The first part of this article investigates the joint behaviour of the maxi
mum 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 likeli
hood ratio approach. This work is an extention of the results in Silvey (19
59), El Barmi and Dykstra (1995) and Silvapulle (1994), and is closely conn
ected with the work of Chernoff (1954); Robertson et al. (1988) and Self an
d Liang (1987). We show that the limiting distribution of the likelihood ra
tio test statistic under quite general conditions is of chi-bar square type
and give an expression for the weighting values.