Functional estimation in log-concave location families
Koltchinskii V, Wahl M (2023)
In: High dimensional probability IX—the ethereal volume. Progr. Probab., 80. Cham: Birkhäuser/Springer: 393--440.
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Autor*in
Koltchinskii, Vladimir;
Wahl, MartinUniBi
Einrichtung
Abstract / Bemerkung
Let $\{P_{\theta}:\theta \in {\mathbb R}^d\}$ be a log-concave location
family with $P_{\theta}(dx)=e^{-V(x-\theta)}dx,$ where $V:{\mathbb R}^d\mapsto
{\mathbb R}$ is a known convex function and let $X_1,\dots, X_n$ be i.i.d. r.v.
sampled from distribution $P_{\theta}$ with an unknown location parameter
$\theta.$ The goal is to estimate the value $f(\theta)$ of a smooth functional
$f:{\mathbb R}^d\mapsto {\mathbb R}$ based on observations $X_1,\dots, X_n.$ In
the case when $V$ is sufficiently smooth and $f$ is a functional from a ball in
a H\"older space $C^s,$ we develop estimators of $f(\theta)$ with minimax
optimal error rates measured by the $L_2({\mathbb P}_{\theta})$-distance as
well as by more general Orlicz norm distances. Moreover, we show that if $d\leq
n^{\alpha}$ and $s>\frac{1}{1-\alpha},$ then the resulting estimators are
asymptotically efficient in H\'ajek-LeCam sense with the convergence rate
$\sqrt{n}.$ This generalizes earlier results on estimation of smooth
functionals in Gaussian shift models. The estimators have the form $f_k(\hat
\theta),$ where $\hat \theta$ is the maximum likelihood estimator and $f_k:
{\mathbb R}^d\mapsto {\mathbb R}$ (with $k$ depending on $s$) are functionals
defined in terms of $f$ and designed to provide a higher order bias reduction
in functional estimation problem. The method of bias reduction is based on
iterative parametric bootstrap and it has been successfully used before in the
case of Gaussian models.
Erscheinungsjahr
2023
Buchtitel
High dimensional probability IX—the ethereal volume
Serientitel
Progr. Probab.
Band
80
Seite(n)
393--440
Page URI
https://pub.uni-bielefeld.de/record/2969765
Zitieren
Koltchinskii V, Wahl M. Functional estimation in log-concave location families. In: High dimensional probability IX—the ethereal volume. Progr. Probab. Vol 80. Cham: Birkhäuser/Springer; 2023: 393--440.
Koltchinskii, V., & Wahl, M. (2023). Functional estimation in log-concave location families. High dimensional probability IX—the ethereal volume, Progr. Probab., 80, 393--440. Cham: Birkhäuser/Springer. https://doi.org/10.1007/978-3-031-26979-0_15
Koltchinskii, Vladimir, and Wahl, Martin. 2023. “Functional estimation in log-concave location families”. In High dimensional probability IX—the ethereal volume, 80:393--440. Progr. Probab. Cham: Birkhäuser/Springer.
Koltchinskii, V., and Wahl, M. (2023). “Functional estimation in log-concave location families” in High dimensional probability IX—the ethereal volume Progr. Probab., vol. 80, (Cham: Birkhäuser/Springer), 393--440.
Koltchinskii, V., & Wahl, M., 2023. Functional estimation in log-concave location families. In High dimensional probability IX—the ethereal volume. Progr. Probab. no.80 Cham: Birkhäuser/Springer, pp. 393--440.
V. Koltchinskii and M. Wahl, “Functional estimation in log-concave location families”, High dimensional probability IX—the ethereal volume, Progr. Probab., vol. 80, Cham: Birkhäuser/Springer, 2023, pp.393--440.
Koltchinskii, V., Wahl, M.: Functional estimation in log-concave location families. High dimensional probability IX—the ethereal volume. Progr. Probab. 80, p. 393--440. Birkhäuser/Springer, Cham (2023).
Koltchinskii, Vladimir, and Wahl, Martin. “Functional estimation in log-concave location families”. High dimensional probability IX—the ethereal volume. Cham: Birkhäuser/Springer, 2023.Vol. 80. Progr. Probab. 393--440.