Testing for the Eigenvector Based on the Multiple Correlation Coefficient

We propose a novel method for testing the hypothesis of an eigenvector based on the exact distribution of the multiple correlationcoefficientunderanormalpopulation. Inparticular, wediscussbothnonsingularandsingularcases, addressing the relationship between sample size and the number of variables. The proposed test has the advantage of being invariant to the ordering of the target eigenvector, focusing only on whether the target vector is an eigenvector. The ordering of the eigenvector is determined by the minimum angle between the target vector and the sample eigenvector. Furthermore, we demonstrated that type I errors is exactly controlled at a particular significance level, and the power under the specified alternative hypothesis can be calculated by the Gauss hypergeometric function in the nonsingular case. Our simulation studies confirm that the empirical distribution of the test statistic is in agreement with theoretical distribution.