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.

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

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