# Phase Transition Unbiased Estimation in High Dimensional Settings

## TL;DR

This paper introduces a new property called phase transition unbiasedness for estimators in high-dimensional statistics, and develops an estimator that achieves this property using simulation techniques, improving bias control in complex models.

## Contribution

The paper proposes a novel phase transition unbiasedness property and constructs an estimator that satisfies it, applicable in high-dimensional settings without requiring initial estimator consistency.

## Key findings

- The estimator converges exponentially fast via iterative bootstrap.
- It is unbiased for all sample sizes above a finite threshold.
- The approach enhances robustness and avoids issues like separability in logistic regression.

## Abstract

An important challenge in statistical analysis concerns the control of the finite sample bias of estimators. For example, the maximum likelihood estimator has a bias that can result in a significant inferential loss. This problem is typically magnified in high-dimensional settings where the number of variables $p$ is allowed to diverge with the sample size $n$. However, it is generally difficult to establish whether an estimator is unbiased and therefore its asymptotic order is a common approach used (in low-dimensional settings) to quantify the magnitude of the bias. As an alternative, we introduce a new and stronger property, possibly for high-dimensional settings, called phase transition unbiasedness. An estimator satisfying this property is unbiased for all $n$ greater than a finite sample size $n^\ast$. Moreover, we propose a phase transition unbiased estimator built upon the idea of matching an initial estimator computed on the sample and on simulated data. It is not required for this initial estimator to be consistent and thus it can be chosen for its computational efficiency and/or for other desirable properties such as robustness. This estimator can be computed using a suitable simulation based algorithm, namely the iterative bootstrap, which is shown to converge exponentially fast. In addition, we demonstrate the consistency and the limiting distribution of this estimator in high-dimensional settings. Finally, as an illustration, we use our approach to develop new estimators for the logistic regression model, with and without random effects, that also enjoy other properties such as robustness to data contamination and are also not affected by the problem of separability. In a simulation exercise, the theoretical results are confirmed in settings where the sample size is relatively small compared to the model dimension.