Testing sufficiency for transfer learning

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Modern statistical analysis often encounters high dimensional models but with limited sample sizes. This makes it difficult to estimate high-dimensional statistical models based on target data with limited sample size. 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. We denote transfer learning sufficiency to be the null hypothesis. It refers to the situation 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.