  We consider random sample splitting for estimation and inference in high
dimensional generalized linear models, where we first apply the lasso to select
a submodel using one subsample and then apply the debiased lasso to fit the
selected model using the remaining subsample. We show that, no matter including
a prespecified subset of regression coefficients or not, the debiased lasso
estimation of the selected submodel after a single splitting follows a normal
distribution asymptotically. Furthermore, for a set of prespecified regression
coefficients, we show that a multiple splitting procedure based on the debiased
lasso can address the loss of efficiency associated with sample splitting and
produce asymptotically normal estimates under mild conditions. Our simulation
results indicate that using the debiased lasso instead of the standard maximum
likelihood estimator in the estimation stage can vastly reduce the bias and
variance of the resulting estimates. We illustrate the proposed multiple
splitting debiased lasso method with an analysis of the smoking data of the
Mid-South Tobacco Case-Control Study.
