报告题目：Approximation of Power Function of Roy's Largest Root Test

报告时间：2018年10月20日，星期六，下午3：30

报告地点：数学楼408

报告人：侯志强博士，东北师范大学，研究方向为随机矩阵及其在高维统计中的应用

报告摘要：

Roy's largest root is a common test statistic in multivariate analysis, statistical signal processing and related fields. According to \cite{Anderson2003}, it is numerically known that compared with the other three tests of linear hypotheses, Roy's largest root test has the highest power under rank-one alternatives. Therefore, it is important to study the asymptotic distribution of the largest root under rank-one alternatives to obtain an estimation of the power. To the best of our knowledge, no one had solved the problem until \cite{JN2017} presented a tractable approximation of the distribution of Roy's largest root test where the alternative is of rank-one and the variance of the noise tends to zero. It is natural to ask how Roy's largest root test performs under other alternatives, for example, rank-finite alternatives. Therefore, we are more interested in the power estimates of Roy's largest root test under wider alternatives, whether its power is still higher than the other three tests of linear hypotheses.

In fact, the distribution of the largest root under rank-finite alternatives can be characterized as the distribution of the largest sample eigenvalue among several spiked eigenvalues in a general spiked eigenvalue model.  In this paper, we employ the asymptotic results of the spiked eigenvalues derived in \cite{BY2008,BY2012} and \cite{WY2017} to obtain the approximate power of Roy's largest root test, which is a more accurate and simpler method than that given by \cite{JN2017}; more importantly, our results apply to cases involving rank-finite alternatives.