Communication-efficient distributed inference and diagnostics for multivariate linear models

With the rapid growth of large-scale data stored across distributed systems, developing multivariate linear regression methods that are both statistically efficient and communication-efficient has become increasingly important. This paper proposes a pilot-subsampling surrogate framework for distributed multivariate linear models under non-random data storage. The proposed estimator, referred to as the distributed multivariate linear model via pilot subsampling (DMLM-S), uses a small random pilot sample to approximate the curvature of the full-data loss and aggregates global gradient information from all workers. For the moderate-dimensional regime studied here, DMLM-S avoids transmitting second-order derivative blocks of dimension d×d, where d=pm+m(m+1)/2, and is asymptotically equivalent to the full-data estimator under the stated regularity conditions; we do not claim a diverging-p or diverging-m theory. We establish consistency and asymptotic normality for the pilot estimator and the refined DMLM-S estimator. We also develop a coefficient-displacement diagnostic based on a centred, pilot-preconditioned perturbation of the full-data regression score. It ranks influential observations and worker blocks using all observations, conditional on the fitted coefficients and pilot curvature. The worker score measures group influence on the fitted regression, not a formal test of distributional equality. Extensive simulation studies and two real data applications are designed to assess estimation accuracy, pilot-size sensitivity, scalability, inference accuracy, and diagnostic performance. The results demonstrate that the proposed framework provides a statistically efficient and interpretable tool for large-scale multivariate linear modelling under non-random distributed storage.