Hypothesis testing in a generic nesting framework with general population distributions

Nested parameter spaces, either in the null or alternative hypothesis, constitute a guarantee for improving the performance of the tests, however in the existing literature on order restricted inference they have been usually skipped for being studied in detail. Divergence based divergence measures provide a flexible tool for creating meaningful test-statistics, which usually contain the likelihood ratio-test statistics as special case. The existing literature on hypothesis testing with inequality constraints using phidivergence measures, is centered in a very specific models with multinomial sampling. The contribution of this paper consists in extending and unifying widely the existing work: new families of test-statistics are presented, valid for nested parameter spaces containing either equality or inequality constraints and general distributions for either single or multiple populations are considered.