We derive multiscale statistics for deconvolution in order to detect qualitative features of the unknown density. An important example covered within this framework is to test for local monotonicity on all scales simultaneously. We investigate the moderately ill-posed setting, where the Fourier transform of the error density in the deconvolution model is of polynomial decay. For multiscale testing, we consider a calibration, motivated by the modulus of continuity of Brownian motion. We investigate the performance of our results from both the theoretical and simulation based point of view. A major consequence of our work is that the detection of qualitative features of a density in a deconvolution problem is a doable task although the minimax rates for pointwise estimation are very slow.
In this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality. The aim of this paper is to establish non-asymptotic minimax rates of testing for goodness-of-fit hypotheses in a heteroscedastic setting. More precisely, we deal with sequences @math of independent Gaussian random variables, having mean @math and variance @math . The set @math will be either finite or countable. In particular, such a model covers the inverse problem setting where few results in test theory have been obtained. The rates of testing are obtained with respect to @math and @math norms, without assumption on @math and on several functions spaces. Our point of view is completely non-asymptotic.  We study non-parametric tests for checking parametric hypotheses about a multivariate density f of independent identically distributed random vectors Z1, Z2,... which are observed under additional noise with density ψ. The tests we propose are an extension of the test due to Bickel and Rosenblatt [On some global measures of the deviations of density function estimates, Ann. Statist. 1 (1973) 1071-1095] and are based on a comparison of a nonparametric deconvolution estimator and the smoothed version of a parametric fit of the density f of the variables of interest Zi. In an example the loss of efficiency is highlighted when the test is based on the convolved (but observable) density g = f * ψ instead on the initial density of interest f. We consider the detection problem of a two-dimensional function from noisy observations of its integrals over lines. We study both rate and sharp asymptotics for the error probabilities in the minimax setup. By construction, the derived tests are non-adaptive. We also construct a minimax rate-optimal adaptive test of rather simple structure.  We propose two test statistics for use in inverse regression problems "Y"e"K""t"p"e", where "K" is a given linear operator which cannot be continuously inverted. Thus, only noisy, indirect observations "Y" for the function "t" are available. Both test statistics have a counterpart in classical hypothesis testing, where they are called the order selection test and the data-driven Neyman smooth test. We also introduce two model selection criteria which extend the classical Akaike information criterion and Bayes information criterion to inverse regression problems. In a simulation study we show that the inverse order selection and Neyman smooth tests outperform their direct counterparts in many cases. The theory is motivated by data arising in confocal fluorescence microscopy. Here, images are observed with blurring, modelled as convolution, and stochastic error at subsequent times. The aim is then to reduce the signal-to-noise ratio by averaging over the distinct images. In this context it is relevant to decide whether the images are still equal, or have changed by outside influences such as moving of the object table. Copyright (c) 2009 Royal Statistical Society. We construct a testing procedure for monotonicity on some interval in the statistical models of density deconvolution and signal deblurring under white noise. The corresponding error probabilities are studied with particular attention to their asymptotic properties. We analyze a new dataset from an electrophysiological recording of transmembrane currents through a bacterial membrane channel to demonstrate the existence of single and multiple channel currents. Protein channels mediate transport through biological membranes; knowledge of the channel properties gained from electrophysiological recordings is important for a targeted drug design. We investigate the bacterial membrane protein SecYEG which is of essential importance for the secretory pathway for sorting of newly synthesized proteins to their place of function in the cell. Our results strongly indicate that in the SecYEG pore the different modes of the density of channel currents are approximately equidistant and correspond to different numbers of open channels in the membrane. A current of ≈12 pA under the present experimental conditions turns out to be characteristic of the presence of a single open SecYEG pore, a fact that had not been electrophysiologically characterized so far. Electrophysiological reco...