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On the wald,lagrangian multiplier and likelihood ratio tests when the information matrix is singular
Authors:Abdalla T. El-Helbawy  Tawfik Hassan
Affiliation:(1) Department of Mathematics, Faculty of Science, Helwan University, Cairo, Egypt
Abstract:Summary Modified formulas for the Wald and Lagrangian multiplier statistics are introduced and considered together with the likelihood ratio statistics for testing a typical null hypothesisH 0 stated in terms of equality constraints. It is demonstrated, subject to known standard regularity conditions, that each of these statistics and the known Wald statistic has the asymptotic chi-square distribution with degrees of freedom equal to the number of equality constraints specified byH 0 whether the information matrix is singular or nonsingular. The results of this paper include a generalization of the results of Sively (1959) concerning the equivalence of the Wald, Lagrange multiplier and likelihood ratio tests to the case of singular information matrices.
Keywords:Likelihood estimation subject to equality constraints  Wald test  Lagrangian multiplier test  likelihood ratio test  modified Wald and Lagrangian multiplier statistics  equivalence of tests
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