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1. Nonparametric generalized fiducial inference for survival functions under censoring

   https://doi.org/10.1093/biomet/asz016  

   Cui, Y; Hannig, J (June 2019, Biometrika)

   Summary Since the introduction of fiducial inference by Fisher in the 1930s, its application has been largely confined to relatively simple, parametric problems. In this paper, we present what might be the first time fiducial inference is systematically applied to estimation of a nonparametric survival function under right censoring. We find that the resulting fiducial distribution gives rise to surprisingly good statistical procedures applicable to both one-sample and two-sample problems. In particular, we use the fiducial distribution of a survival function to construct pointwise and curvewise confidence intervals for the survival function, and propose tests based on the curvewise confidence interval. We establish a functional Bernstein–von Mises theorem, and perform thorough simulation studies in scenarios with different levels of censoring. The proposed fiducial-based confidence intervals maintain coverage in situations where asymptotic methods often have substantial coverage problems. Furthermore, the average length of the proposed confidence intervals is often shorter than the length of confidence intervals for competing methods that maintain coverage. Finally, the proposed fiducial test is more powerful than various types of log-rank tests and sup log-rank tests in some scenarios. We illustrate the proposed fiducial test by comparing chemotherapy against chemotherapy combined with radiotherapy, using data from the treatment of locally unresectable gastric cancer. 

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2. On Differentially Private U Statistics

   Chaudhuri, Kamalika; Loh, Po-Ling; Pandey, Shourya; Sarkar, Purnamrita (July 2024, https://doi.org/10.48550/arXiv.2407.04945)

   We consider the problem of privately estimating a parameter 𝔼[h(X1,…,Xk)], where X1, X2, …, Xk are i.i.d. data from some distribution and h is a permutation-invariant function. Without privacy constraints, standard estimators are U-statistics, which commonly arise in a wide range of problems, including nonparametric signed rank tests, symmetry testing, uniformity testing, and subgraph counts in random networks, and can be shown to be minimum variance unbiased estimators under mild conditions. Despite the recent outpouring of interest in private mean estimation, privatizing U-statistics has received little attention. While existing private mean estimation algorithms can be applied to obtain confidence intervals, we show that they can lead to suboptimal private error, e.g., constant-factor inflation in the leading term, or even Θ(1/n) rather than O(1/n²) in degenerate settings. To remedy this, we propose a new thresholding-based approach using local Hájek projections to reweight different subsets of the data. This leads to nearly optimal private error for non-degenerate U-statistics and a strong indication of near-optimality for degenerate U-statistics. 

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3. Post-Contextual-Bandit Inference

   Bibaut, Aurelien; Dimakopoulou, Maria; Kallus, Nathan; Chambaz, Antoine; van der Laan, Mark (January 2021, Advances in neural information processing systems)

   Contextual bandit algorithms are increasingly replacing non-adaptive A/B tests in e-commerce, healthcare, and policymaking because they can both improve outcomes for study participants and increase the chance of identifying good or even best policies. To support credible inference on novel interventions at the end of the study, nonetheless, we still want to construct valid confidence intervals on average treatment effects, subgroup effects, or value of new policies. The adaptive nature of the data collected by contextual bandit algorithms, however, makes this difficult: standard estimators are no longer asymptotically normally distributed and classic confidence intervals fail to provide correct coverage. While this has been addressed in non-contextual settings by using stabilized estimators, variance stabilized estimators in the contextual setting pose unique challenges that we tackle for the first time in this paper. We propose the Contextual Adaptive Doubly Robust (CADR) estimator, a novel estimator for policy value that is asymptotically normal under contextual adaptive data collection. The main technical challenge in constructing CADR is designing adaptive and consistent conditional standard deviation estimators for stabilization. Extensive numerical experiments using 57 OpenML datasets demonstrate that confidence intervals based on CADR uniquely provide correct coverage. 

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4. Differentially private estimation of U statistics

   Kamalika Chaudhuri, Po-Ling Loh (January 2024, Proceedings of Machine Learning Research)

   Under Review for COLT 2024 (Ed.)

   In this paper, we consider the problem of private estimation of U statistics. U statistics are widely used estimators that naturally arise in a broad class of problems, from nonparametric signed rank tests to subgraph counts in random networks. They are simply averages of an appropriate kernel applied to all subsets of a given size (also known as the degree) of sample size n. However, despite the recent outpouring of interest in private mean estimation, private algorithms for more general U statistics have received little attention. We propose a framework where, for a broad class of U statistics, one can use existing tools in private mean estimation to obtain confidence intervals where the private error does not overwhelm the irreducible error resulting from the variance of the U statistics. However, in specific cases that arise when the U statistics degenerate or have vanishing moments, the private error may be of a larger order than the non-private error. To remedy this, we propose a new thresholding-based approach that uses Hajek projections to re-weight different subsets. As we show, this leads to more accurate inference in certain settings. 

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5. Confidence intervals for the Cox model test error from cross‐validation

   https://doi.org/10.1002/sim.9873  

   Sun, Min Woo; Tibshirani, Robert (November 2023, Statistics in Medicine)

   Summary Cross‐validation (CV) is one of the most widely used techniques in statistical learning for estimating the test error of a model, but its behavior is not yet fully understood. It has been shown that standard confidence intervals for test error using estimates from CV may have coverage below nominal levels. This phenomenon occurs because each sample is used in both the training and testing procedures during CV and as a result, the CV estimates of the errors become correlated. Without accounting for this correlation, the estimate of the variance is smaller than it should be. One way to mitigate this issue is by estimating the mean squared error of the prediction error instead using nested CV. This approach has been shown to achieve superior coverage compared to intervals derived from standard CV. In this work, we generalize the nested CV idea to the Cox proportional hazards model and explore various choices of test error for this setting. 

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