In this paper, we propose and study an L ²-norm based test for the two-sample Behrens Fisher problem for functional data, which tests the equality of mean functions of two Gaussian processes with possibly unequal covariance functions. The distributions of the proposed test statistic under the null hypothesis and a sequence of local alternatives are derived. Under certain regularity conditions, the proposed test is shown to be χn-consistent. To well approximate the null distribution, we propose a Welch-type (two-cumulant matched χ²-approximation) method with the unknown parameters estimated using a simple naive method. When the two functional samples have the same covariance functions, some existing testing procedures for functional data can be adaptively applied. However, when this assumption is violated, we illustrate, via a simulation study, that the effect of unequal co-variance functions can be very significant. The methodologies are motivated by and illustrated with a real data example.

It is natural to assume that a regression relationship is monotone. But the existing monotone smoothers either may not result in desirable results or are very complicated in implementation. In this article, we propose a simple and efficient monotone smoother based on estimation of derivatives using smoothing splines. The main idea is to shift the monotonicity constraint on an underlying function to the positiveness or negativeness constraint on its associated derivative curve. The simplicity of the smoother is in the sense that closed form formulas for estimation of a monotone function and its derivative curve are available, together with fast construction of their approximate standard deviation bands, and quick selection of smoothing parameter using approximate cross-validation or generalized cross-validation rules. The efficiency of the smoother is demonstrated via a small scale simulation study and successful applications of the smoother to several real data examples. Testing of non-monotonicity is also discussed via applying the recently developed smoothing spline-based SiZer directly to the data or indirectly to the residuals with the monotone effects removed.

Tests of Linear Hypotheses in the ANOVA under Heteroscedasticity