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Robust integral compounding criteria for trend and correlation structures

Abstract

Optimal design is a crucial issue in Environmental 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 particular, 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 information 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.

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Acknowledgments

This work was partially done while V. Casero-Alonso visited the Institute of Statistics of Johannes Kepler University. He wants to thank their hospitality and ideas. This work has been supported by Ministerio de Educación y Ciencia and Fondos FEDER MTM2010-20774-C03-01 and Junta de Comunidades de Castilla la Mancha PEII10-0291-1850 and Amadee, Project Nr. FR 11/2010. The work of E. Bukina was partially supported by the EU through a Marie-Curie Fellowship (EST-SIGNAL program: http://est-signal.i3s.unice.fr) under the contract Nb. MEST-CT-2005-021175. Milan Stehlík was supported by ANR project Desire FWF I 833-N18. The authors are thankful for helpful comments of Werner G. Müller, Luc Pronzato and Joao Rendas. We also thank the editor and reviewers, whose insightful comments helped us to sharpen the paper considerably.

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Appendix: Proofs and technicalities

Appendix: Proofs and technicalities

Proposition 1

Let us have

$$\begin{aligned} Y(x_i)=\eta (x_i,\vartheta )+\varepsilon (x_i),\ i=1,2,\ x_1,x_2\in [0,1], \gamma (d)=1-\exp (-\theta d), \end{aligned}$$

where \(\gamma \) stands for the semi-variogram and only the covariance parameter \(\theta \) is of interest. Then the maximal FIM is obtained for \(d=0.\)

Proof

We have the log-likelihood function \(L=K-\frac{1}{2}\log | \Sigma (\theta )|-\frac{1}{2}v^T \Sigma (\theta )^{-1}v,\) where \(v=(Y(x_1)-\eta (x_1,\vartheta ),Y(x_2)-\eta (x_2,\vartheta ))^T.\) The FIM for the covariance parameter \(\theta \) is

$$\begin{aligned} M_{\theta }=E\left( -\frac{\partial ^2L}{\partial \theta ^2}\right) =\frac{1}{2} \frac{\partial ^2\log |\Sigma (\theta )|}{\partial \theta ^2}+\frac{1}{2}E\left( v^T\frac{\partial ^2\Sigma (\theta )^{-1}}{\partial \theta ^2} v\right) , \end{aligned}$$

and we have \(\frac{\partial ^2\{A_{i,j}\}}{\partial \theta ^2} \{\frac{\partial ^2A_{i,j}}{\partial \theta ^2}\}\) and \(|\Sigma (\theta )|=1-\exp (-2\theta d).\) Further we have

$$\begin{aligned} \frac{1}{2} \frac{\partial ^2\log |\Sigma (\theta )|}{\partial \theta ^2}=-\frac{2d^2\exp (-2d\theta )}{(1-\exp (-2d\theta ))^2} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}E\left( v^T\frac{\partial ^2\Sigma (\theta )^{-1}}{\partial \theta ^2} v\right) =\frac{d^2\exp (-2d\theta )(\exp (-2d\theta )+3)}{(1-\exp (-2d\theta ))^2} \end{aligned}$$

and finally

$$\begin{aligned} M_{\theta }=\frac{d^2\exp (-2\theta d)(1+\exp (-2\theta d))}{(1-\exp (-2\theta d))^2}. \end{aligned}$$

Note that for every \(\theta >0\) the maximum \(\frac{1}{2\theta ^2}\) is attained for \(d=0\).

Proposition 2

Let us have

$$\begin{aligned} Y(x_i)=\eta (x_i,\vartheta )+\varepsilon (x_i),\ i=1,2,\ x_1,x_2\in [0,1] \end{aligned}$$
$$\begin{aligned} \text{ cov }(x_1,x_2)=1-\theta \vert x_1-x_2 \vert , \end{aligned}$$

and only covariance parameter \(\theta \) is parameter of interest. For regularity assumption we suppose that \(\theta d<2, \theta >0.\) Then the maximal FIM is obtained for maximal \(d.\)

Proof

We have \(|\Sigma (\theta )|=\theta d(2-\theta d),\)

$$\begin{aligned} \frac{1}{2} \frac{\partial ^2\log | \Sigma (\theta )|}{\partial \theta ^2}=-\frac{-2\theta d+\theta ^2d^2+2}{\theta ^2(\theta d-2)^2} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{2}E\left( v^T\frac{\partial ^2\Sigma (\theta )^{-1}}{\partial \theta ^2} v\right) =2\frac{-2\theta d+\theta ^2d^2+2}{\theta ^2(\theta d-2)^2} \end{aligned}$$

and finally

$$\begin{aligned} M_{\theta }=\frac{-2\theta d+\theta ^2d^2+2}{\theta ^2(\theta d-2)^2}. \end{aligned}$$

We have

$$\begin{aligned} \frac{\partial M_{\theta }}{\partial d}=\frac{2d}{(2-\theta d)^3} \end{aligned}$$

So \(M_{\theta }\) is increasing function for every (acceptable) \(0<d<\min \{\frac{2}{\theta },1\}.\)

Proposition 3

By direct integration we obtain the following:

$$\begin{aligned} Lu^*&= \int \limits _0^1 \varphi (M_{\exp }(\theta ),M_{lin}(\theta ),\theta )u^*(\theta )d\theta \\&= A/B, \end{aligned}$$

where

$$\begin{aligned} A=& -6\exp (2\theta d)\theta ^2d^2 + 4\exp (2\theta d)\theta d-\theta ^4d^4\exp (2\theta d) + 4\theta ^3d^3 \exp (2\theta d)\\&-2\theta d\exp (4\theta d) + \theta ^2d^2\exp (4\theta d) + 2 + 2\exp (4\theta d)-4\exp (2 \theta d)-\theta ^4d^4 + 4\theta ^3d^3-3\theta ^2d^2-2\theta d,\\ B=& (-\log ((-2\theta d + \theta ^2d^2+2)/\theta ^2/(\theta ^2d^2-4\theta d+4))\theta ^2d^2\\& +\,4\log ((-2\theta d+\theta ^2d^2 + 2)/\theta ^2/(\theta ^2d^2-4\theta d + 4) )\theta d\\&-\,4\log ((-2\theta d + \theta ^2d^2+2)/\theta ^2/(\theta ^2d^2-4\theta d + 4))\\& + \log (-d^2(\exp (2\theta d) + 1)/(-\exp (4\theta d) + 2\exp (2\theta d)-1))\theta ^2d^2\\&-\,4\log (-d^2(\exp (2\theta d) + 1)/( -\exp (4\theta d) + 2\exp (2\theta d)-1))\theta d\\& +\,4\log (-d^2(\exp (2\theta d)+1)/(-\exp (4\theta d) + 2\exp (2\theta d)-1))). \end{aligned}$$

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Stehlík, M., López-Fidalgo, J., Casero-Alonso, V. et al. Robust integral compounding criteria for trend and correlation structures. Stoch Environ Res Risk Assess 29, 379–395 (2015). https://doi.org/10.1007/s00477-014-0892-5

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Keywords

  • Correlated errors
  • Efficiency
  • Equidistant design
  • Experimental design
  • Fredholm equation
  • Parameterized covariance functions
  • Regularization.