Index coefficients estimation in single-index models: the generalized maximum rank correlation estimator

We propose a new estimator, called the Generalized Maximum Rank Correlation Estimator (GMRC), of the index coefficients in the context of the so-called Single-Index Model : Yi = g(β′0 Xi)+ε i, with g and β0 unknown. The underlying idea is very simple: given a pair of observations (Xi, Yi) and (Xj , Yj), if g(β′0 Xi) is greater than g(β′0 Xj), it is likely that Yi be greater than Yj . In other words, the ranks of the Yi’s and the ranks of the g(β′0 Xi)’s would be highly positively correlated. The clue is thus to estimate β0 by the value of β which maximizes an estimated version of the rank correlation. Han (1987) proposed such kind of estimation method, but assuming the strict monotonicity of the link function g. We relax this assumption. The estimator is shown to be root-n consistent and asymptotically normal, and has multiple advantages. In particular, an extensive simulation study shows its very good finite-sample behavior : in most of the situations, it seems that the GMRC estimator represents the best choice in practice