Statistical Estimation with Heterogenous Data Structures

Statistical estimation in modern data science often encounters heterogeneous data structures, arise from diverse sources, irregular sampling patterns, or harbor latent subpopulations. In these cases, traditional methods that rely on strong regularity assumptions may struggle to provide reliable estimates. This dissertation addresses these challenges by developing robust statistical methodologies for heterogeneous data structures across three domains: ranking from pairwise comparisons, parameter estimation in item response theory, and shared subspace estimation in multi-matrix settings. Each problem is characterized by challenges such as non-uniform sampling, imbalanced data, or misaligned subspaces, necessitating novel theoretical and algorithmic advancements. In the first part, we study ranking from pairwise comparison with general or semi-random comparison graph. We propose a weighted maximum likelihood estimator that utilizes a semi-definite programming based reweighting to restore the spectral properties in the comparison graphs and achieves near-optimal sample complexity for top-$K$ ranking. In the second part, we design the random pairing MLE (RP-MLE) that addresses sparse and imbalanced item-response data. We derive minimax-optimal error bounds as well as asymptotic normality for this algorithm. In the third part, we give an analysis of the estimation of shared subspace in the presence of unique and misaligned component using Angle-based Joint and Individual Variation Explained (AJIVE), revealing its strengths in high signal-to-noise regimes and fundamental limitations in signal-to-noise settings. Integrating innovative algorithms and novel analytical tools, this work advances efficient, theoretically grounded solutions for statistical estimation with heterogeneity in ranking, item response theory, and multi-view learning.