Conference Agenda

We present test of symmetry of distribution and test of time symmetry for functional data. These test are Cramér - von Mises type tests based on empirical characteristic functionals. Specific variants of time symmetry including time symmetry of Wiener process are proposed. In general, the test statistics assume a relatively simple form if we use a Gaussian measure to construct the test. Then, we use bootstrap or permutation techniques to estimate the asymptotic critical values for the test statistics.

Event study plots are the centerpiece of Difference-in-Differences (DiD) analysis, but current plotting methods cannot provide honest causal inference when the parallel trends and/or no-anticipation assumption fails. We introduce a novel functional data approach to DiD that directly enables honest causal inference via event study plots. Our DiD estimator converges to a Gaussian process in the Banach space of continuous functions, enabling powerful simultaneous confidence bands. This theoretical contribution allows us to turn an event study plot into a rigorous, honest causal inference tool through equivalence and relevance testing: Honest reference bands can be validated using equivalence testing in the pre-treatment period, and honest causal effects can be tested using relevance testing in the post-treatment period. We demonstrate the performance of our method in simulations and two case studies.

Mercer's celebrated theorem is refined and extended for (weakly) differentiable symmetric kernels by associating not the common $L^2$-integral operator but a slightly more complex operator, that additionally takes into account information encoded in the (weak) derivatives of the kernel. The natural domain for this associated operator is the Sobolev Space $H^k(Theta) = W^{k,2}(Theta) subset L^2(Theta)$, where $Theta subset R^d$ is some bounded domain and $kinN_0$ depends on the order of weak differentiability. The spectral decomposition of this operator then leads to a Mercer-type expansion of the kernel, which converges with respect to the $H^k$-norm and, if $k>d$, also uniformly emph{without} requiring the kernel to be positive-definite. In case the kernel is also positive-definite and differentiable in the strong sense, a refinement of Mercer's theorem is obtained that additionally provides uniform convergence of the term-wise derivatives of the expansion to the respective derivatives of the kernel as well.
Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these results provide a novel Karhunen-Loève-type representation that simultaneously approximates the process as well as its (weak) derivatives in a mean square optimal sense.

We revisit the classic situation in functional data analysis in which data items such as curves are observed at discrete (possibly sparse and irregular) arguments with observation noise. We focus on the reconstruction of individual curves, especially on prediction intervals and prediction bands for them. The standard approach is to proceed in two steps: First, one estimates the mean and covariance function of curves and observation noise variance function by smoothing techniques such as penalized splines. Second, under Gaussian assumptions, one derives the conditional distribution of a curve given its noisy discrete observations and constructs prediction sets with required properties (usually employing sampling from the predictive distribution). This approach is indeed well established, commonly used and theoretically valid but practically, it surprisingly fails in its key property: prediction sets constructed this way often do not have the required coverage. The actual coverage is lower than the nominal one. This has been little reported and studied in the literature. We investigate the cause of this issue and propose a remedy.