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Semiparametric regressions can be used to model data when covariables and the response variable have a nonlinear relationship. In this work, we propose three flexible regression models for bimodal data called the additive, additive partial and semiparametric regressions, basing on the  odd log-logistic generalized inverse Gaussian  distribution under three types of penalized smoothers, where the main idea is not to confront the three forms of smoothings but to show the versatility of the distribution with three types of penalized smoothers. We present several Monte Carlo simulations carried out for different configurations of the parameters and some sample sizes to verify the precision of the penalized maximum-likelihood estimators. The usefulness of the proposed regressions is proved empirically through three applications to climatology, ethanol and air quality data.

The semiparametric regression model for bimodal data with different penalized smoothers applied to climatology, ethanol and air quality data