Strongly Consistent Determination of the Rank of Matrix

In this paper, we develop methods of the determination of the rank of random matrix. Using the matrix perturbation theory to construct or find a suitable bases of the kernel (null space) of the matrix and to determine the limiting distribution of the estimator of the smallest singular values. We propose a new rank test for an unobserved matrix for which a root-N-consistent estimator is available and construct a Wald- type test statistic (generalized Wald test). The test, based on matrix perturbation theory, enable to determine how many singular values of the estimated matrix are insignificantly different from zero and we fully characterise the asymptotic distribution of the generalized Wald statistic under the most general conditions. We show that it is chi- square distribution under the null. In particular case, when the asymptotic covariance matrix has a Kronecker product form, the test statistic is equivalent to likelihood ratio test statistic and to Multiplier Lagrange test statistic. Two approaches to be considered are sequential testing strategy and information theoretic criterion. We establish a strongly consistent of the determination of the rank of matrix using the two approaches.Rank Testing; Matrix Perturbation Theory; Rank Estimation; Subspace Methods; Singular Value Decomposition; Weighting Matrices; Sequential Testing Strategy; Information Theoretic Criterion.

Application of Adaptive Νeuro-Fuzzy Inference System in Interest Rates Effects on Stock Returns

In the current study we examine the effects of interest rate changes on common stock returns of Greek banking sector. We examine the Generalized Autoregressive Heteroskedasticity (GARCH) process and an Adaptive Neuro-Fuzzy Inference System (ANFIS). The conclusions of our findings are that the changes of interest rates, based on GARCH model, are insignificant on common stock returns during the period we examine. On the other hand, with ANFIS we can get the rules and in each case we can have positive or negative effects depending on the conditions and the firing rules of inputs, which information is not possible to be retrieved with the traditional econometric modelling. Furthermore we examine the forecasting performance of both models and we conclude that ANFIS outperforms GARCH model in both in-sample and out-of-sample periods

Computation of generalized inverses using Php/MySql environment

The main aim of this paper is to develop a client/server-based model for computing the weighted Moore-Penrose inverse using the partitioning method as well as for storage of generated results. The web application is developed in the PHP/MySQL environment. The source code is open and free for testing by using a web browser. Influence of different matrix representations and storage systems on the computational time is investigated. The CPU time for searching the previously stored pseudo-inverses is compared with the CPU time spent for new computation of the same inverses.Comment: International Journal of Computer Mathematics, Volume 88, Issue 11, 201

Slepian functions and their use in signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla

Rank Test Based On Matrix Perturbation Theory

In this paper, we propose methods of the determination of the rank of matrix. We consider a rank test for an unobserved matrix for which an estimate exists having normal asymptotic distribution of order N1/2 where N is the sample size. The test statistic is based on the smallest estimated singular values. Using Matrix Perturbation Theory, the smallest singular values of random matrix converge asymptotically to zero in the order O(N-1) and the corresponding left and right singular vectors converge asymptotically in the order O(N-1/2). Moreover, the asymptotic distribution of the test statistic is seen to be chi-squared. The test has advantages over standard tests in being easier to compute. Two approaches are be considered sequential testing strategy and information theoretic criterion. We establish a strongly consistent of the determination of the rank of matrix using both the two approaches. Some economic applications are discussed and simulation evidence is given for this test. Its performance is compared to that of the LDU rank tests of Gill and Lewbel (1992) and Cragg and Donald (1996).Rank Testing; Matrix Perturbation Theory; Rank Estimation; Singular Value Decomposition; Sequential Testing Procedure; Information Theoretic Criterion.