Multi-index regression models with missing covariates at random

AMS subject classifications: 62H12 62G20 Keywords: Covariates missing at random Inverse selection probability Multi-index model Single-index model a b s t r a c t This paper considers estimation of the semiparametric multi-index model with missing covariates at random. A weighted estimating equation is suggested by invoking the inverse selection probability approach, and estimators of the indices are respectively defined when the selection probability is known in advance, is estimated parametrically and nonparametrically. The consistency is provided. For the single-index model, the large sample properties show that the estimators with both parametric and nonparametric plug-in estimations can play an important role to achieve smaller limiting variances than the estimator with the true selection probability. Simulation studies are carried out to assess the finite sample performance of the proposed estimators. The proposed methods are applied to an AIDS clinical trials dataset to examine which method could be more efficient. A horse colic dataset is also analyzed for illustration. The multi-index regression model (MIM) for the scalar outcome variable Y and covariate X of dimension p has the form Y = g(θ τ 0 X) + ε, (1) where g(·) is the unknown mean function defined on R d , and θ 0 is an unknown regression parameter matrix of dimension p × d. In this paper, we assume the dimension d is known in advance. The conditional expectation of ε given X equals zero, and the superscript τ in (1) denotes transposition. For identifiability consideration, we assume that the direction parameter θ 0 satisfies θ τ 0 θ 0 = I d an identity matrix. When d = 1, the model is the well-known single-index model (SIM). In this case, the first component of θ 0 is assumed to be positive without loss of generality. The multi-index model is widely used, in several areas such as statistics and econometrics, as a reasonable compromise between fully parametric and fully nonparametric modeling. A remarkable amount of research has already been carried out for statistical inference in the context of completed samples. For estimation of the parameter θ 0 as well as of the link function g(·), there are a number of proposals in the literature. For multi-index and more general models, the examples of methods are sliced inverse regression (SIR, Li, [9]), sliced average variance estimation (SAVE, Cook and Weisberg, [3]). Zhu and Ng [23] and Zhu and Fang [22] gave general results …