# Testing the regularity of a smooth signal

@article{Carpentier2015TestingTR,
title={Testing the regularity of a smooth signal},
author={A. Carpentier},
journal={Bernoulli},
year={2015},
volume={21},
pages={465-488}
}
We develop a test to determine whether a function lying in a fixed $L_2$-Sobolev-type ball of smoothness $t$, and generating a noisy signal, is in fact of a given smoothness $s\geq t$ or not. While it is impossible to construct a uniformly consistent test for this problem on every function of smoothness $t$, it becomes possible if we remove a sufficiently large region of the set of functions of smoothness $t$. The functions that we remove are functions of smoothness strictly smaller than $s… ## Figures from this paper Minimax$L_2$-Separation Rate in Testing the Sobolev-Type Regularity of a function In this paper we study the problem of testing if an$L_2-$function$f$belonging to a certain$l_2$-Sobolev-ball$B_t(R)$of radius$R>0$with smoothness level$t>0$indeed exhibits a higher The smoothness test for a density function • Mathematics • 2014 Abstract The problem of testing hypothesis that a density function has no more than μ derivatives versus it has more than μ derivatives is considered. For a solution, the L 2 norms of wavelet Hypothesis Testing For Densities and High-Dimensional Multinomials: Sharp Local Minimax Rates • Mathematics, Computer Science The Annals of Statistics • 2019 The goodness-of-fit testing problem of distinguishing whether the data are drawn from a specified distribution, versus a composite alternative separated from the null in the total variation metric is considered, and the first local minimax lower bounds for this problem are provided. Minimax Euclidean separation rates for testing convex hypotheses in$\mathbb{R}^{d}$• Mathematics • 2017 We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a convex subset$\mathcal{C}$of$\mathbb{R}^d$. We adopt Optimal sparsity testing in linear regression model • Mathematics • 2019 We consider the problem of sparsity testing in the high-dimensional linear regression model. The problem is to test whether the number of non-zero components (aka the sparsity) of the regression HIGH DIMENSIONAL ASYMPTOTICS OF LIKELIHOOD RATIO TESTS IN THE GAUSSIAN SEQUENCE MODEL UNDER CONVEX CONSTRAINTS • 2021 In the Gaussian sequence model Y = μ+ξ, we study the likelihood ratio test (LRT) for testing H0 : μ = μ0 versus H1 : μ ∈K, where μ0 ∈K, and K is a closed convex set in Rn. In particular, we show that Adaptive estimation of the sparsity in the Gaussian vector model • Mathematics The Annals of Statistics • 2019 Consider the Gaussian vector model with mean value {\theta}. We study the twin problems of estimating the number |{\theta}|_0 of non-zero components of {\theta} and testing whether |{\theta}|_0 is Distributed function estimation: adaptation using minimal communication • Mathematics • 2020 We investigate whether in a distributed setting, adaptive estimation of a smooth function at the optimal rate is possible under minimal communication. It turns out that the answer depends on the risk Honest and adaptive confidence sets in Lp We consider the problem of constructing honest and adaptive confidence sets in Lp-loss (with p>=1 and p =2, we identify two main regimes, (i) one where adaptation is possible without any restrictions Mathematical Foundations of Infinite-Dimensional Statistical Models • Mathematics • 2015 1. Nonparametric statistical models 2. Gaussian processes 3. Empirical processes 4. Function spaces and approximation theory 5. Linear nonparametric estimators 6. The minimax paradigm 7. ## References SHOWING 1-10 OF 37 REFERENCES Minimax Nonparametric Hypothesis Testing: The Case of an Inhomogeneous Alternative • Mathematics • 1999 We study the problem of testing a simple hypothesis for a nonparametric "signal + white-noise" model. It is assumed under the null hypothesis that the "signal" is completely specified, e.g., that no Adaptive hypothesis testing using wavelets Let a function f be observed with a noise. We wish to test the null hypothesis that the function is identically zero, against a composite nonparametric alternative: functions from the alternative set Adaptive minimax testing in the discrete regression scheme • Mathematics • 2005 We consider the problem of testing hypotheses on the regression function from n observations on the regular grid on [0,1]. We wish to test the null hypothesis that the regression function belongs to On nonparametric tests of positivity/monotonicity/convexity • Mathematics • 2002 We consider the problem of estimating the distance from an unknown signal, observed in a white-noise model, to convex cones of positive/monotone/convex functions. We show that, when the unknown On adaptive inference and confidence bands • Mathematics • 2011 The problem of existence of adaptive confidence bands for an unknown density$f$that belongs to a nested scale of H\"{o}lder classes over$\mathbb{R}$or$[0,1]\$ is considered. Whereas honest
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