On Second order correctness of Bootstrap in Logistic Regression

TL;DR

This paper introduces a new perturbation resampling bootstrap method for logistic regression that achieves second order correctness, providing more accurate inference than traditional asymptotic approaches.

Contribution

The paper develops a novel bootstrap technique with smoothing and studentization to attain second order correctness in logistic regression inference.

Findings

Bootstrap method is more accurate than asymptotic normality.

Smoothing ensures the MLE has a density despite binary response.

Direct bootstrapping fails without smoothing.

Abstract

In the fields of clinical trials, biomedical surveys, marketing, banking, with dichotomous response variable, the logistic regression is considered as an alternative convenient approach to linear regression. In this paper, we develop a novel bootstrap technique based on perturbation resampling method for approximating the distribution of the maximum likelihood estimator (MLE) of the regression parameter vector. We establish second order correctness of the proposed bootstrap method after proper studentization and smoothing. It is shown that inferences drawn based on the proposed bootstrap method are more accurate compared to that based on asymptotic normality. The main challenge in establishing second order correctness remains in the fact that the response variable being binary, the resulting MLE has a lattice structure. We show the direct bootstrapping approach fails even after studentization. We adopt smoothing technique developed in Lahiri (1993) to ensure that the smoothed studentized version of the MLE has a density. Similar smoothing strategy is employed to the bootstrap version also to achieve second order correct approximation.