The $\ell$-test: leveraging sparsity in the Gaussian linear model for improved inference
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| author | Sengupta, Souhardya Janson, Lucas |
| author_facet | Sengupta, Souhardya Janson, Lucas |
| contents | We develop novel LASSO-based methods for coefficient testing and confidence interval construction in the Gaussian linear model with $n\ge d$. Our methods' finite-sample validity is identical to that of their ubiquitous ordinary-least-squares-$t$-test-based analogues, yet have substantially higher power when the true coefficient vector is sparse. In particular, under sparsity our coefficient test, which we call the $\ell$-test, performs like the \emph{one-sided} $t$-test (despite not being given any information about the sign), and $\ell$-test-based confidence intervals are correspondingly shorter than the standard $t$-test-based intervals. The nature of the $\ell$-test directly provides a novel exact adjustment conditional on LASSO selection for post-selection inference, allowing for the construction of post-selection $p$-values and confidence intervals. None of our methods require resampling or Monte Carlo estimation. We perform a variety of simulations and a real data analysis on an HIV drug resistance data set to demonstrate the benefits of the $\ell$-test. We additionally show that the $\ell$-test can be applied to a large class of asymptotically Gaussian estimators, dramatically expanding its applicability beyond linear models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_18390 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $\ell$-test: leveraging sparsity in the Gaussian linear model for improved inference Sengupta, Souhardya Janson, Lucas Methodology We develop novel LASSO-based methods for coefficient testing and confidence interval construction in the Gaussian linear model with $n\ge d$. Our methods' finite-sample validity is identical to that of their ubiquitous ordinary-least-squares-$t$-test-based analogues, yet have substantially higher power when the true coefficient vector is sparse. In particular, under sparsity our coefficient test, which we call the $\ell$-test, performs like the \emph{one-sided} $t$-test (despite not being given any information about the sign), and $\ell$-test-based confidence intervals are correspondingly shorter than the standard $t$-test-based intervals. The nature of the $\ell$-test directly provides a novel exact adjustment conditional on LASSO selection for post-selection inference, allowing for the construction of post-selection $p$-values and confidence intervals. None of our methods require resampling or Monte Carlo estimation. We perform a variety of simulations and a real data analysis on an HIV drug resistance data set to demonstrate the benefits of the $\ell$-test. We additionally show that the $\ell$-test can be applied to a large class of asymptotically Gaussian estimators, dramatically expanding its applicability beyond linear models. |
| title | The $\ell$-test: leveraging sparsity in the Gaussian linear model for improved inference |
| topic | Methodology |
| url | https://arxiv.org/abs/2406.18390 |