Alternative test criteria in covariance structure analysis: A unified approach

In the context of covariance structure analysis, a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided. The discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal, and allows the null and alternative hypothesis to be only approximations of the true model. Also, the equivalent of the information matrix, and the asymptotic covariance matrix of the vector of summary statistics, are allowed to be singular. When the fitting function is not asymptotically optimal, test statistics which have asymptotically a chi-square distribution are developed as a natural generalization of more classical ones. Issues relevant for power analysis, and the asymptotic theory of a testing related statistic, are also investigated.This research has been supported by the U.S.-Spanish Joint Committee for Cultural and Educational Cooperation, grant number V-B.854020. The author wishes to express his gratitude to P. M. Bentler who provided very helpful suggestions and research facilities—with an stimulating working environment—at the University of California, Los Angeles, where this work was undertaken. Thanks are also due to W. E. Saris who provided very valuable comments to earlier versions of this paper. Finally, it has also to be acknowledged the editor's and reviewers suggestions which led to substantial improvements of this article.