The arrival process of bidders and bids in online auctions is important for
studying and modeling supply and demand in the online marketplace. A popular
assumption in the online auction literature is that a Poisson bidder arrival
process is a reasonable approximation. This approximation underlies theoretical
derivations, statistical models and simulations used in field studies. However,
when it comes to the bid arrivals, empirical research has shown that the
process is far from Poisson, with early bidding and last-moment bids taking
place. An additional feature that has been reported by various authors is an
apparent self-similarity in the bid arrival process. Despite the wide evidence
for the changing bidding intensities and the self-similarity, there has been no
rigorous attempt at developing a model that adequately approximates bid
arrivals and accounts for these features. The goal of this paper is to
introduce a family of distributions that well-approximate the bid time
distribution in hard-close auctions. We call this the BARISTA process (Bid
ARrivals In STAges) because of its ability to generate different intensities at
different stages. We describe the properties of this model, show how to
simulate bid arrivals from it, and how to use it for estimation and inference.
We illustrate its power and usefulness by fitting simulated and real data from
eBay.com. Finally, we show how a Poisson bidder arrival process relates to a
BARISTA bid arrival process.Comment: Published in at http://dx.doi.org/10.1214/07-AOAS117 the Annals of
Applied Statistics (http://www.imstat.org/aoas/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Let U
1,U
2,… be a sequence of i.i.d. random vectors distributed uniformly in a compact plane region A of unit area. Sufficient conditions on the geometry of A are provided under which the Euclidean diameter D
n
of the first n of the points converges weakly upon suitable rescaling.
On independent random points U1,· ··,Un distributed uniformly on [0, 1]d, a random graph Gn(x) is constructed in which two distinct such points are joined by an edge if the l∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn} are provided which guarantee the random graph to be empty of edges, a.s.
On independent random points U
1
,· ··,Un
distributed uniformly on [0, 1]
d
, a random graph Gn
(x) is constructed in which two distinct such points are joined by an edge if the l
∞-distance between them is at most some prescribed value 0 ≦ x ≦ 1. Almost-sure asymptotic rates of convergence/divergence are obtained for the maximum vertex degree of the random graph and related quantities, including the clique number, chromatic number and independence number, as the number n of points becomes large and the edge distance x is allowed to vary with n. Series and sequence criteria on edge distances {xn
} are provided which guarantee the random graph to be empty of edges, a.s.
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