Matias D. Cattaneo, Michael Jansson, Kenichi Nagasawa
arXiv 23 Mar 2023 · Mathematics — Statistics Theory · publishedThe Annals of Statistics (2024) · 2 citations (OpenAlex)
arXiv:2303.13598 · PDF · DOI · OpenAlex · Extracted main text
Westling and Carone (2020) proposed a framework for studying the large sample distributional properties of generalized Grenander-type estimators, a versatile class of nonparametric estimators of monotone functions. The limiting distribution of those estimators is representable as the left derivative of the greatest convex minorant of a Gaussian process whose monomial mean can be of unknown order (when the degree of flatness of the function of interest is unknown). The standard nonparametric bootstrap is unable to consistently approximate the large sample distribution of the generalized Grenander-type estimators even if the monomial order of the mean is known, making statistical inference a challenging endeavour in applications. To address this inferential problem, we present a bootstrap-assisted inference procedure for generalized Grenander-type estimators. The procedure relies on a carefully crafted, yet automatic, transformation of the estimator. Moreover, our proposed method can be made “flatness robust” in the sense that it can be made adaptive to the (possibly unknown) degree of flatness of the function of interest. The method requires only the consistent estimation of a single scalar quantity, for which we propose an automatic procedure based on numerical derivative estimation and the generalized jackknife. Under random sampling, our inference method can be implemented using a computationally attractive exchangeable bootstrap procedure. We illustrate our methods with examples and we also provide a small simulation study. The development of formal results is made possible by some technical results that may be of independent interest.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | barticle[author] van der Vaart, Aad W.A. W. van der Laan, Mark J.M. J (2006) ) | 1.000 | 6 | 3 | 100% |
| 2 | bbook[author] van der Vaart, Aad W.A. W. Wellner, Jon A.J. A (1996) ) | 0.874 | 10 | 2 | 100% |
| 3 | barticle[author] Westling, TedT. Carone, MarcoM (2020) ) | 0.843 | 3 | 3 | 100% |
| 4 | barticle[author] Pollard, DavidD (1989) ) | 0.693 | 8 | 1 | 100% |
| 5 | barticle[author] Cattaneo, Matias D.M. D., Farrell, MaxM. Feng, Ying… (2020) ) | 0.693 | 5 | 1 | 100% |
| 6 | barticle[author] Kim, JeankyungJ. Pollard, DavidD (1990) ) | 0.644 | 4 | 1 | 100% |
| 7 | barticle[author] Lo, Shaw-HwaS.-H. Singh, KesarK (1986) ) | 0.644 | 2 | 2 | 100% |
| 8 | barticle[author] Cattaneo, Matias D.M. D., Chandak, RajitaR., Jansso… (2024) ) | 0.511 | 2 | 1 | 100% |
| 9 | bincollection[author] Kosorok, Michael R.M. R (2008) ) | 0.511 | 2 | 1 | 100% |
| 10 | barticle[author] Burke, M. D.M. D., Csörgo, S.S. Horváth, LL (1988) ) | 0.405 | 1 | 1 | 100% |
Showing the top 10 of 18 scored citations.
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| 1 | Continuity of the Distribution Function of the $arg\,max$ of a Gaussian Process | 0.811 | 4 | 2 |
| 2 | Bridging Root-$n$ and Non-standard Asymptotics: Adaptive Inference in M-Estimation | 0.585 | 3 | 1 |
| 3 | Robust Inference for Convex Pairwise Difference Estimators | 0.405 | 1 | 1 |
| 4 | Root-$n$ Asymptotically Normal Maximum Score Estimation | 0.405 | 1 | 1 |