Sufficient dimension reduction in the presence of controlling variable and missing multivariate response

In this paper, we focus on partial dimension reduction for conditional mean function in the presence of controlling variable when responses are multivariate and missing at random. A weighted-profile least squares estimate based on inverse probability weighted technique is proposed to estimate the central mean subspace. The profile least squares method does not need any distributional assumptions on the covariates, which is different from existing methods in sufficient dimension reduction. It turns out that under some mild conditions, the estimator of central mean subspace is asymptotically normal and root-n consistent. The proposed test statistic as well as its asymptotic distribution for parametric hypothesis tests are obtained. In addition, a BIC-type criterion is used to determine the structural dimension of the central mean subspace. Its consistency is also established. Both simulated and real data analysis results demonstrate promising performance of the proposed sufficient dimension reduction estimation method.