  We develop uniform inference for high-dimensional threshold regression
parameters and valid inference for the threshold parameter in this paper. We
first establish oracle inequalities for prediction errors and $\ell_1$
estimation errors for the Lasso estimator of the slope parameters and the
threshold parameter, allowing for heteroskedastic non-subgaussian error terms
and non-subgaussian covariates. Next, we derive the asymptotic distribution of
tests involving an increasing number of slope parameters by debiasing (or
desparsifying) the scaled Lasso estimator. The asymptotic distribution of tests
without the threshold effect is identical to that with a fixed effect.
Moreover, we perform valid inference for the threshold parameter using
subsampling method. Finally, we conduct simulation studies to demonstrate the
performance of our method in finite samples.
