Generalized Inverses, Wald's Method, and the Construction of Chi-Squared Tests of Fit

Abstract Wald's method constructs test statistics having chi-squared limiting distributions from estimators having nonsingular multivariate normal limiting distributions, using the inverse of the covariance matrix of the limiting distribution. This method is here extended to singular multivariate normal limiting distributions by use of generalized inverses. The generalized Wald's method is then applied to give a quite general solution to the problem of constructing chi-squared type statistics for testing fit to a parametric family when unknown parameters are estimated in various ways.

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