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In this paper, we propose nonparametric locally and asymptotically optimal tests for the problem of detecting randomness in the coefficient of a linear regression model (in the Le Cam and H´ajek sense). That is, the problem of testing the null hypothesis of a Standard Linear Regression (SLR) model against the alternative of a Random Coefficient Regression (RCR) model. A Local Asymptotic Normality (LAN) property, which allows for constructing locally asymptotically optimal tests, is therefore established for RCR models in the vicinity of SLR ones. Rank and signed-rank based versions of the optimal parametric tests are provided. These tests are optimal, most powerful and valid under a wide class of densities. A Monte-Carlo study confirms the performance of the proposed tests.

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