Besov Bodies Research Articles

A model of Poissonian interactions and detection of dependence

This paper proposes a model of interactions between two point processes, ruled by a reproduction function h, which is considered as the intensity of a Poisson process. In particular, we focus on the context of neurosciences to detect possible interactions in the cerebral activity associated with two neurons. To provide a mathematical answer to this specific problem of neurobiologists, we address so the question of testing the nullity of the intensity h. We construct a multiple testing procedure obtained by the aggregation of single tests based on a wavelet thresholding method. This test has good theoretical properties: it is possible to guarantee the level but also the power under some assumptions and its uniform separation rate over weak Besov bodies is adaptive minimax. Then, some simulations are provided, showing the good practical behavior of our testing procedure.

The two-sample problem for Poisson processes: Adaptive tests with a nonasymptotic wild bootstrap approach

Considering two independent Poisson processes, we address the question of testing equality of their respective intensities. We first propose single tests whose test statistics are U-statistics based on general kernel functions. The corresponding critical values are constructed from a non-asymptotic wild bootstrap approach, leading to level \alpha tests. Various choices for the kernel functions are possible, including projection, approximation or reproducing kernels. In this last case, we obtain a parametric rate of testing for a weak metric defined in the RKHS associated with the considered reproducing kernel. Then we introduce, in the other cases, an aggregation procedure, which allows us to import ideas coming from model selection, thresholding and/or approximation kernels adaptive estimation. The resulting multiple tests are proved to be of level \alpha, and to satisfy non-asymptotic oracle type conditions for the classical L2-norm. From these conditions, we deduce that they are adaptive in the minimax sense over a large variety of classes of alternatives based on classical and weak Besov bodies in the univariate case, but also Sobolev and anisotropic Nikol'skii-Besov balls in the multivariate case.

Adaptive estimation in the nonparametric random coefficients binary choice model by needlet thresholding

In the random coefficients binary choice model, a binary variable equals 1 iff an index $X^\top\beta$ is positive. The vectors $X$ and $\beta$ are independent and belong to the sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$. We prove lower bounds on the minimax risk for estimation of the density $f_{\beta}$ over Besov bodies where the loss is a power of the $L^p(\mathbb{S}^{d-1})$ norm for $1\le p\le \infty$. We show that a hard thresholding estimator based on a needlet expansion with data-driven thresholds achieves these lower bounds up to logarithmic factors.

Adaptive quadratic functional estimation of a weighted density by model selection

We consider the problem of estimating the integral of the square of a probability density function f on the basis of a random sample from a weighted distribution. Specifically, using model selection via a penalized criterion, an adaptive estimator for ∫ f 2 based on weighted data is proposed for probability density functions which are uniformly bounded and belong to certain Besov bodies. We show that the proposed estimator attains the minimax rate of convergence that is optimal in the case of direct data. Additionally, we obtain the information bound for the problem of estimating ∫ f 2 when weighted data are available and compare it with the information bound for the case of direct data. A small simulation study is conducted to illustrate the usefulness of the proposed estimator in finite sample situations.

Estimating the joint distribution of independent categorical variables via model selection

Assume one observes independent categorical variables or, equivalently, one observes the corresponding multinomial variables. Estimating the distribution of the observed sequence amounts to estimating the expectation of the multinomial Sequence. A new estimator for this mean is proposed that is nonparametric, non-asymptotic and implementable even for large sequences. It is a penalized least-squares estimator based oil wavelets, with a penalization term inspired by papers of Birge and Massart. The estimator is proved to satisfy ail oracle inequality and to be adaptive in the minimax sense over a class of Besov bodies. The method is embedded in a general framework which allows us to recover also an existing method for segmentation. Beyond theoretical results, a simulation Study is reported and an application on real data is provided.

A data-driven block thresholding approach to wavelet estimation

A data-driven block thresholding procedure for wavelet regression is proposed and its theoretical and numerical properties are investigated. The procedure empirically chooses the block size and threshold level at each resolution level by minimizing Stein’s unbiased risk estimate. The estimator is sharp adaptive over a class of Besov bodies and achieves simultaneously within a small constant factor of the minimax risk over a wide collection of Besov Bodies including both the “dense” and “sparse” cases. The procedure is easy to implement. Numerical results show that it has superior finite sample performance in comparison to the other leading wavelet thresholding estimators.

Adaptive confidence balls

Adaptive confidence balls are constructed for individual resolution levels as well as the entire mean vector in a multiresolution framework. Finite sample lower bounds are given for the minimum expected squared radius for confidence balls with a prespecified confidence level. The confidence balls are centered on adaptive estimators based on special local block thresholding rules. The radius is derived from an analysis of the loss of this adaptive estimator. In addition adaptive honest confidence balls are constructed which have guaranteed coverage probability over all of $\mathbb{R}^N$ and expected squared radius adapting over a maximum range of Besov bodies.

Minimax nonparametric hypothesis testing for ellipsoids and Besov bodies

We observe an infinitely dimensional Gaussian random vector x = ξ + v where ξ is a sequence of standard Gaussian variables and v ∈ l2 is an unknown mean. We consider the hypothesis testing problem H0 : v = 0 versus alternatives for the sets . The sets Ve are l q -ellipsoids of semi-axes ai = i-s R/e with l p -ellipsoid of semi-axes bi = i-r pe /e removed or similar Besov bodies Bq,t;s (R/e) with Besov bodies Bp,h;r (pe /e) removed. Here or are the parameters which define the sets Ve for given radii pe → 0 , 0 0 ; e → 0 is the asymptotical parameter. We study the asymptotics of minimax second kind errors and construct asymptotically minimax or minimax consistent families of tests , if it is possible. We describe the partition of the set of parameters κ into regions with different types of asymptotics: classical, trivial, degenerate and Gaussian (of various types). Analogous rates have been obtained in a signal detection problem for continuous variant of white noise model: alternatives correspond to Besov or Sobolev balls with Besov or Sobolev balls removed. The study is based on an extension of methods of constructions of asymptotically least favorable priors. These methods are applicable to wide class of “convex separable symmetrical infinite-dimensional hypothesis testing problems in white Gaussian noise model. Under some assumptions these methods are based on the reduction of hypothesis testing problem to convex extreme problem: to minimize specially defined Hilbert norm over convex sets of sequences of measures πi on the real line. The study of this extreme problem allows to obtain different types of Gaussian asymptotics. If necessary assumptions do not hold, then we obtain other types of asymptotics.

On nonparametric estimation in nonlinear AR(1)-models

We estimate the mean function and the conditional variance (the volatility function) of a nonlinear first-order autoregressive model nonparametrically. Minimax rates of convergence are established over a scale of Besov bodies Bspq and a range of global Lp′ error measurements, for 1⩽p′<∞. We propose an estimating procedure based on a martingale regression approximation scheme. This enables us to implement wavelet thresholding and obtain adaptation results with respect to an unknown degree of smoothness.

Minimax estimation via wavelet shrinkage

We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets, we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints and asymptotically mini-max over Besov bodies with $p \leq q$. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with $p<2$, so the method can significantly outperform every linear method (e.g., kernel, smoothing spline, sieve in a minimax sense). Variants of our method based on simple threshold nonlinear estimators are nearly minimax. Our method possesses the interpretation of spatial adaptivity; it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper, which was first drafted in November 1990, discuss practical implementation, spatial adaptation properties, universal near minimaxity and applications to inverse problems.