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A brief introduction to detection problems and illustration

Abstract
In this paper, some basics concepts in statistical approaches for detection of signals embedded in noise are briefly reviewed. Though most material presented in the first part of the paper is well documented in the litterature (see references), some effort is made to provide a self-contained and concise introduction to the subject. The importance and relevance of the likelihood ratio is highlighted from a Bayesian formulation of the problem; the optimal (in the sense of maximized detection probability for given false alarm rates) Neyman Pearson test and likelihood ratio test are discussed in the general framework of test performance studies. Therefore, the usefullness of the ROCs (Receiver Operating Characteristics) is illustrated on simple examples. When some unknown parameters must be taken into account, extension of the previous approaches are mentionned, with some emphasis put on the GLRT (generalized likelihood ratio test). Most results introduced in this tutorial are then applied and discussed in the framework of the extra-solar planet detection problem. A observation model involving Poisson random variables is studied. A thorough study in performed, and the Gaussian asymptotics are discussed in the case where all parameters of the model are assumed. In a more realistic situation where some parameters must be estimated, the GLRT is derived and its performances are evaluated by Monte Carlo simulation.