Functional Data Analysis for Sparse Auction Data

Bid arrivals of eBay auctions often exhibit “bid sniping”, a phenomenon where “snipers” place their bids at the last moments of an auction. This is one reason why bid histories for eBay auctions tend to have sparse data in the middle and denser data both in the beginning and at the end of the auction. Time spacing of the bids is thus irregular and sparse. For nearly identical products that are auctioned repeatedly, one may view the price history of each of these auctions as realization of an underlying smooth stochastic process, the price process. While the traditional Functional Data Analysis (FDA) approach requires that entire trajectories of the underlying process are observed without noise, this assumption is not satisfied for typical auction data. We provide a review of a recently developed version of functional principal component analysis (Yao et al., 2005), which is geared towards sparse, irregularly observed and noisy data, the principal analysis through conditional expectation (PACE) method. The PACE method borrows and pools information from the sparse data in all auctions. This allows the recovery of the price process even in situations where only few bids are observed. In a modified approach, we adapt PACE to summarize the bid history for varying current times during an ongoing auction through time-varying principal component scores. These scores then serve as time-varying predictors for the closing price. We study the resulting time-varying predictions using both linear regression and generalized additive modelling, with current scores as predictors. These methods will be illustrated with a case study for 157 Palm M515 PDA auctions from e-Bay, and the proposed methods are seen to work reasonably well. Other related issues will also be discussed.