Functional Data Analysis for Sparse Auction Data

Abstract

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.

Functional Additive Models

“…For the case of a scalar response, the combination of FPC scores with an additive model, emphasizing applied modeling with readily available software tools, has been demonstrated in concurrent work with applied emphasis (Foutz and Jank 2008;Liu and Müller 2008;Sood, James and Tellis 2008). As we show, a key to both analysis and implementation of this combination is the uncorrelatedness of the functional predictor scores.…”

Modeling time‐varying effects with generalized and unsynchronized longitudinal data

“…Senturk and Mueller [28] give explicit expressions of the local least squares estimators. For a computationally efficient bandwidth choice in the proposed onedimensional and two-dimensional smoothing, we adopt the generalized cross-validation algorithm of Liu and Mueller [31].…”

Classical testing in functional linear models

“…Nevertheless, as Jank and Shmueli (2006) argues “bidding in eBay auctions tends to be concentrated at the end, resulting in very sparse bid-arrivals during most of the auction except for its final moments, when the bidding volume can be extremely high”. The dynamics of the bids has attracted large interest, especially in the literature of functional data (Liu and Müller 2008). Here we investigate whether the dynamics of the bids in the first part of the auction duration is related to the auction’s closing price.…”