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Abstract:
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In this article, we proposed a U-statistic based approach for testing the mean of a high dimensional stationary time series. Under the framework of high dimensional linear processes, we derived the asymptotic normality of our U-statistic with the convergence rate dependent upon the order of the Frobenius norm of the long run variance matrix. To avoid the bandwidth choice involved in the consistent long run variance estimation, we opt to use the self-normalized approach and show that the corresponding recursive process converges to a time-changed Brownian motion. This is the first time self-normalization is used for the inference of a parameter of growing dimension in a high-dimensional setting. Interesting, we do not have to know the convergence rate of the Froenious norm of long run variance matrix, since that is canceled out in our SN-based test statistic. Our theory also allow the dimension to grow as either a polynomial or exponential function of sample size, thus accommodating a wide range of dimensions. Simulation studies are conducted to compare with the recently proposed maximum type tests, and the empirical size is quite accurate.
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