CHI-SQUARE APPROXIMATIONS FOR EIGENVALUE DISTRIBUTIONS AND CONFIDENTIAL INTERVAL CONSTRUCTION ON POPULATION EIGENVALUES

Abstract

In this paper, first we consider the chi-square approximation of Takemura & Sheena (2005) and extend it to derive an approximate distribution of higher numerical precision. Furthermore, we demonstrate that this extension provides almost the same result as that found by Sugiyama (1972) for the distribution of the largest eigenvalue. We then consider the confidence interval of each eigenvalue and perform a numerical comparison between Takemura and Sheena's chi-square approximation and the approximated distribution found with the extended method to demonstrate that the latter has higher accuracy. Next, we consider the simultaneous confidence interval for all population eigenvalues discussed by Anderson (1965). Anderson employed the chi-square approximation of the largest and of the smallest eigenvalues of a Wishart matrix to construct the simultaneous confidence interval. This method can be interpreted as a different viewpoint of employing Takemura and Sheena's results, namely that the distributions of the largest and of the smallest eigenvalue of a Wishart matrix can be approximated with a chi-square distribution. Accordingly, an extension of Takemura & Sheena (2005) can express the simultaneous confidence interval of all population eigenvalues obtained by the procedure of Anderson (1965). When the proposed approximation distribution is used, one must establish the confidence intervals using all the estimate eigenvalues of the population; even though the issue of correcting the bias in the estimates must subsequently be addressed, ultimately, our proposed simultaneous confidence interval is reasonably good in terms of having type I errors that are closer to 5%.