On the population least?squares criterion in the monotone single index model

Monotone single index models have gained over the past decades increasing popularity due to their flexibility and versatile use in diverse areas. Semi-parametric estimators such as least squares maximum likelihood of unknown monotone ridge function were considered make inference without having choose some tuning parameter. Description asymptotic behavior those crucially depends on acquiring a good understanding optimization problems associated with corresponding population criteria. In this paper, we give several insights into these criteria by proving existence minimizers thereof general classes parameters. order describe minimizers, prove different results which direction variation special case where common distribution covariates is Gaussian. A complementary simulation study was performed whose support our main theorems.