On the estimation of correlation in a binary sequence model

TL;DR

This paper investigates the challenges of estimating correlation in binary sequence models derived from thresholded continuous variables, revealing fundamental limits and conditions for consistent estimation.

Contribution

It proves the nonestimability of correlation from binary data and demonstrates that trinary data allows consistent estimation, highlighting a phase transition phenomenon.

Findings

Likelihood maximization does not produce consistent estimates for binary data.

Trinary data enables consistent correlation estimation with parametric rate.

A phase transition exists between binary and trinary data in correlation estimability.

Abstract

We consider a binary sequence generated by thresholding a hidden continuous sequence. The hidden variables are assumed to have a compound symmetry covariance structure with a single parameter characterizing the common correlation. We study the parameter estimation problem under such one-parameter models. We demonstrate that maximizing the likelihood function does not yield consistent estimates for the correlation. We then formally prove the nonestimability of the parameter by deriving a non-vanishing minimax lower bound. This counter-intuitive phenomenon provides an interesting insight that one-bit information of each latent variable is not sufficient to consistently recover their common correlation. On the other hand, we further show that trinary data generated from the hidden variables can consistently estimate the correlation with parametric convergence rate. Thus we reveal a phase transition phenomenon regarding the discretization of latent continuous variables while preserving the estimability of the correlation. Numerical experiments are performed to validate the conclusions.