Evaluability of paired comparison data in stochastic paired comparison models: Necessary and sufficient condition

TL;DR

This paper establishes a necessary and sufficient condition for the existence and uniqueness of maximum likelihood estimators in three-option stochastic paired comparison models, enhancing data evaluability understanding.

Contribution

It provides the first complete condition for estimator existence and uniqueness in three-option paired comparison models, improving upon previous sufficient-only conditions.

Findings

The new condition accurately predicts estimator evaluability.

Simulation shows the condition's efficiency over older criteria.

Results improve the reliability of parameter estimation in these models.

Abstract

In this paper, paired comparison models with stochastic background are investigated. We focus on the models that allow three options for choice. We estimate all parameters, the strength of the objects and the boundaries of equal decision, by maximum likelihood method. The existence and uniqueness of the estimator are key issues of the evaluation. Although a necessary and sufficient condition for the general case of three options has not been known until now, there are some different sufficient conditions that are formulated in the literature. In this paper, we provide a necessary and sufficient condition for the existence of a maximum and the uniqueness of the argument that maximizes the value, i.e. for the evaluability of the data in models of these types. By computer simulation, we present the efficiency of the condition, comparing it to the previously known sufficient conditions.