A Note on Wald's Type Chi-squared Tests of Fit

Abstract

The Wald's method for constructing chi-squared tests of fit has been formulated more accurately. It is shown that Wald's type statistics will follow the central chi-squared distribution if and only if the limit covariance matrix of standardized frequencies will not depend on unknown parameters. Several examples that illustrate this important fact are presented. In particular, it is shown that the goodness-of-fit statistic developed by Moore and Stubblebine does not follow the chi-squared limit distribution, and, hence, cannot be used for testing multivariate normality.

Keywords:

• Khatri's lemma
• Matrix generalized inverse
• Maximum likelihood estimator (MLE)
• Method of moments estimator (MME)
• Modified chi-squared tests
• Standardized frequencies
• Wald's method.

Mathematics Subject Classification:

• 62F03
• 62F05.

Acknowledgments

The author is grateful to Natalie Pya and Rashid Makarov for producing Monte Carlo simulations used in Fig. 1.