Conference Agenda

We suggest a dependence coefficient between a categorical variable and some general variable taking values in a metric space. In particular, this framework includes functional data.

We derive important theoretical properties and study the large sample behaviour of our suggested estimator. Moreover, we develop an independence test and prove that it is consistent against any violation of independence. The test is also applicable to the classical $K$-sample problem with possibly high- or infinite-dimensional distributions.

We obtain minimax-optimal convergence rates in the supremum norm,
including information-theoretic lower bounds,
for estimating the covariance kernel as well as principle component basis functions
of a stochastic process which is repeatedly observed at discrete,
synchronous design points.
We focus on the supremum norm instead of the simpler $L_2$ norm,
since it corresponds to the visualization of the estimation error and
forms the basis for the construction of uniform confidence bands.
For dense design, assuming Hölder-smooth sample paths we obtain
the $\sqrt n$-rate of convergence in the supremum norm without
additional logarithmic factors which typically occur in the results
in the literature.
Surprisingly, for the covariance kernel, in the transition from dense to
sparse design the rates do not reflect the two-dimensional nature
of the covariance kernel but correspond to those for univariate mean function
estimation.
Our estimation method can make use of higher-order smoothness of the
covariance kernel away from the diagonal, and does not require the same smoothness
on the diagonal itself. Hence, our results
cover covariance kernels of processes with rough, non-differentiable sample paths.
Moreover, the estimator does not use mean function estimation to form residuals,
and no smoothness assumptions on the mean have to be imposed.
In the dense case we also obtain central limit theorems in the supremum norm,
both for the covariance kernel and the principle component basis functions,
which can be used as the basis for the construction of uniform confidence sets.
Simulations and real-data applications illustrate the practical usefulness of the methods.

The two-sample problem for functional data is investigated for discrete, synchronous designs in each sample, in settings in which one sample is densely observed while the other is only relatively sparsely observed. This is motivated by comparing historical and more current daily temperature patterns, where more recent devices take measurements every 10 minutes, while historical measurements in the time period 1952 to 1972 are available only every hour. We use recently developed methods from transfer learning for functional data to estimate the difference of the mean functions at optimal rates in the supremum norm. Further, we derive a central limit theorem in the space of continuous functions and discuss the construction of uniform confidence bands using the multiplier bootstrap. We also show how our methods can be extended to functional time series.
In the application to daily temperature patterns we decompose the mean difference function into a daily average - the normalized integral of the mean difference function - as well as into the deviation from the average value. Using the developed inferential methodology we show that not only the daily average temperatures in each month have increased significantly, but also that the daily temperature patterns have changed for most months, with night temperature remaining relatively stable while daily temperatures increased beyond the daily average increase.