  Ridge regression with random coefficients provides an important alternative
to fixed coefficients regression in high dimensional setting when the effects
are expected to be small but not zeros. This paper considers estimation and
prediction of random coefficient ridge regression in the setting of transfer
learning, where in addition to observations from the target model, source
samples from different but possibly related regression models are available.
The informativeness of the source model to the target model can be quantified
by the correlation between the regression coefficients. This paper proposes two
estimators of regression coefficients of the target model as the weighted sum
of the ridge estimates of both target and source models, where the weights can
be determined by minimizing the empirical estimation risk or prediction risk.
Using random matrix theory, the limiting values of the optimal weights are
derived under the setting when $p/n \rightarrow \gamma$, where $p$ is the
number of the predictors and $n$ is the sample size, which leads to an explicit
expression of the estimation or prediction risks. Simulations show that these
limiting risks agree very well with the empirical risks. An application to
predicting the polygenic risk scores for lipid traits shows such transfer
learning methods lead to smaller prediction errors than the single sample ridge
regression or Lasso-based transfer learning.
