DeepAI

# Functional estimation in log-concave location families

Let {P_θ:θ∈ℝ^d} be a log-concave location family with P_θ(dx)=e^-V(x-θ)dx, where V:ℝ^d↦ℝ is a known convex function and let X_1,…, X_n be i.i.d. r.v. sampled from distribution P_θ with an unknown location parameter θ. The goal is to estimate the value f(θ) of a smooth functional f:ℝ^d↦ℝ based on observations X_1,…, X_n. In the case when V is sufficiently smooth and f is a functional from a ball in a Hölder space C^s, we develop estimators of f(θ) with minimax optimal error rates measured by the L_2(ℙ_θ)-distance as well as by more general Orlicz norm distances. Moreover, we show that if d≤ n^α and s>1/1-α, then the resulting estimators are asymptotically efficient in Hájek-LeCam sense with the convergence rate √(n). This generalizes earlier results on estimation of smooth functionals in Gaussian shift models. The estimators have the form f_k(θ̂), where θ̂ is the maximum likelihood estimator and f_k: ℝ^d↦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.

• 8 publications
• 11 publications
12/18/2019

### Estimation of Smooth Functionals in Normal Models: Bias Reduction and Asymptotic Efficiency

Let X_1,..., X_n be i.i.d. random variables sampled from a normal distri...
10/25/2017

### Asymptotically Efficient Estimation of Smooth Functionals of Covariance Operators

Let X be a centered Gaussian random variable in a separable Hilbert spac...
11/07/2020

### Estimation of smooth functionals in high-dimensional models: bootstrap chains and Gaussian approximation

Let X^(n) be an observation sampled from a distribution P_θ^(n) with an ...
05/20/2022

### Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank

Let E be a separable Banach space and let X, X_1,…, X_n, … be i.i.d. Gau...
02/19/2020

### Asymptotically Optimal Bias Reduction for Parametric Models

An important challenge in statistical analysis concerns the control of t...
01/20/2020

### Asymptotic normality of a linear threshold estimator in fixed dimension with near-optimal rate

Linear thresholding models postulate that the conditional distribution o...
09/29/2020

### On Smooth Change-Point Location Estimation for Poisson Processes

We are interested in estimating the location of what we call "smooth cha...