Smoothing sparse and unevenly sampled curves using semiparametric mixed models: An application to online auctions

Functional data analysis can be challenging when the functional
objects are sampled only very sparsely and unevenly. Most
approaches rely on smoothing to recover the underlying functional
object from the data which can be difficult if the data is
irregularly distributed. In this paper we present a new approach
that can overcome this challenge. The approach is based on the
ideas of mixed models. Specifically, we propose a semiparametric
mixed model with boosting to recover the functional object. While
the model can handle sparse and unevenly distributed data, it also
results in conceptually more meaningful functional objects. In
particular, we motivate our method within the framework of eBay's
online auctions. Online auctions produce monotonic increasing price
curves that are often correlated across two auctions. The
semiparametric mixed model accounts for this correlation in a
parsimonious way. It also estimates the underlying increasing trend
from the data without imposing model-constraints. Our application
shows that the resulting functional objects are conceptually more
appealing. Moreover, when used to forecast the outcome of an online
auction, our approach also results in more accurate price
predictions compared to standard approaches. We illustrate our
model on a set of 183 closed auctions for Palm M515 personal
digital assistants.