# The Lagrange approach in the monotone single index model

## TL;DR

This paper introduces a Lagrange-based method for estimating parameters in the monotone single index model, avoiding reparametrization, and compares various estimators including score and least squares methods.

## Contribution

It proposes a novel Lagrange-type approach to estimate parameters without reparametrization and applies it to several existing estimators in the monotone single index model.

## Key findings

- The Lagrange approach simplifies the estimation process.
- The estimators are compared with existing methods like LSE, MRE, and EDR.
- Effects of random starting values on the algorithms are analyzed.

## Abstract

The finite-dimensional parameters of the monotone single index model are often estimated by minimization of a least squares criterion and reparametrization to deal with the non-unicity. We avoid the reparametrization by using a Lagrange-type method and replace the minimization over the finite-dimensional parameter alpha by a `crossing of zero' criterion at the derivative level. In particular, we consider a simple score estimator (SSE), an efficient score estimator (ESE), and a penalized least squares estimator (PLSE) for which we can apply this method. The SSE and ESE were discussed in Balabdaoui, Groeneboom and Hendrickx (2018}, but the proofs still used reparametrization. Another version of the PLSE was discussed in Kuchibhotla and Patra (2017), where also reparametrization was used. The estimators are compared with the profile least squares estimator (LSE), Han's maximum rank estimator (MRE), the effective dimension reduction estimator (EDR) and a linear least squares estimator, which can be used if the covariates have an elliptically symmetric distribution. We also investigate the effects of random starting values in the search algorithms.