Estimating Derivatives for Samples of Sparsely Observed Functions, With Application to Online Auction Dynamics

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Functional Data Analysis (2016, background): “…variation with some variation in the timing of the peaks, many of which have a bimodal appearance, indicating that when the boom cycle peaked, there was a ﬁrst possible in the case of sparsely sampled data, where direct diﬀerentiation of trajectories would not be possible (Liu & M¨uller 2009).”
Functional quadratic regression (2010, unknown): “Liu & Müller ( 2009 ) .”
Varying-coefficient functional linear regression (2010, methodology): “Even for the case of fitted trajectories, conditional approaches have been implemented successfully, even allowing reasonable derivative estimates to be obtained from very sparse data (Liu and Müller (2009)).”