ORIGINAL PAPER Robust integral compounding criteria for trend and correlation structures M. Stehlı ´k • J. Lo ´pez-Fidalgo • V. Casero-Alonso • Elena Bukina Ó Springer-Verlag Berlin Heidelberg 2014 Abstract Optimal design is a crucial issue in Environ- mental measurement with typical time–space correlated observations. A modified Arrhenius model with a particular correlation structure will be applied to the methane removal in the atmosphere, a very important environmental issue at this moment. We introduce a class of integrated compound criteria for obtaining robust designs. In partic- ular, the paper provides an insight into the relationship of a compound D-optimality criterion for both the trend and covariance parameters, and the Integrated Mean Squared Prediction Error (IMSPE) criterion. In general, if there are two or more approaches of a given problem, e.g. two rival models or two different parts of a model, an integral relationship may be constructed with the aim of finding a suitable compromise between them. The Fisher informa- tion matrix (FIM) will be used in both cases. Then the integral compound criterion with respect to a density from a given parametric family of distributions is optimized. We also discuss some general conditions around the behavior of the introduced approach for comparing the FIMs and provide computing methods. Keywords Correlated errors  Efficiency  Equidistant design  Experimental design  Fredholm equation  Parameterized covariance functions  Regularization. 1 Introduction Comparison of FIMs from different models have been considered in different ways in the statistical literature [see e.g. Ahmadi and Arghami (2003), Hofmann (2004) or Alshunnar et al. (2012)]. If there are two or more approa- ches of a given problem, e.g. two rival models or two different parts of a model, an integral relationship may be constructed with the aim of finding a suitable compromise between them. The purpose of this is to obtain robust designs for both matrices. The usual compound criteria may not be powerful enough to do this in a proper way. Observations from various environmental measurements are often approximated as realizations of correlated random fields. Such an approach is interesting for assessing e.g. drought and flood risks [see e.g. Unami et al. (2010)]. The robust designs obtained in the current paper can be of interest for irregular sampling, studied e.g. in Tandeo et al. (2011) for the case of aggregation of many meteorological and oceanographic variables from satellites. Rodrı ´guez- Dı ´az et al. (2012) gave optimal designs for a modified Arrhenius model, used for modeling a flux of methane in troposphere. This has an important impact in greenhouse gas emission. Kinetics of chemical reactions are usually M. Stehlı ´k (&) Departamento de Matema ´tica, Universidad Te ´cnica Federico Santa Marı ´a, Casilla 110-V, Valparaı ´so, Chile e-mail: mlnstehlik@gmail.com; milan.stehlik@usm.cl M. Stehlı ´k Department of Applied Statistics, Johannes Kepler University in Linz, Linz, Austria e-mail: milan.stehlik@jku.at J. Lo ´pez-Fidalgo  V. Casero-Alonso Department of Mathematics, University of Castilla-La Mancha, Ciudad Real, Spain e-mail: jesus.lopezfidalgo@uclm.es V. Casero-Alonso e-mail: victormanuel.casero@uclm.es E. Bukina Laboratoire I3S, CNRS/Universite ´ de Nice-Sophia Antipolis, Sophia Antipolis, France e-mail: bukina@i3s.unice.fr 123 Stoch Environ Res Risk Assess DOI 10.1007/s00477-014-0892-5