  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.
