  We investigate hypothesis testing in nonparametric additive models estimated
using simplified smooth backfitting (Huang and Yu, Journal of Computational and
Graphical Statistics, \textbf{28(2)}, 386--400, 2019). Simplified smooth
backfitting achieves oracle properties under regularity conditions and provides
closed-form expressions of the estimators that are useful for deriving
asymptotic properties. We develop a generalized likelihood ratio (GLR) and a
loss function (LF) based testing framework for inference. Under the null
hypothesis, both the GLR and LF tests have asymptotically rescaled chi-squared
distributions, and both exhibit the Wilks phenomenon, which means the scaling
constants and degrees of freedom are independent of nuisance parameters. These
tests are asymptotically optimal in terms of rates of convergence for
nonparametric hypothesis testing. Additionally, the bandwidths that are
well-suited for model estimation may be useful for testing. We show that in
additive models, the LF test is asymptotically more powerful than the GLR test.
We use simulations to demonstrate the Wilks phenomenon and the power of these
proposed GLR and LF tests, and a real example to illustrate their usefulness.
