An exact unbiased semi-parametric maximum quasi-likelihood framework which is complete in the presence of ties

TL;DR

This paper develops a comprehensive quasi-likelihood framework for correlation estimation that handles ties and weak orderings in ranked data, providing consistent inference and extending classical models.

Contribution

It introduces a novel quasi-likelihood extension of Kendall's tau, supporting inference with ties and weak orderings, and unifies several classical ranking models.

Findings

Supports inference on population correlation with ties

Provides analytic standard errors for the proposed tie model

Establishes equivalence to Bradley-Terry and Thurstone models

Abstract

This paper introduces a novel quasi-likelihood extension of the generalised Kendall \(\tau_{a}\) estimator, together with an extension of the Kemeny metric and its associated covariance and correlation forms. The central contribution is to show that the U-statistic structure of the proposed coefficient \(\tau_{\kappa}\) naturally induces a quasi-maximum likelihood estimation (QMLE) framework, yielding consistent Wald and likelihood ratio test statistics. The development builds on the uncentred correlation inner-product (Hilbert space) formulation of Emond and Mason (2002) and resolves the associated sub-Gaussian likelihood optimisation problem under the \(\ell_{2}\)-norm via an Edgeworth expansion of higher-order moments. The Kemeny covariance coefficient \(\tau_{\kappa}\) is derived within a novel likelihood framework for pairwise comparison-continuous random variables, enabling direct inference on population-level correlation between ranked or weakly ordered datasets. Unlike existing approaches that focus on marginal or pairwise summaries, the proposed framework supports sample-observed weak orderings and accommodates ties without information loss. Drawing parallels with Thurstone's Case V latent ordering model, we derive a quasi-likelihood-based tie model with analytic standard errors, generalising classical U-statistics. The framework applies to general continuous and discrete random variables and establishes formal equivalence to Bradley-Terry and Thurstone models, yielding a uniquely identified linear representation with both analytic and likelihood-based estimators.