Random coefficient regressions: parametric goodness-of-fit tests

Random coefficient regression models have been applied in different fields during recent years and they are a unifying frame for many statistical models. Recently, Beran and Hall (Ann. Statist. 20 (1992) 1970) raised the question of the nonparametric study of the coefficients distribution. Nonparametric goodness-of-fit tests were considered in Delicado and Romo (Ann. Inst. Statist. Math. 51 (1999) 125). In this nonparametric framework, the study of parametric families for the coefficient distributions was started by Beran (Ann. Inst. Statist. Math. (1993) 639). Here we propose statistics for parametric goodness-of-fit tests and we obtain their asymptotic distributions. Moreover, we construct bootstrap approximations to these distributions, proving their validity. Finally, a simulation study illustrates our results. (C) 2002 Elsevier B.V. All rights reserved.