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A bootstrap lasso + partial ridge method to construct confidence intervals for parameters in high-dimensional sparse linear models


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”.

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