Constrained Parameters in Applications: Review of Issues and Approaches

This review article provides an introduction to statistical issues that arise when some statistical model parameters are constrained. This often happens in applications, in particular in testing for variance components (e.g., genomics) and construction of one-sided confidence intervals (e.g., environmental risk analysis). Heuristic explanations are provided, and a number of general and recent statistical results that appeared in statistical literature are summarized for use in applications. Simulation results are shown for illustration of consequences of ignoring parameters on the boundary. Special attention is paid to likelihood ratio tests, but other approaches to confidence interval construction, such as Wald, bootstrap, and Bayesian are also briefly discussed. This paper presents examples from the risk assessment field and genomics, but all conclusions apply to whenever one-sided testing is conducted. Recommendations are provided for dealing with parameters on the boundary for a range of situations.