Random coefficient regression (RCR) models are the regression versions of random effects models in analysis of variance and panel data analysis. Optimal detection of the presence of random coefficients (equivalently, optimal testing of the hypothesis of constant regression coefficients) has been an open problem for many years. The simple regression case has been solved recently and the multiple regression case is considered here. The latter poses several theoretical challenges: (a) a nonstandard ULAN structure, with log-likelihood gradients vanishing at the null; (b) cone-shaped alternatives under which traditional optimality concepts are no longer adequate; (c) nuisance parameters that are not identified under the null but have a significant impact on local powers. We propose a new (local and asymptotic) concept of optimality for this problem and, for specified error densities, derive parametrically optimal procedures. A suitable modification of the Gaussian version of the latter is shown to qualify as a pseudo-Gaussian test. The asymptotic performances of those pseudo-Gaussian tests, however, are quite poor under skewed and heavy-tailed densities. We therefore also construct rank-based tests, possibly based on data-driven scores, the asymptotic relative efficiencies of which are remarkably high with respect to their pseudo-Gaussian counterparts.
"Optimal pseudo-Gaussian and rank-based random coefficient detection in multiple regression." Electron. J. Statist. 14 (2) 4207 - 4243, 2020. https://doi.org/10.1214/20-EJS1770