The asymptotically optimal estimating equation for longitudinal data. Strong Consistency

TL;DR

This paper develops an asymptotically optimal estimation method for longitudinal data using martingale difference sequences, providing strong consistency results under certain conditions.

Contribution

It introduces a new class of estimating equations for longitudinal data and establishes conditions for their asymptotic optimality and strong consistency.

Findings

Identification of sufficient conditions for asymptotic optimality.

Proof of strong consistency of the estimators.

Extension of existing models to a broader class of estimating equations.

Abstract

In this article, we introduce a conditional marginal model for longitudinal data, in which the residuals form a martingale difference sequence. This model allows us to consider a rich class of estimating equations, which contains several estimating equations proposed in the literature. A particular sequence of estimating equations in this class contains a random matrix $\mathcal{R}_{i-1}^*(\beta)$, as a replacement for the ``true'' conditional correlation matrix of the $i$-th individual. Using the approach of [12], we identify some sufficient conditions under which this particular sequence of equations is asymptotically optimal (in our class). In the second part of the article, we identify a second set of conditions, under which we prove the existence and strong consistency of a sequence of estimators of $\beta$, defined as roots of estimation equations which are martingale transforms (in particular, roots of the sequence of asymptotically optimal equations).