  In many applications, we collect independent samples from interconnected
populations. These population distributions share some latent structure, so it
is advantageous to jointly analyze the samples. One effective way to connect
the distributions is the semiparametric density ratio model (DRM). A key
ingredient in the DRM is that the log density ratios are linear combinations of
prespecified functions; the vector formed by these functions is called the
basis function. A sensible basis function can often be chosen based on
knowledge of the context, and DRM-based inference is effective even if the
basis function is imperfect. However, a data-adaptive approach to the choice of
basis function remains an interesting and important research problem. We
propose an approach based on the classical functional principal component
analysis (FPCA). Under some conditions, we show that this approach leads to
consistent basis function estimation. Our simulation results show that the
proposed adaptive choice leads to an efficiency gain. We use a real-data
example to demonstrate the efficiency gain and the ease of our approach.
