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Asymptotic Expansions and Bootstrap Approximations in Factor Analysis

By Masanori Ichikawa and Sadanori Konishi

Abstract

We derive asymptotic expansions for the distributions of the normal theory maximum likelihood estimators of unique variances and uniquenesses (standardized unique variances) in the factor analysis model. Asymptotic expansions are given for the distributions of non-Studentized and also Studentized statistics to construct accurate confidence intervals. In the case of Studentized statistics, we investigate the accuracy of the asymptotic approximations to the exact distributions that are determined by Monte Carlo simulations. The results show that, compared with normal approximations, the asymptotic expansions generally improve the accuracy of the approximations in the tail area except for the cases of the uniqueness estimators whose true values are close to their upper bounds unity. We also compare three types of confidence intervals that are based on the distributions of the Studentized statistics; each of which employs normal approximation, asymptotic expansion of the percentile points of the Studentized statistic, and further modification using the bootstrap. The results show that while the first type of confidence intervals were far from equal-tailed, the latter two achieved better balance in both sides.factor analysis asymptotic expansion Edgeworth expansion Cornish-Fisher expansion bootstrap confidence interval

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