Asymptotic Properties of ML Estimators of Parameters Subject to Linear Constraints Under a Class of Local Alternatives when the Information Matrix is Singular

SYNOPTIC ABSTRACT

Tests of hypotheses stated in terms of linear constraints on the parameters involved in the distribution functions of random variables are considered. A sequence of local alternatives and a set of assumptions incorporating the possibility that the information matrix is singular are presented. Restricted likelihood equations are derived and existence of likelihood estimators emerging from solving these equations is demonstrated. Asymptotic properties of the obtained estimators, together with the asymptotic noncentral chi-square distribution of the likelihood ratio statistic −2 log λ, are developed. Numerical examples illustrate the applications of the results.

Key Words and Phrases:

• linear constraints
• restricted likelihood equations
• asymptotic properties
• likelihood ratio tests
• noncentrality parameter