Likelihood ratio test against a set of inequality constraints

The first part of this article investigates the joint behaviour of the maximum likelihood estimators and lagrange multipliers corresponding to a null and alternative hypotheses defined by equality constraints. The results are then used to test against a set of inequality constraints using the likelihood ratio approach. This work is an extention of the results in Silvey (1959), El Barmi and Dykstra (1995) and Silvapulle (1994), and is closely connected with the work of Chernoff (1954); Robertson et al. (1988) and Self and Liang (1987). We show that the limiting distribution of the likelihood ratio test statistic under quite general conditions is of chi-bar square type and give an expression for the weighting values.