Online Debiasing for Adaptively Collected High-dimensional Data with Applications to Time Series Analysis
This work addresses bias issues in statistical inference for adaptive data collection, which is common in science and engineering, but it appears incremental as it builds on existing debiasing techniques for LASSO.
The paper tackles the problem of bias in high-dimensional linear regression with adaptively collected data, where sample size can be smaller than the number of covariates, by proposing an online debiasing procedure for estimators like LASSO. The result shows that this method optimally debiases the LASSO estimate under sparsity conditions, enabling computation of p-values and confidence intervals of optimal size in contexts such as time series analysis and batched data collection.
Adaptive collection of data is commonplace in applications throughout science and engineering. From the point of view of statistical inference however, adaptive data collection induces memory and correlation in the samples, and poses significant challenge. We consider the high-dimensional linear regression, where the samples are collected adaptively, and the sample size $n$ can be smaller than $p$, the number of covariates. In this setting, there are two distinct sources of bias: the first due to regularization imposed for consistent estimation, e.g. using the LASSO, and the second due to adaptivity in collecting the samples. We propose "online debiasing", a general procedure for estimators such as the LASSO, which addresses both sources of bias. In two concrete contexts $(i)$ time series analysis and $(ii)$ batched data collection, we demonstrate that online debiasing optimally debiases the LASSO estimate when the underlying parameter $θ_0$ has sparsity of order $o(\sqrt{n}/\log p)$. In this regime, the debiased estimator can be used to compute $p$-values and confidence intervals of optimal size.