# Multivariate spline analysis for multiplicative models: Estimation, testing and application to climate change

## Azaïs, Jean-Marc ; Ribes, Aurélien

This paper presents multiplicative, or bi-additive, models with some spline-type regularity for a rectangular array of data, for example in space and time. The one-dimensional smoothing spline model is extended to this multiplicative model including regularity in each dimension. For estimation, we prove the existence of the maximum penalized likelihood estimates (MPLEs), and introduce a numerical algorithm that converges in a weak sense to a critical point of the penalized likelihood. Explicit MPLEs are given in two important particular cases. With regard to hypothesis testing, we focus on the “no effect” test and prove that the null distribution of the penalized likelihood ratio test (PLRT) does not depend on the nuisance parameters under H0H0, leading to easy Monte-Carlo techniques. Numerical results are presented for both simulated data and climate data. For simulated data, our estimation algorithm is shown to have a good behavior. The application to climate data illustrates how multivariate spline analysis for multiplicative models may be of interest in climate change detection.

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