Testing the regularity of a smooth signal

Carpentier, Alexandra

doi:10.3150/13-bej575

Cited by 14 publications

(31 citation statements)

References 35 publications

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“…the cone of positive, monotone, or convex functions, see e.g. [16], or also balls for some metrics [17,8]. These papers exhibit the minimaxoptimal separation distance -or near optimal distance, in some cases of [16] and [17] -for the specific convex shapes that are considered, namely cones and smoothness balls.…”

Section: Introductionmentioning

confidence: 99%

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Blanchard¹,

Carpentier²,

Gutzeit³

2018

Electron. J. Statist.

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We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R d . We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1 n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C.MSC 2010 subject classifications: Primary 60K35, 60K35; secondary 60K35.

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Section: Introductionmentioning

confidence: 99%

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Blanchard¹,

Carpentier²,

Gutzeit³

2018

Electron. J. Statist.

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“…As mentioned earlier, although the results appear to be of similar flavor to those in Bull and Nickl (2013);Carpentier (2015), the rigorous derivations require careful understanding and modifications to accommodate for the effect of estimating an unknown density. A possible approach to the testing problem (1.3) can be the method of Carpentier (2015) without further modification. However, such an approach results in unbiased estimation of…”

Section: Consider the Testing Problemmentioning

confidence: 86%

“…Instead, our proof shows that under the alternative, the quantity Π(f ĝ g |L) 2 2 is also large enough for suitable subspaces L. This quantity is easier to estimate modulo the availability of a nice estimatorĝ-which is in turn guaranteed by Theorem 1.1. However, this also necessitates modifying the testing procedure of Carpentier (2015) suitably to incorporate the effect of estimating g. We make this more clear in the proof of Theorem 2.…”

Section: Consider the Testing Problemmentioning

confidence: 99%

“…The second point to note is that we have not considered the fixed design case in our set up. The fixed design problem can be addressed similarly with more straightforward generalization of ideas from Robins and Van Der Vaart (2006), Bull andNickl (2013), andCarpentier (2015) due to lack of extra nuisance parameter g. We omit this for the sake of brevity.…”

Section: Choice Of Binary Regression Modelmentioning

confidence: 99%

“…We therefore extend the exponential inequality obtained in Bull and Nickl (2013) to more general second order U-statistics based on wavelet projection kernels (See Lemma 4.1). For the case of testing of composite alternatives 1.3, this also adds to the chi-square type empirical wavelet coefficient procedure of Carpentier (2015).…”

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Optimal adaptive inference in random design binary regression

Mukherjee¹,

Sen²

2018

Bernoulli

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We construct confidence sets for the regression function in nonparametric binary regression with an unknown design density-a nuisance parameter in the problem. These confidence sets are adaptive in L 2 loss over a continuous class of Sobolev type spaces. Adaptation holds in the smoothness of the regression function, over the maximal parameter spaces where adaptation is possible, provided the design density is smooth enough. We identify two key regimes -one where adaptation is possible, and one where some critical regions must be removed. We address related questions about goodness of fit testing and adaptive estimation of relevant infinite dimensional parameters.In many epidemiological studies, a binary response variable Y is independently observed on a population of individuals along with multiple covariates X to explain the variability in the response. In the context of epidemiological studies, the probability of observing a specific outcome conditional on the covariates is often referred to as the propensity score. Estimating propensity score type functions from observed data is often of interest, and these estimates are subsequently used in multiple inferential procedures such as propensity score matching (Rosenbaum and Rubin, 1983), inverse probability weighted inference (Robins, Rotnitzky and Zhao, 1994) etc. In the context of semiparametric inference for missing data type problems, a nice exposition to the importance of understanding questions of similar flavor can be found in Tsiatis (2007).Historically, regression models with binary outcomes have been approached through both parametric (McCullagh and Nelder, 1989) and nonparametric lenses (Antoniadis and Leblanc, 2000;Signorini and Jones, 2004). Although parametric regression has the natural advantage of being simpler in interpretation and implementation, it often lacks the desired complexity required to capture varieties of dependence between covariates and outcomes. Nonparametric binary regression attempts to address this question, but it has its own share of shortcomings-the two major concerns being dependence on a priori knowledge about the true underlying regression function class and ease of implementation. Motivated by these, in this paper we study inference (estimation, testing, and confidence sets) in binary regression problems under nonparametric models having random covariates with unknown design density, with primary focus on adaptation over function classes.To fix ideas, suppose we observe data (x i , y i ) n i=1 , where x i ∈ [0, 1] d and y i ∈ {0, 1}. Consider the binary regression model E(y|x) = P (y = 1|x) = f (x), y ∈ {0, 1}, x ∼ g.(0.1)AMS 2000 subject classifications: Primary 62G10, 62G20, 62C20

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Mathematical Foundations of Infinite-Dimensional Statistical Models

Giné¹,

Nickl

2015

396

703

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In nonparametric and high-dimensional statistical models, the classical Gauss-Fisher-Le Cam theory of the optimality of maximum likelihood and Bayesian posterior inference does not apply, and new foundations and ideas have been developed in the past several decades. This book gives a coherent account of the statistical theory in infinite-dimensional parameter spaces. The mathematical foundations include self-contained 'mini-courses' on the theory of Gaussian and empirical processes, on approximation and wavelet theory and on the basic theory of function spaces. The theory of statistical inference in such models-hypothesis testing, estimation and confidence sets-is then presented within the minimax paradigm of decision theory. This includes the basic theory of convolution kernel and projection estimation, as well as Bayesian nonparametrics and nonparametric maximum likelihood estimation. In the final chapter, the theory of adaptive inference in nonparametric models is developed, including Lepski's method, wavelet thresholding and adaptive confidence regions for self-similar functions. EVARIST GINÉ (1944-2015) was the head of the Department of Mathematics at the University of Connecticut. Giné was a distinguished mathematician who worked on mathematical statistics and probability in infinite dimensions. He was the author of two books and more than a hundred articles.

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Testing the regularity of a smooth signal

Cited by 14 publications

References 35 publications

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Minimax Euclidean separation rates for testing convex hypotheses in $\mathbb{R}^{d}$

Optimal adaptive inference in random design binary regression

Mathematical Foundations of Infinite-Dimensional Statistical Models

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