  Modern statistical analysis often encounters high dimensional models but with
limited sample sizes. This makes the target data based statistical estimation
very difficult. Then how to borrow information from another large sized source
data for more accurate target model estimation becomes an interesting problem.
This leads to the useful idea of transfer learning. Various estimation methods
in this regard have been developed recently. In this work, we study transfer
learning from a different perspective. Specifically, we consider here the
problem of testing for transfer learning sufficiency. By transfer learning
sufficiency (denoted as the null hypothesis), we mean that, with the help of
the source data, the useful information contained in the feature vectors of the
target data can be sufficiently extracted for predicting the interested target
response. Therefore, the rejection of the null hypothesis implies that
information useful for prediction remains in the feature vectors of the target
data and thus calls for further exploration. To this end, we develop a novel
testing procedure and a centralized and standardized test statistic, whose
asymptotic null distribution is analytically derived. Simulation studies are
presented to demonstrate the finite sample performance of the proposed method.
A deep learning related real data example is presented for illustration
purpose.
