Asymptotics of bivariate penalised splines

Nonparametric trend estimation in functional time series with application to annual mortality rates

We address the problem of trend estimation for functional time series. Existing contributions either deal with detecting a functional trend or assuming a simple model. They consider neither the estimation of a general functional trend nor the analysis of functional time series with a functional trend component. Similarly to univariate time series, we propose an alternative methodology to analyze functional time series, taking into account a functional trend component. We propose to estimate the functional trend by using a tensor product surface that is easy to implement, to interpret, and allows to control the smoothness properties of the estimator. Through a Monte Carlo study, we simulate different scenarios of functional processes to show that our estimator accurately identifies the functional trend component. We also show that the dependency structure of the estimated stationary time series component is not significantly affected by the error approximation of the functional trend component. We apply our methodology to annual mortality rates in France.

Evaluation of longitudinal surrogate markers

The use of surrogate markers to examine the effectiveness of a treatment has the potential to decrease study length and identify effective treatments more quickly. Most available methods to investigate the usefulness of a surrogate marker involve restrictive parametric assumptions and tend to focus on settings where the surrogate is measured at a single point in time. However, in many clinical settings, the potential surrogate marker is often measured repeatedly over time, and thus, the surrogate marker information is a trajectory of measurements. In addition, it is often difficult in practice to correctly specify the relationship between a treatment, primary outcome, and surrogate marker trajectory. In this paper, we propose a model-free definition for the proportion of the treatment effect on the primary outcome that is explained by the treatment effect on the longitudinal surrogate markers. We propose three novel flexible methods to estimate this proportion, develop the asymptotic properties of our estimators, and investigate the robustness of the estimators under multiple settings via a simulation study. We apply our proposed procedures to an AIDS clinical trial dataset to examine a trajectory of CD4 counts as a potential surrogate.