Over the last two decades inference problems on high-dimensional data that arise in finance, genetics and information technology have gained huge momentum. In this work, the main focus will be on developing bootstrap testing procedures under high dimensional set up for the following two hypotheses testing problems.i) High-dimensional Multivariate Analysis of Variance ii) Testing the equality of two covariance matrices in the two sample set up. The statistics considered for testing are infinity norm based statistics over either weighted sums or differences across various samples. We provide Gaussian approximation results for normalized sums of high dimensional random vectors and U-statistics under some weak conditions on moments and tails of their marginal distributions. The obtained results are free from the assumption of sparsity and correlation structures among the components of the random vectors. For the implementation of these tests, we develop multiplier bootstrap and jackknifed multiplier bootstrap procedures. These newly developed bootstrap techniques ensure first order accuracy of the asymptotic level and power of the formulated tests, enhancing their applicability. We also provide consistency of the proposed test against both fixed and local alternatives.
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