On the Maximal Invariant Statistic for Adaptive Radar Detection in Partially-Homogeneous Disturbance with Persymmetric Covariance

This letter deals with the problem of adaptive signal detection in partially-homogeneous and persymmetric Gaussian disturbance within the framework of invariance theory. First, a suitable group of transformations leaving the problem invariant is introduced and the Maximal Invariant Statistic (MIS) is derived. Then, it is shown that the (Two-step) Generalized-Likelihood Ratio test, Rao and Wald tests can be all expressed in terms of the MIS, thus proving that they all ensure a Constant False-Alarm Rate (CFAR).Comment: submitted for journal publicatio

Detecting a stochastic background of gravitational waves with non-standard polarizations

In this thesis work I consider the detection of a stochastic background of gravitational waves (SGWB) produced in the context of a generic theory of gravity. I will show in the second chapter that most theories admit solutions in terms of gravitational waves(GWs); these may differ in their propagation speed or, most relevant for the present work, in their polarization modes. For example, it is well-known that many theories of gravity, obtained for example as low-energy limits of string theories, predict a propagating scalar mode of polarization in addition to the usual tensor ones of General Relativity (GR). I will present a thorough discussion of the classification of GWs polarizations according to the E(2) scheme, which comprises the analysis of the non-vanishing components of the Riemann tensor as measured by a locally inertial observer, and thereafter the interaction of GWs with a detector. In the third chapter I present a comprehensive characterization of an SGWB with non-standard polarizations in terms of the detector response to it. Some considerations are made on the most general form that the corresponding signal may have according to alternative theories of gravity and the production mechanisms described before. It follows then the discussion about some “first order approximations” that will be useful for its study in these preliminary phases. The aim of the present work is to relax some of the usual constraints adopted in standard literature to include also the possibility of non-standard polarizations, and open up to a new more general class of possible SGWBs. In the second part of this chapter I construct and study an optimal detection algorithm for a generic SGWB. In particular, I will give importance to a procedure that is as less dependent as possible on the details of the model; for example, I begin without introducing any assumptions about the shape of the power spectrum densities of the stochastic signal. Only later they will be considered cases where it becomes necessary to include further assumptions, for example, in order to obtain some estimates on the parameters characterizing a certain model. This choice is motivated by the desire of understanding how much sensitivity is lost when a not well defined model is available, which is even more true when we extend the framework to include also alternative theories. In this sense, the present work is meant as an upgrade to those already present in literature and commonly adopted in the standard data analysis for the research of an SGWB. I will recover the known results from the literature adding only later some further assumptions. This treatment has some advantages over the standard one, in particular from a theoretical point of view. Finally, in Chapter 4, I make use of the proposed algorithm to study real data from Virgo and LIGO. The current upper limit on the intensity of the (standard) SGWB, published in 2009, is reconsidered. As it was reasonable to expect, it is not possible to improve this limit or, even more so, to perform a detection of a non-standard SGWB. Anyway, the upper limits on the non-standard polarization modes are computed and compared with the standard one. Also, several related quantities are computed and analysed from the point of view of the detection. The important news comes from the study of the predicted sensitivities that will be achieved by the advanced detectors with the scheduled upgrades (2015-2021). I will show that these sensitivities will become good enough to test several mechanisms of production of an SGWB, both of cosmological and of astrophysical origin, or at least to determine further upper limits on them. Therefore, we can expect that the tools provided by the study of GWs within an SGWB will become worth for testing alternative theories of gravity, as well as early Universe cosmological models and astrophysical ones

On the invariance, coincidence, and statistical equivalence of the GLRT, Rao test, and wald test

Three common techniques to discriminate between alternatives in a binary hypothesis testing problem are: the generalized likelihood ratio test (GLRT), the Rao test, and the Wald test. In this paper, we investigate some characteristics of the corresponding decision statistics and provide their expressions for some problems of particular interest in statistical signal processing. First of all, we focus on the invariance of the Rao and Wald tests with respect to transformations leaving the testing problem unaltered. Then, we introduce necessary and sufficient conditions in order for their decision statistics to coincide with twice the logarithm of the GLRT statistic. Finally, we present some detection problems, usually encountered in practical signal processing applications, where the decision variables of the three quoted tests are equivalent, namely related by strictly monotonic transformations. © 2006 IEEE