# Parameter Estimation for Thurstone Choice Models

## TL;DR

This paper analyzes the accuracy of parameter estimation in Thurstone choice models, including the Luce and Bradley-Terry models, providing theoretical bounds for maximum likelihood and rank-breaking estimators.

## Contribution

It offers a tight theoretical characterization of the mean squared error for maximum likelihood and rank-breaking estimators in Thurstone models, extending understanding of estimation accuracy.

## Key findings

- Derived tight bounds for MLE estimation error
- Analyzed rank-breaking method under independence assumptions
- Connected estimation accuracy to classification error in binary parameters

## Abstract

We consider the estimation accuracy of individual strength parameters of a Thurstone choice model when each input observation consists of a choice of one item from a set of two or more items (so called top-1 lists). This model accommodates the well-known choice models such as the Luce choice model for comparison sets of two or more items and the Bradley-Terry model for pair comparisons. We provide a tight characterization of the mean squared error of the maximum likelihood parameter estimator. We also provide similar characterizations for parameter estimators defined by a rank-breaking method, which amounts to deducing one or more pair comparisons from a comparison of two or more items, assuming independence of these pair comparisons, and maximizing a likelihood function derived under these assumptions. We also consider a related binary classification problem where each individual parameter takes value from a set of two possible values and the goal is to correctly classify all items within a prescribed classification error.