A Note on the Wald, Lagrangian Multiplier and Likelihood Ratio Tests when the Information Matrix is Singular | ||||
| The Egyptian Statistical Journal | ||||
| Article 5, Volume 32, Issue 2, December 1988, Page 135-152 | ||||
| Document Type: Original Article | ||||
| DOI: 10.21608/esju.1988.316559 | ||||
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| Authors | ||||
| Abdalla T. El-Helbawy; Tawfik Hassan | ||||
| Abstract | ||||
| Modified formulas for the Wald and Lagrangian multiplier statistics are introduced and considered together with the likelihood ratio statistic for testing a typical null hypothesis Ho 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 by Ho whether the information matrix is singular or nonsingular. The results of this paper include a generalization of the results of Silvey (1959) concerning the equivalence of the Wald, Lagrange multiplier and likelihood ratio tests to the case of singular information matrices. A numerical example, using the data of the multinomial experiment given in Example 3 of Aitchison and Silvey (1960), is given to compare the obtained values of our modified test statistics with the corresponding values of the test statistics obtained in that reference of Aitchison and Sivley. | ||||
| 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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