# Optimal Linear Discriminators For The Discrete Choice Model In Growing Dimensions

## TL;DR

This paper investigates the behavior of Manski's maximum score estimator for discrete choice models in high-dimensional settings, deriving convergence rates, bounds, and optimal estimators for different growth regimes of the dimension.

## Contribution

It extends the analysis of the maximum score estimator to scenarios where the number of predictors grows with the sample size, providing convergence rates, bounds, and computational methods.

## Key findings

- Derived $ ext{ell}_2$ convergence rates under different growth regimes.
- Established minimax bounds for estimation error in high dimensions.
- Proposed algorithms for computing the maximum score estimator in large dimensions.

## Abstract

Manski's celebrated maximum score estimator for the discrete choice model, which is an optimal linear discriminator, has been the focus of much investigation in both the econometrics and statistics literatures, but its behavior under growing dimension scenarios largely remains unknown. This paper addresses that gap. Two different cases are considered: $p$ grows with $n$ but at a slow rate, i.e. $p/n \rightarrow 0$; and $p \gg n$ (fast growth). In the binary response model, we recast Manski's score estimation as empirical risk minimization for a classification problem, and derive the $\ell_2$ rate of convergence of the score estimator under a \emph{transition condition} in terms of our margin parameter that calibrates the level of difficulty of the estimation problem. We also establish upper and lower bounds for the minimax $\ell_2$ error in the binary choice model that differ by a logarithmic factor, and construct a minimax-optimal estimator in the slow growth regime. Some extensions to the general case -- the multinomial response model -- are also considered. Last but not least, we use a variety of learning algorithms to compute the maximum score estimator in growing dimensions.