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Principal component analysis (PCA) is a well-known tool for dimension reduction. It can summarise the data in fewer than the original number of dimensions without losing essential information. However, when data are dispersed across multiple servers, communication cost can't make PCA useful in this situation. Thus distributed algorithms for PCA are needed. Fan et al. [Annals of statistics 47(6) (2019) 3009-3031] proposed a distributed PCA algorithm to settle this problem. On each server, They computed the K leading eigenvectors V ( ) K = v ( ) 1 , . . . , v ( ) K ∈ R d×K of the sample covariance matrix Σ and sent V ( ) K to the data center. In this paper, we introduce robust covariance matrix estimators respectively proposed by Minsker [Annals of statistics 46(6A) (2018) 2871-2903] and Ke et al. [Statistical Science 34(3) (2019) 454-471] into the distributed PCA algorithm and compute its top K eigenvectors on each server and transmit them to the central server. We investigate the statistical error of the resulting distributed estimator and derive the rate of convergence for

This paper provides some extended results on estimating the parameter matrix of high-dimensional regression model when the covariate or response possess weaker moment condition. We investigate the M -estimator of Fan et al. ( AnnStat 49(3):1239-1266, 2021) for matrix completion model with (1 + ǫ)-th moments. The corresponding phase transition phenomenon is observed. When ǫ ≥ 1, the robust estimator possesses the same convergence rate as previous literature. While 1 > ǫ > 0, the rate will be slower. For high dimensional multiple index coefficient model, we also apply the element-wise truncation method to construct a robust estimator which handle missing and heavy-tailed data with finite fourth moment.