Invariant hypothesis testing with applications in signal processing
Statistical hypothesis testing has application in many areas of signal processing such as signal detection, classification, wireless communications, and image processing. The applications usually have unknown parameters which result in hypothesis testing problems that are composite, and for which an optimal test, such as a uniformly most powerful or minimum probability of error test, may not exist. ^ The problem addressed by this dissertation, is to apply the Neyman-Pearson theory of hypothesis testing, in which tests are derived as solutions of a stated optimum problem, to signal processing applications. In composite hypothesis testing applications, the problem itself may be invariant, in which case it is reasonable to impose the additional requirement that tests for the problem be invariant. This limits the class of tests to be considered, from which an optimal one may exist. ^ Problem invariances are identified, and tests such as the uniformly most powerful invariant (UMPI) test, which are optimal among the class of all invariant tests, are derived. One of the methods used, Stein's method, does not require the explicit identification of a maximal invariant statistic and can be applied to general problem statements for which a maximal invariant cannot be specified. Using this method, general results are obtained for the detection of signals with deterministic unknown parameters. ^ The generalized likelihood ratio test (GLRT) is commonly used in composite hypothesis testing applications. The GLRT belongs to the class of tests for which the UMPI test is optimal. The invariance of the GLRT is investigated and the connection between invariance and the desirable constant false alarm rate (CFAR) property is discussed. A new interpretation of the GLRT is obtained in which it is written in terms of the transformation group used to describe the problem invariances. Using this, an expression is found showing the UMPI test statistic to be the sum of two terms, one of which is the GLRT. It is found that the GLRT becomes UMPI as signal-to-noise ratio approaches infinity, which is shown to be a relevant property for certain signal detection applications. ^ Examples are provided to illustrate the theoretical results. ^
Statistics|Engineering, Electronics and Electrical
Joseph R Gabriel,
"Invariant hypothesis testing with applications in signal processing"
Dissertations and Master's Theses (Campus Access).