This paper investigates generalized pilot estimators of covariance matrix in the presence of missing data. When the random samples have only bounded fourth moment, two kinds of generalized pilot estimators are provided, the generalized Huber estimator and the generalized truncated mean estimator. In addition, we construct thresholding generalized pilot estimator for a kind of sparse covariance matrices and establish the convergence rates in terms of probability under spectral and Frobenius norms respectively. Moreover, the convergence rates in sense of expectation are also given under an extra condition. Finally, simulation studies are conducted to demonstrate the superiority of our method.

This paper considers the minimax perturbation bounds of the low-rank matrix under Ky Fan norm. We first explore the upper bounds via the best rank- $r$ approximation ${\hat{A}}_r$ of the observation matrix $\hat{A}$. Next, the lower bounds are established by constructing special matrix groups to show the upper bounds are tight on the low-rank matrix estimation error. In addition, we derive the rate-optimal perturbation bounds for the left and right singular subspaces under Ky Fan norm $\sin \mathrm{\Theta }$ distance. Finally, some simulations have been carried out to support our theories.