High Dimensional Time Series: Mean Vector Testing and Testing for Autocorrelation Matrices

Multivariate analysis has undergone radical changes in the recent past with the advent of the so-called ultra high-dimensional data sets. Standard procedures cannot be applied for analysis of such data sets as they are all developed based on the assumption that the sample size is larger than the dimension of the data. Two different families of tests have been proposed so far for mean vector testing in high-dimensional case, but they work only when the observations are assumed to be independently and identically distributed. We propose a new testing procedure when the observations are dependent. Asymptotic normality of the proposed test statistic is derived under the assumption that the data is a realization of a M-dependent strictly stationary process. The proposed test is also extended to the two sample case. Another aspect of multivariate analysis is testing independence of the variates. Likelihood ratio based tests for testing independence of variates assume independence amongst the observations. An extension of this problem to the time series setting is considered, with emphasis on testing the equality of correlation matrices at lag zero. A Wald-type test statistic is proposed for testing equality of correlation matrices between two dependent groups. Construction of a MANOVA model for Resting State Networks(RSN) in brain imaging is discussed. The derived results are applied to validate the reproducibility and reliability of RSNs and compare several data pre-processing techniques. In multiple testing, estimation of the number of hypotheses accepted is of importance in certain problems. Measuring the proportion of accepted nulls gives a measure of sparsity of the noise. When the number of hypotheses to be tested is in millions, testing each hypothesis can be disregarded to estimate the proportion of accepted hypotheses. We study the properties of an empirical characteristic function based estimator for exponentially distributed signals. An iterative bias correction procedure is applied to improve the performance of the estimator. A non-parametric approach based on level set estimators is proposed