When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916468499677184 |
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| author | Lin, Licong Su, Fangzhou Mou, Wenlong Ding, Peng Wainwright, Martin |
| author_facet | Lin, Licong Su, Fangzhou Mou, Wenlong Ding, Peng Wainwright, Martin |
| contents | Resampling methods are especially well-suited to inference with estimators that provide only "black-box'' access. Jackknife is a form of resampling, widely used for bias correction and variance estimation, that is well-understood under classical scaling where the sample size $n$ grows for a fixed problem. We study its behavior in application to estimating functionals using high-dimensional $Z$-estimators, allowing both the sample size $n$ and problem dimension $d$ to diverge. We begin showing that the plug-in estimator based on the $Z$-estimate suffers from a quadratic breakdown: while it is $\sqrt{n}$-consistent and asymptotically normal whenever $n \gtrsim d^2$, it fails for a broad class of problems whenever $n \lesssim d^2$. We then show that under suitable regularity conditions, applying a jackknife correction yields an estimate that is $\sqrt{n}$-consistent and asymptotically normal whenever $n\gtrsim d^{3/2}$. This provides strong motivation for the use of jackknife in high-dimensional problems where the dimension is moderate relative to sample size. We illustrate consequences of our general theory for various specific $Z$-estimators, including non-linear functionals in linear models; generalized linear models; and the inverse propensity score weighting (IPW) estimate for the average treatment effect, among others. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_02909 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators Lin, Licong Su, Fangzhou Mou, Wenlong Ding, Peng Wainwright, Martin Statistics Theory Methodology Resampling methods are especially well-suited to inference with estimators that provide only "black-box'' access. Jackknife is a form of resampling, widely used for bias correction and variance estimation, that is well-understood under classical scaling where the sample size $n$ grows for a fixed problem. We study its behavior in application to estimating functionals using high-dimensional $Z$-estimators, allowing both the sample size $n$ and problem dimension $d$ to diverge. We begin showing that the plug-in estimator based on the $Z$-estimate suffers from a quadratic breakdown: while it is $\sqrt{n}$-consistent and asymptotically normal whenever $n \gtrsim d^2$, it fails for a broad class of problems whenever $n \lesssim d^2$. We then show that under suitable regularity conditions, applying a jackknife correction yields an estimate that is $\sqrt{n}$-consistent and asymptotically normal whenever $n\gtrsim d^{3/2}$. This provides strong motivation for the use of jackknife in high-dimensional problems where the dimension is moderate relative to sample size. We illustrate consequences of our general theory for various specific $Z$-estimators, including non-linear functionals in linear models; generalized linear models; and the inverse propensity score weighting (IPW) estimate for the average treatment effect, among others. |
| title | When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2411.02909 |