Time-frequency analysis deals with signals for which the underlying spectral characteristics change over time. The essential tool is the short-time Fourier transform, which localizes the Fourier transform in time by means of a window function. In a white noise model, we derive rate-optimal and adaptive estimators of signals in modulation spaces, which measure smoothness in terms of decay properties of the short-time Fourier transform. The estimators are based on series expansions by means of Gabor frames and on thresholding the coefficients. The minimax rates have interesting new features, and the derivation of the lower bounds requires the use of test functions which approximately localize both in time and in frequency. Simulations and applications to audio recordings illustrate the practical relevance of our methods. We also discuss the best$N$-term approximation and the approximation of variational problems in modulation spaces by Gabor frame expansions.

We show nonparametric identification of the parameters in the dynamic stochastic block model as recently introduced by Matias and Miele (2017) in the case of binary, finitely weighted, and general edge states. We formulate conditions on the true parameters, which guarantee actual point identification instead of mere generic identification, and which also lead to novel conclusions in the static case. In particular, our results justify in terms of the identification of the applications by Matias and Miele to finitely weighted edges with three edge states. We also give numerical illustrations via the variational EM algorithm in simulation settings covered by our identification analysis.

Statistical tests are introduced for checking whether an image functionf(x,y) defined on the unit discD={(x,y):x2+y2les 1} is invariant under certain symmetry transformations ofD, given that discrete and noisy data are observed. Invariance under reflections or under rotations by rational angles is considered, as well as rotational invariance. These symmetry relations can be naturally expressed as restrictions for the Zernike moments off(x,y). Therefore, the test statistics are based on theL2distance between Zernike series estimates of the image function itself and its version obtained after applying the symmetry transformation. The asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives is derived. Furthermore, the quality of the asymptotic approximations via simulation studies is investigated. The usefulness of our theory is verified by examining an important problem in confocal microscopy, i.e., possible imprecise alignments in the optical path of the microscope are investigated. For optical systems with rotational symmetry, the theoretical point-spread function (PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. The tests are used to investigate whether the required symmetries can indeed be detected in the empirical PSF.

This paper considers the problem of testing parametric assumptions on signals f from which only noisy observations y/sub k/=f(/spl tau/k)+/spl epsi//sub k/ are available, and where the signal is assumed to be either band-limited or time-limited. To this end, the signal is reconstructed by an estimator based on the Whittaker-Shannon (WS) sampling theorem with oversampling. As test statistic, the minimal L/sub 2/ distance between the estimated signal and the parametric model is used. To construct appropriate tests, the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives. As a byproduct, the asymptotic distribution of the integrated square error of the estimator is computed, which is of interest by itself, e.g., for the analysis of a cross-validated bandwidth selector.