Simultaneous Inference for High Dimensional Mean Vectors

TL;DR

This paper develops a method for constructing simultaneous confidence intervals for high-dimensional mean vectors using a truncated sample mean and Gaussian approximation, applicable when the dimension grows exponentially with the sample size.

Contribution

It introduces a novel resampling approach based on Gaussian approximation for ultra high-dimensional mean inference under mild moment conditions.

Findings

Gaussian approximation holds under polynomial moment conditions

Method allows for exponential growth of dimension with sample size

Provides a practical resampling technique for simultaneous inference

Abstract

Let $X_1, \ldots, X_n\in\mathbb{R}^p$ be i.i.d. random vectors. We aim to perform simultaneous inference for the mean vector $\mathbb{E} (X_i)$ with finite polynomial moments and an ultra high dimension. Our approach is based on the truncated sample mean vector. A Gaussian approximation result is derived for the latter under the very mild finite polynomial ($(2+\theta)$-th) moment condition and the dimension $p$ can be allowed to grow exponentially with the sample size $n$. Based on this result, we propose an innovative resampling method to construct simultaneous confidence intervals for mean vectors.