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Figure . . Creating functional objects: price curves using data preprocessing via interpolation of the
bids and penalized smoothing splines with p
=
andλ
=
carefully checking the results), then very unrepresentative functional objects would
be obtained and valuable information would be lost.
Onereasonforthepoorrepresentativeness oftheobjectsinFig. . isthelowvalue
ofthesmoothingparameter.Increasing λ to (Fig. . )resultsinmuchsmoother(i.e.,
less wiggly) price curves. However, there are still some partial functional objects (# ,
# ) and missing functional objects (# , # ). Moreover, while the higher value of λ
resultsincurvesthataremuchlesswiggly,someofthefunctionalobjectsnowappear
to be too inflexible (e.g., # may be too similar to a straight line).
We can achieve a better fit (i.e., one with more flexibility, but only a little extra
wiggliness) by increasing the order of the spline together with the magnitude of the
smoothing parameter. We can also solve the problem of partial and missing func-
tional objects by using a preprocessing step via interpolation. hat is, we interpolate
the observed bidding data and fit the smoothing spline to a discretized grid of the
interpolating function. Specifically, let t ij denote the time that the jth bid is placed
in auction i,andlety ij denote the corresponding bid amount. We interpolate the
y ij values linearly to obtain the interpolating function y i
(
t
)
.Letnow
t j
be
acommongridoftimepoints.Weevaluateall y i
(
t
)
valuesatthecommongridpoints
y i
t j to obtain y ij
and fit the smoothing spline to the y ij values. In this way,
we can ensure that we estimate the smoothing spline based on a su cient number of
=
(
t j
)
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