arXiv subjects

Cecile Durot

Publications and source records attributed to Cecile Durot.

4 recordsLinked to original sources

arXiv2026

Inference for concave distribution functions under measurement error

We propose an estimator of a concave cumulative distribution function under the measurement error model, where the non-negative variables of interest are perturbed by additive independent random noise. The estimator is defined as the least concave majorant on the positive half-line of the deconvolution estimator of the distribution function. We show its uniform consistency and its square root convergence in law in $\ell_\infty(\mathbb R)$. To assess the validity of the concavity assumption, we construct a test for the nonparametric null hypothesis that the distribution function is concave on the positive half-line, against the alternative that it is not. We calibrate the test using bootstrap methods. The theoretical justification for calibration led us to establish a bootstrap version of Theorem 1 in S\"ohl and Trabs (2012), a Donsker-type result from which we obtain, as a special case, the limiting behavior of the deconvolution estimator of the distribution function in a bootstrap setting with measurement error. Combining this Donsker-type theorem with the functional delta method, we show that the test statistic and its bootstrap version have the same limiting distribution under the null hypothesis, whereas under the alternative, the bootstrap statistic is stochastically smaller. Consequently, the power of the test tends to one, for any fixed alternative, as the sample size tends to infinity. In addition to the theoretical results for the estimator and the test, we investigate their finite-sample performance in simulation studies.

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arXiv2024

Minimax Optimal rates of convergence in the shuffled regression, unlinked regression, and deconvolution under vanishing noise

Shuffled regression and unlinked regression represent intriguing challenges that have garnered considerable attention in many fields, including but not limited to ecological regression, multi-target tracking problems, image denoising, etc. However, a notable gap exists in the existing literature, particularly in vanishing noise, i.e., how the rate of estimation of the underlying signal scales with the error variance. This paper aims to bridge this gap by delving into the monotone function estimation problem under vanishing noise variance, i.e., we allow the error variance to go to $0$ as the number of observations increases. Our investigation reveals that, asymptotically, the shuffled regression problem exhibits a comparatively simpler nature than the unlinked regression; if the error variance is smaller than a threshold, then the minimax risk of the shuffled regression is smaller than that of the unlinked regression. On the other hand, the minimax estimation error is of the same order in the two problems if the noise level is larger than that threshold. Our analysis is quite general in that we do not assume any smoothness of the underlying monotone link function. Because these problems are related to deconvolution, we also provide bounds for deconvolution in a similar context. Through this exploration, we contribute to understanding the intricate relationships between these statistical problems and shed light on their behaviors when subjected to the nuanced constraint of vanishing noise.

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arXiv2018

Removing the Curse of Superefficiency: an Effective Strategy For Distributed Computing in Isotonic Regression

We propose a strategy for computing the isotonic least-squares estimate of a monotone function in a general regression setting where the data are distributed across different servers and the observations across servers, though independent, can come from heterogeneous sub-populations, thereby violating the identically distributed assumption. Our strategy fixes the super-efficiency phenomenon observed in prior work on distributed computing in the isotonic regression framework, where averaging several isotonic estimates (each computed at a local server) on a central server produces super-efficient estimates that do not replicate the properties of the global isotonic estimator, i.e. the isotonic estimate that would be constructed by transferring all the data to a single server. The new estimator proposed in this paper works by smoothing the data on each local server, communicating the smoothed summaries to the central server, and then computing an isotonic estimate at the central server, and is shown to replicate the asymptotic properties of the global estimator, and also overcome the super-efficiency phenomenon exhibited by earlier estimators. For data on $N$ observations, the new estimator can be constructed by transferring data just over order $N^{1/3}$ across servers [as compared to transferring data of order $N$ to compute the global isotonic estimator], and requires the same order of computing time as the global estimator.

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arXiv2016

Divide and Conquer in Non-standard Problems and the Super-efficiency Phenomenon

We study how the divide and conquer principle --- partition the available data into subsamples, compute an estimate from each subsample and combine these appropriately to form the final estimator --- works in non-standard problems where rates of convergence are typically slower than $\sqrt{n}$ and limit distributions are non-Gaussian, with a special emphasis on the least squares estimator (and its inverse) of a monotone regression function. We find that the pooled estimator, obtained by averaging non-standard estimates across the mutually exclusive subsamples, outperforms the non-standard estimator based on the entire sample in the sense of pointwise inference. We also show that, under appropriate conditions, if the number of subsamples is allowed to increase at appropriate rates, the pooled estimator is asymptotically normally distributed with a variance that is empirically estimable from the subsample-level estimates. Further, in the context of monotone function estimation we show that this gain in pointwise efficiency comes at a price --- the pooled estimator's performance, in a uniform sense (maximal risk) over a class of models worsens as the number of subsamples increases, leading to a version of the super-efficiency phenomenon. In the process, we develop analytical results for the order of the bias in isotonic regression, which are of independent interest.

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