Modern multivariate statistical analysis : a graduate course and handbook

Testing correlation structures has attracted extensive attention in the literature due to both its importance in real applications and several major theoretical challenges. The aim of this paper is to develop a general framework of testing correlation structures for the one-, two-, and multiple sample testing problems under a high-dimensional setting when both the sample size and data dimension go to infinity. Our test statistics are designed to deal with both the dense and sparse alternatives. We systematically investigate the asymptotic null distribution, power function, and unbiasedness of each test statistic. Theoretically, we make great efforts to deal with the non-independency of all random matrices of the sample correlation matrices. We use simulation studies and real data analysis to illustrate the versatility and practicability of our test statistics.

Test for high dimensional correlation matrices.

Jennrich Jennrich statistic is one of the existing statistics which is used for testing the equality of several independent samples of correlation matrices. The statistic is
gaining considerable importance in several areas of economics and financial markets. In these research areas, the number of variables, p, is usually larger than the sample size, n, which is known as high dimension data p > n. Subsequently, the estimation of correlation and covariance determinant will breakdown due to singularity problem. When this happens, Jennrich statistic is unable to function as the calculation involves the inversion of correlation matrix. Therefore, to resolve the aforementioned problem, this study develops an alternative statistic for testing several independent samples of correlation matrices in high dimension data. For this reason, the algebraic approach on the basis of vec operator, commutation matrix and Frobenius norm of
upper-off-diagonal elements are used to derive the new asymptotic distribution for the new alternative statistic, namely * Z statistic. Simulation study was conducted by considering different number of variables, sample sizes, and correlation shifts to evaluate the performance of the new statistic. In addition, real data on Asia Pacific currencies structure during the Tohoku earthquake were applied to validate the new * Z statistic. The power of the * Z statistic is compared with the existing Jennrich
statistic, and * T statistic through simulation study. As a result, the power of * Z statistic dominates the power of Jennrich statistic and * T statistic in all conditions, especially, when the shift in correlation matrix is at least 0.3 As a conclusion, the theoretical and simulation results are established and supported by desirable power of test. Meanwhile, investigation on real data indicates that the new alternative statistic can accommodate high dimension data.

New alternative statistic for testing several independent samples of correlation matrices in high dimension data

/pdf/forensic-intelligence-applications-in-illegal-drug-1xxnywfsq2.pdf

Forensic intelligence : applications in illegal drug trafficking

Bivariate data consist of 2 random variables that are obtained from the same population. The relationship between 2 bivariate data can be measured by correlation coefficient. A correlation coefficient computed from the sample data is used to measure the strength and direction of a linear relationship between 2 variables. However, the classical correlation coefficient results are inadequate in the presence of outliers. Therefore, this study focuses on the performance of different correlation coefficient under contaminated bivariate data to determine the strength of their relationships. We compared the performance of 5 types of correlation, which are classical correlations such as Pearson correlation, Spearman correlation and Kendall’s Tau correlation with other robust correlations, such as median correlation and median absolute deviation correlation. Results show that when there is no contamination in data, all 5 correlation methods show a strong relationship between 2 random variables. However, under the condition of data contamination, median absolute deviation correlation denotes a strong relationship compared to other methods.

/pdf/the-performance-of-different-correlation-coefficient-under-355a1nky4j.pdf

The Performance of Different Correlation Coefficient under Contaminated Bivariate Data

In this article we show that the Kullback’s statistic for testing equality of several correlation matrices may be considered a modiﬁed likelihood ratio statistic when sampling from multivariate normal populations. We derive the asymptotic null distribution of L∗ in series involving independent chisquare variables by expanding L∗ in terms of other random variables and then inverting the expansion term by term. An example is also given to exhibit the procedure to be used when testing the equality of correlation matrices using the statistic L∗.

http://www.scielo.org.co/pdf/rce/v36n2/v36n2a04.pdf

Testing Equality of Several Correlation Matrices

In this article we show that the Kullback’s statistic for testing equality of several correlation matrices may be considered a modified likelihood ratio statistic when sampling from multivariate normal populations. We derive the asymptotic null distribution of L in series involving independent chisquare variables by expanding L in terms of other random variables and then inverting the expansion term by term. An example is also given to exhibit the procedure to be used when testing the equality of correlation matrices using the statistic L .

/pdf/testing-equality-of-several-correlation-matrices-prueba-de-awfv0z9vox.pdf

Testing Equality of Several Correlation Matrices Prueba de Igualdad de Varias Matrices de Correlación

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Testing Equality of Several Correlation Matrices

Testing Equality of Several Correlation Matrices Prueba de Igualdad de Varias Matrices de Correlación