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.
Hypothesis testing for deconvolution and related inverse problems is a relatively new area. Current methods cover testing of parametric assumptions (cf. @cite_24 @cite_39 @cite_41 ) and, more recently, testing for certain smoothness classes such as Sobolev balls in a Gaussian sequence model (Laurent @cite_7 @cite_18 and Ingster @cite_32 ). All these papers focus on regression deconvolution models. Exceptions for density deconvolution are Holzmann @cite_42 , Balabdaoui @cite_23 , and Meister @cite_43 who developed tests for various global hypotheses, such as global monotonicity. The latter test has been derived for one fixed interval and allows to check whether a density is monotone on that interval at a preassigned level of significance.
@cite_18 @cite_7 @cite_41 @cite_42 @cite_32 @cite_39 @cite_24 @cite_43 @cite_23
2018808115 2068523437 2103374230 2076148642
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 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 introduce a simple technique for analyzing the iterative decoder that is broadly applicable to different classes of codes defined over graphs in certain fading as well as additive white Gaussian noise (AWGN) channels. The technique is based on the observation that the extrinsic information from constituent maximum a posteriori (MAP) decoders is well approximated by Gaussian random variables when the inputs to the decoders are Gaussian. The independent Gaussian model implies the existence of an iterative decoder threshold that statistically characterizes the convergence of the iterative decoder. Specifically, the iterative decoder converges to zero probability of error as the number of iterations increases if and only if the channel E sub b N sub 0 exceeds the threshold. Despite the idealization of the model and the simplicity of the analysis technique, the predicted threshold values are in excellent agreement with the waterfall regions observed experimentally in the literature when the codeword lengths are large. Examples are given for parallel concatenated convolutional codes, serially concatenated convolutional codes, and the generalized low-density parity-check (LDPC) codes of Gallager and Cheng-McEliece (1996). Convergence-based design of asymmetric parallel concatenated convolutional codes (PCCC) is also discussed. 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.