The detection of two-component mixture alternatives

In this thesis, we deal with one of the facets of the statistical detection problem. We study a particular type of alternative, the mixture model. We consider testing where the null hypothesis corresponds to the absence of a signal, represented by some known distribution, e.g., Gaussian white noise, while in the alternative one assumes that among observations there might be a cluster of points carrying a signal, which is characterized by some distribution G. The main research objective is to determine a detectable set of alternatives in the parameter space combining the parameters of G and the mixture proportion p. We focus is on the finite sample sizes and wish to study the possibility of detecting alternatives for fixed n, given pre-specified error levels.

The first part of the thesis covers theoretical results. Specifically, we introduce a parametrization which relates the parameter space of an alternative to the sample size, and present the regions of detectability and non-detectability. The regions of detectability are the subsets of the new parameter space (induced by the parametrization) where for a prespecified type I error rate, the type II error rate of the likelihood ratio test (LRT) is bounded from above by some constant. To move towards the real data applications, we also check the performance of some non-parametric testing procedures proposed for this problem and some widely used distributions.

In the second part of the thesis, we use this argument to develop a framework for clinical trial designs aimed at detecting a sensitive-to-therapy subpopulation. The idea of modeling treatment response as a mixture of subpopulations originates from treatment effect heterogeneity. Methods studying the effects of heterogeneity in the clinical data are referred to as subgroup analyses. However, designs accounting for possible response heterogeneity are rarely discussed, though in some cases they might help to avoid trial failure due to the lack of efficacy. In our work, we consider two possible subgroups of patients, drug responders and drug non-responders. Given no preliminary information about patients' memberships, we propose a framework for designing randomized clinical trials that are able to detect a responders' subgroup of desired characteristics. We also propose strategies to minimize the number of enrolled patients whilst preserving the testing errors below given levels and suggest how the design along with all testing metrics can be generalized to the case of multiple centers.

The last part of the thesis is not directly related to the preceding parts. We present two supervised classification algorithms for real-data applications.

MULAN: A Blind and Off-Grid Method for Multichannel Echo Retrieval

Rémi Gribonval, Helena Peic Tukuljac

This paper addresses the general problem of blind echo retrieval, i.e., given M sensors measuring in the discrete-time domain M mixtures of K delayed and attenuated copies of an unknown source signal, can the echo locations and weights be recovered? This problem has broad applications in fields such as sonars, seismology, ultrasounds or room acoustics. It belongs to the broader class of blind channel identification problems, which have been intensively studied in signal processing. Existing methods in the literature proceed in two steps: (i) blind estimation of sparse discrete-time filters and (ii) echo information retrieval by peak-picking on filters. The precision of these methods is fundamentally limited by the rate at which the signals are sampled: estimated echo locations are necessary on-grid, and since true locations never match the sampling grid, the weight estimation precision is impacted. This is the so-called basis-mismatch problem in compressed sensing. We propose a radically different approach to the problem, building on the framework of finite-rate-of-innovation sampling. The approach operates directly in the parameter-space of echo locations and weights, and enables near-exact blind and off-grid echo retrieval from discrete-time measurements. It is shown to outperform conventional methods by several orders of magnitude in precision.

2018

MULAN: A Blind and Off-Grid Method for Multichannel Echo Retrieval

Rémi Gribonval, Helena Peic Tukuljac

NEURAL INFORMATION PROCESSING SYSTEMS (NIPS)2018

Multiple testing with test statistics following heavy-tailed distributions

Zhiwen Jiang

In multiple testing problems where the components come from a mixture model of noise and true effect, we seek to first test for the existence of the non-zero components, and then identify the true alternatives under a fixed significance level

\alpha

. Two parameters, namely the fraction of the non-null components

\varepsilon

and the size of the effects

\mu

, characterise the two-point mixture model under the global alternative. When the number of hypotheses

m

goes to infinity, we are interested in an asymptotic framework where the fraction of the non-null components is vanishing, and the true effects need to be sizable to be detected. Donoho and Jin give an explicit form of the asymptotic detectable boundary based on the Gaussian mixture model under the classic calibration of the parameters of the mixture model. We prove the analogous results for the Cauchy mixture distribution as an example heavy-tailed case. This requires a different formulation of the parameters, which reflects the added difficulties. We also propose a multiple testing procedure based on a filtering approach that can discover the true alternatives. Benjamini and Hochberg (BH) compare the observed

p

-values to a linear threshold curve and reject the null hypotheses from the minimum up to the last up-crossing, and prove the false discovery rate (FDR) is controlled. However, there is an intrinsic difference in heavy-tailed settings. Were we to use the BH procedure we would get a highly variable positive false discovery rate (pFDR). In our study we analyse the distribution of the

p

-values and devise a new multiple testing procedure to combine the usual case and the heavy-tailed case based on the empirical properties of the

p

-values. The filtering approach is designed to eliminate most

p

-values that are more likely to be uniform, while preserving most of the true alternatives. Based on the filtered

p

-values, we estimate the mode

\vartheta

and define the rejection region

\mathscr{R}(\vartheta, \delta)=\left[ \vartheta -\delta/2, \vartheta +\delta/2 \right]

such that the most informative

p

-values are included. The length

\delta

is chosen by controlling the data-dependent estimation of FDR at a desired level.

EPFL2021