Asymptotics for minimisers of convex processes

TL;DR

This paper introduces a simple convexity-based method for establishing the consistency and asymptotic normality of estimators derived from convex criterion functions, applicable to various statistical models.

Contribution

It presents a new, simplified proof technique for estimator asymptotics and extends results to weaker regularity conditions, especially in Cox and logistic regression.

Findings

Simpler proofs for estimator asymptotics in multiple models

Weaker regularity conditions for Cox regression asymptotics

Extended asymptotic results to logistic regression

Abstract

By means of two simple convexity arguments we are able to develop a general method for proving consistency and asymptotic normality of estimators that are defined by minimisation of convex criterion functions. This method is then applied to a fair range of different statistical estimation problems, including Cox regression, logistic and Poisson regression, least absolute deviation regression outside model conditions, and pseudo-likelihood estimation for Markov chains. Our paper has two aims. The first is to exposit the method itself, which in many cases, under reasonable regularity conditions, leads to new proofs that are simpler than the traditional proofs. Our second aim is to exploit the method to its limits for logistic regression and Cox regression, where we seek asymptotic results under as weak regularity conditions as possible. For Cox regression in particular we are able to weaken previously published regularity conditions substantially.