Debiased Estimators in High-Dimensional Regression: A Review and Replication of Javanmard and Montanari (2014)

High-dimensional statistical settings ($p \gg n$) pose fundamental challenges for classical inference, largely due to bias introduced by regularized estimators such as the LASSO. To address this, Javanmard and Montanari (2014) propose a debiased estimator that enables valid hypothesis testing and confidence interval construction. This report examines their debiased LASSO framework, which yields asymptotically normal estimators in high-dimensional settings. We present the key theoretical results underlying this approach, specifically, the construction of an optimized debiased estimator that restores asymptotic normality, which enables the computation of valid confidence intervals and $p$-values. To evaluate the claims of Javanmard and Montanari, a subset of the original simulation study and a re-examination of their real-data analysis are presented. Building on this baseline, we extend the empirical analysis to include the desparsified LASSO, a closely related method referenced but not implemented in the original study. The results demonstrate that while the debiased LASSO achieves reliable coverage and controls Type I error, the LASSO projection estimator can offer improved power in low-signal settings without compromising error rates. Our findings highlight a critical practical trade-off: while the LASSO projection estimator demonstrates superior statistical power in an idealized simulated low-signal setting, the estimation procedure employed by Javanmard and Montanari adapts more robustly to complex correlation networks, yielding superior precision and signal detection in real-world genomic data.