The advantage of decomposing elaborate hypotheses on covariance matrices into conditionally independent hypotheses in building near-exact distributions for the test statistics

AbstractThe aim of this paper is to show how the decomposition of elaborate hypotheses on the structure of covariance matrices into conditionally independent simpler hypotheses, by inducing the factorization of the overall test statistic into a product of several independent simpler test statistics, may be used to obtain near-exact distributions for the overall test statistics, even in situations where asymptotic distributions are not available in the literature and adequately fit ones are not easy to obtain

Multivariate Analysis - A Great Tool for a Multivariate World

In this short report the author tries to introduce in simple terms some Multivariate Analysis techniques and show their usefulness in the analysis of econometric data, namely in studies which aim at studying and comparing different regions inside a given country.publishersversionpublishe

Near-Exact Distributions – Needing Them and Building Them

Near-exact distributions are asymptotic distributions built using a different concept and technique in what concerns the approximation of the distribution of given statistics whose exact distribution has a complex structure and expression. In the present paper the author introduces, in simple terms and through two examples, the near-exact distributions as an alternative to common asymptotic distributions. First are introduced the characteristics and the general building process of these distributions. Then, through a couple of examples, a very simple first one and a more elaborate second one, we try to illustrate how in practice these distributions may be developed and to show their very good performance.As distribuições quase-exactas são distribuições assimptóticas construídas sobre uma abordagem diferente no que diz respeito ao princípio e à técnica da aproximação da distribuição de estatísticas cuja distribuição exacta tem uma estrutura e expressão complexas. No presente artigo o autor apresenta, em termos simples e através de dois exemplos, as distribuições quase-exactas como alternativa vantajosa às usuais distribuições assimptóticas. Em primeiro lugar são apresentadas as características e forma de construção destas distribuições e daí intuídas as suas vantagens em relação às usuais distribuições assimptóticas. Depois, através de dois exemplos, um primeiro muito simples e um segundo mais elaborado, tenta-se ilustrar como se constroem na prática estas distribuições e mostrar o seu excelente desempenho

Near-Exact Distributions for Likelihood Ratio Statistics Used in the Simultaneous Test of Conditions on Mean Vectors and Patterns of Covariance Matrices

The authors address likelihood ratio statistics used to test simultaneously conditions on mean vectors and patterns on covariance matrices. Tests for conditions on mean vectors, assuming or not a given structure for the covariance matrix, are quite common, since they may be easily implemented. But, on the other hand, the practical use of simultaneous tests for conditions on the mean vectors and a given pattern for the covariance matrix is usually hindered by the nonmanageability of the expressions for their exact distribution functions. The authors show the importance of being able to adequately factorize the c.f. of the logarithm of likelihood ratio statistics in order to obtain sharp and highly manageable near-exact distributions, or even the exact distribution in a highly manageable form. The tests considered are the simultaneous tests of equality or nullity of means and circularity, compound symmetry, or sphericity of the covariance matrix. Numerical studies show the high accuracy of the near-exact distributions and their adequacy for cases with very small samples and/or large number of variables. The exact and near-exact quantiles computed show how the common chi-square asymptotic approximation is highly inadequate for situations with small samples or large number of variables

One-Bit Spectrum Sensing for Cognitive Radio

Spectrum sensing in cognitive radio necessitates effective monitoring of wide bandwidths, which requires high-rate sampling. Traditional spectrum sensing methods employing high-precision analog-to-digital converters (ADCs) result in increased power consumption and expensive hardware costs. In this paper, we explore blind spectrum sensing utilizing one-bit ADCs. We derive a closed-form detector based on Rao's test and demonstrate its equivalence with the second-order eigenvalue-moment-ratio test. Furthermore, a near-exact distribution based on the moment-based method, and an approximate distribution in the low signal-to-noise ratio (SNR) regime with the use of the central limit theorem, are obtained. Theoretical analysis is then performed and our results show that the performance loss of the proposed detector is approximately $2$ dB ($\pi/2$) compared to detectors employing $\infty$-bit ADCs when SNR is low. This loss can be compensated for by using approximately $2.47$ ($\pi^2/4$) times more samples. In addition, we unveil that the efficiency of incoherent accumulation in one-bit detection is the square root of that of coherent accumulation. Simulation results corroborate the correctness of our theoretical calculations

Permutation test for the structure of a covariance matrix

Scope and Method of Study: Many statistical procedures, such as repeated measures analysis, time-series, structural equation modeling, and factor analysis, require an assessment of the structure of the underlying covariance matrix. The classical parametric method of testing such a hypothesis involves the use of a likelihood ratio test (LRT). These tests have many limitations, including the need for very large sample sizes and the requirement of a random sample from a multivariate normal population. The LRT is also undefined for cases in which the sample size is not greater than the number of repeated measures. In such situations, researchers could benefit from a non-parametric testing procedure. In particular, permutation tests have no distributional assumptions and do not require random samples of any particular size. This research involves the development and analysis of a permutation/randomization test for the structure of a covariance matrix. Samples of various sizes and number of measures on each subject were simulated from multiple distributions. In each case, the type I error rates and power were examined.Findings and Conclusions: When testing for sphericity, compound symmetry, type H structure, and serial correlation, the LRT clearly performs best with regard to type I error rates for normally distributed data, but for uniform data, it is too conservative, and for double exponential data, it results in extremely large type I error rates. The randomization test, however, is consistent regardless of the data distribution and performs better than the LRT, in most cases, for non-normally distributed data. In most situations, the LRT is more powerful than the randomization test, but the power of the randomization test is comparable to that of the LRT in many situations

Spectrum Sensing Algorithms for Cognitive Radio Applications

Future wireless communications systems are expected to be extremely dynamic, smart and capable to interact with the surrounding radio environment. To implement such advanced devices, cognitive radio (CR) is a promising paradigm, focusing on strategies for acquiring information and learning. The first task of a cognitive systems is spectrum sensing, that has been mainly studied in the context of opportunistic spectrum access, in which cognitive nodes must implement signal detection techniques to identify unused bands for transmission. In the present work, we study different spectrum sensing algorithms, focusing on their statistical description and evaluation of the detection performance. Moving from traditional sensing approaches we consider the presence of practical impairments, and analyze algorithm design. Far from the ambition of cover the broad spectrum of spectrum sensing, we aim at providing contributions to the main classes of sensing techniques. In particular, in the context of energy detection we studied the practical design of the test, considering the case in which the noise power is estimated at the receiver. This analysis allows to deepen the phenomenon of the SNR wall, providing the conditions for its existence and showing that presence of the SNR wall is determined by the accuracy of the noise power estimation process. In the context of the eigenvalue based detectors, that can be adopted by multiple sensors systems, we studied the practical situation in presence of unbalances in the noise power at the receivers. Then, we shift the focus from single band detectors to wideband sensing, proposing a new approach based on information theoretic criteria. This technique is blind and, requiring no threshold setting, can be adopted even if the statistical distribution of the observed data in not known exactly. In the last part of the thesis we analyze some simple cooperative localization techniques based on weighted centroid strategies

Review of Top Quark Physics

We present an overview of Top Quark Physics - from what has been learned so far at the Tevatron, to the searches that lie ahead at present and future colliders. We summarize the richness of the measurements and discuss their possible impact on our understanding of the Standard Model by pointing out their key elements and limitations. When possible, we discuss how the top quark may provide a connection to new or unexpected physics.Comment: 84 pp. With permission from the Annual Review of Nuclear & Particle Science. Final version of this material is scheduled to appear in the Annual Review of Nuclear & Particle Science Vol. 53, to be published in December 2003 by Annual Reviews (http://www.annualreviews.org