# Score estimation in the monotone single index model

We consider estimation of the regression parameter in the single index model where the link function ψ is monotone. For this model it has been proposed to estimate the link function nonparametrically by the monotone least square estimate ψ̂_nα for a fixed regression parameter α and to estimate the regression parameter by minimizing the sum of squared deviations ∑_i{Y_i-ψ̂_nα(α^TX_i)}^2 over α, where Y_i are the observations and X_i the corresponding covariates. Although it is natural to propose this least squares procedure, it is still unknown whether it will produce √(n)-consistent estimates of α. We show that the latter property will hold if we solve a score equation corresponding to this minimization problem. We also compare our method with other methods such as Han's maximum rank correlation estimate, which has been proved to be √(n)-consistent.

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