Additive models, as a natural generalization of linear regression, have played an important role in studying nonlinear relationships. Despite of a rich literature and many recent advances on the topic, the statistical inference problem in additive models is still relatively poorly understood. Motivated by the inference for the exposure effect and other applications, we tackle in this paper the statistical inference problem for $f_1(x_0)$ in additive models, where $f_1$ denotes the univariate function of interest and $f_1(x_0)$ denotes its first order derivative evaluated at a specific point $x_0$. The main challenge for this local inference problem is the understanding and control of the additional uncertainty due to the need of estimating other components in the additive model as nuisance functions. To address this, we propose a decorrelated local linear estimator, which is particularly useful in reducing the effect of the nuisance function estimation error on the estimation accuracy of $f_1(x_0)$. We establish the asymptotic limiting distribution for the proposed estimator and then construct confidence interval and hypothesis testing procedures for $f_1(x_0)$. The variance level of the proposed estimator is of the same order as that of the local least squares in nonparametric regression, or equivalently the additive model with one component, while the bias of the proposed estimator is jointly determined by the statistical accuracies in estimating the nuisance functions and the relationship between the variable of interest and the nuisance variables. The method is developed for general additive models and is demonstrated in the high-dimensional sparse setting.
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