  Constructing confidence intervals for the coefficients of high-dimensional
sparse linear models remains a challenge, mainly because of the complicated
limiting distributions of the widely used estimators, such as the lasso.
Several methods have been developed for constructing such intervals. Bootstrap
lasso+ols is notable for its technical simplicity, good interpretability, and
performance that is comparable with that of other more complicated methods.
However, bootstrap lasso+ols depends on the beta-min assumption, a theoretic
criterion that is often violated in practice. Thus, we introduce a new method,
called bootstrap lasso+partial ridge, to relax this assumption. Lasso+partial
ridge is a two-stage estimator. First, the lasso is used to select features.
Then, the partial ridge is used to refit the coefficients. Simulation results
show that bootstrap lasso+partial ridge outperforms bootstrap lasso+ols when
there exist small, but nonzero coefficients, a common situation that violates
the beta-min assumption. For such coefficients, the confidence intervals
constructed using bootstrap lasso+partial ridge have, on average, $50\%$ larger
coverage probabilities than those of bootstrap lasso+ols. Bootstrap
lasso+partial ridge also has, on average, $35\%$ shorter confidence interval
lengths than those of the de-sparsified lasso methods, regardless of whether
the linear models are misspecified. Additionally, we provide theoretical
guarantees for bootstrap lasso+partial ridge under appropriate conditions, and
implement it in the R package "HDCI."
