lagrange (1776) expressions $\widehat{p}\pm\Phi^{-1}(\beta)\sqrt{\widehat{p}\widehat{q}/n}$.

The goal here is to enable rough but valid mental or back-of-the-envelope calculations for a most elementary and important statistical procedure.

Our result improves a bit on the one of Short (2023), and for example roughly reproduces the well-known upper bound $\frac{3}{n}$ in the case of $\beta=0.95$ and $\widehat{p}=0$. But further simplifications or sharpenings are desirable.