Thin plate spline model under skew-normal random errors: estimation and diagnostic analysis for spatial data

Abstract

Expected Maximization (EM) algorithm is often used for estimation in semiparametric models with non-normal observations. However, the EM algorithm’s main disadvantage is its slow convergence rate. In this paper, we propose the Laplace approximation to maximize the marginal likelihood, given a non-linear function assumed as a spline random effect for a skew-normal thin plate spline model. For this, we used automatic differentiation to get the derivatives and provide a numerical evaluation of the Hessian matrix. Comparative simulations and applications between the EM algorithm for thespatial dimension and Laplace approximation were carried out to illustrate the proposed method’s performance. We show that the Laplace approximation is an efficient method, has flexibility to express log-likelihood in a semiparametric model and obtain a fast estimation process for non-normal models. In addition, a local influence analysis was carried out to evaluate the estimation sensitivity.

Keywords:

• Semiparametric modelling
• thin plate spline model (TPS)
• Laplace approximation
• automatic differentiation
• local influence

Acknowledgments

The author thanks the editor and an anonymous referee for their helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Joaquin Cavieres was supported by Universidad de Valparaíso (Chile) grant FIB-UV (Res. Ex. 1529 and 1448). Contreras-Reyes's research was fully supported by FONDECYT (Chile) grant No. 11190116. G. Ibacache-Pulgar acknowledges funding support by FONDECYT (Chile) grant No. 11130704.