There is a well-developed statistical inference theory for classical one-dimensional models. However, many important inference problems are still unanswered for high dimensional models where the dimension or the number of involved parameters can be much larger than the sample size. In the thesis, we solve three problems on hypothesis testing for high dimensional data based on quadratic form test statistics. In the first work, we investigate high dimensional two sample mean test for independent observations. The theoretical contribution is the new distributional theory of quadratic forms of mean vectors by Gaussian approximation. The primary methodological contribution is a new half sampling calibration procedure as a model-free tool for valid inference which are not susceptible to model choice or model misspecification. In the second work, we test general linear hypotheses of the multivariate linear regression model with independent observations. We study asymptotic behaviors of the conventional MANOVA test statistic and find its asymptotic distribution is dichotomous in high dimensions, which creates much difficulty to use in practice. We solved this open problem by proposing a new U type test statistic and laying a theoretical foundation for high dimensional MANOVA. In the third work, we study the portmanteau test for high dimensional white noises. It is a popular choice to test for temporal dependence in low dimensional processes. The methodology and theory in the high dimensional case is less investigated. We propose a test statistic that is workable for high dimensional processes. A key technical component of our analysis is a new Gaussian approximation result for quadratic forms of high dimensional martingale differences, which is of independent interest in the statistical inference of high dimensional time series.




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