# 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…

## 12 Citations

Minimax $L_2$-Separation Rate in Testing the Sobolev-Type Regularity of a function

- Mathematics
- 2019

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 ScienceThe 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

- MathematicsThe 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

- Mathematics
- 2013

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.…

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