  We propose two semiparametric versions of the debiased Lasso procedure for
the model $Y_i = X_i\beta_0 + g_0(Z_i) + \epsilon_i$, where $\beta_0$ is high
dimensional but sparse (exactly or approximately). Both versions are shown to
have the same asymptotic normal distribution and do not require the minimal
signal condition for statistical inference of any component in $\beta_0$. Our
method also works when $Z_i$ is high dimensional provided that the function
classes $E(X_{ij} |Z_i)$s and $E(Y_i|Z_i)$ belong to exhibit certain sparsity
features, e.g., a sparse additive decomposition structure. We further develop a
simultaneous hypothesis testing procedure based on multiplier bootstrap. Our
testing method automatically takes into account of the dependence structure
within the debiased estimates, and allows the number of tested components to be
exponentially high.
