Non-Standard Tests through a Composite Null and Alternative in Point-Identified Parameters

Abstract We propose a new approach to statistical inference on parameters that depend on population parameters in a non-standard way. As examples we consider a parameter that is interval identified and a parameter that is the maximum (or minimum) of population parameters. In both examples we transform the inference problem into a test of a composite null against a composite alternative hypothesis involving point identified population parameters. We use standard tools in this testing problem. This setup substantially simplifies the conceptual basis of the inference problem. By inverting the Likelihood Ratio test statistic for the composite null and composite alternative inference problem, we obtain a closed form expression for the confidence interval that does not require any tuning parameter and is uniformly valid. We use our method to derive a confidence interval for a regression coefficient in a multiple linear regression with an interval censored dependent variable.