  We consider statistical inference for a single coordinate of regression
coefficients in high-dimensional linear models. Recently, the debiased
estimators are popularly used for constructing confidence intervals and
hypothesis testing in high-dimensional models. However, some representative
numerical experiments show that they tend to be biased for large coefficients,
especially when the number of large coefficients dominates the number of small
coefficients. In this paper, we propose a modified debiased Lasso estimator
based on bootstrap. Let us denote the proposed estimator BS-DB for short. We
show that, under the irrepresentable condition and other mild technical
conditions, the BS-DB has smaller order of bias than the debiased Lasso in
existence of a large proportion of strong signals. If the irrepresentable
condition does not hold, the BS-DB is guaranteed to perform no worse than the
debiased Lasso asymptotically. Confidence intervals based on the BS-DB are
proposed and proved to be asymptotically valid under mild conditions. Our study
on the inference problems integrates the properties of the Lasso on variable
selection and estimation novelly. The superior performance of the BS-DB over
the debiased Lasso is demonstrated via extensive numerical studies.
