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Influence diagnostics in elliptical spatial linear models

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Abstract

In recent years, there has been a growing interest in statistical methods for the analysis of spatially referenced data. The spatial dependence structure modeling is an indispensable tool to estimate the parameters that define this structure. In this paper, we use the family of elliptical distributions to estimate the spatial dependence in referenced data. Thus we extend the Gaussian spatial linear model. Also we use the local influence methodology to assess the sensitivity of the maximum likelihood estimators to small perturbations in the data and/or in the spatial linear model assumptions. The methodology is illustrated with a real data set. The results allowed us to conclude that the presence of atypical values in the sample data have a strong influence, changing the spatial dependence structure. Also we have included a small simulation study.

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Acknowledgments

We would like to thank the Associate Editor and two referees for their helpful comments and suggestions, leading to improvement of the paper. Also, we acknowledge the partial financial support from Fundaç\(\tilde{\text {a}}\)o Araucária of Paraná State, Capes, CNPq and FACEPE, Brazil, and Project FONDECYT 1110318, Chile.

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Correspondence to Manuel Galea.

Appendices

Appendix A: The observed information matrix for elliptical spatial linear models

The log-likelihood function is given by

$$\begin{aligned} {\mathcal L}({\varvec{\theta }})= -\frac{1}{2}\log |{\varvec{\varSigma }}|+\log g(\delta ), \end{aligned}$$

where \(\delta =(\mathbf {Z}- \mathbf {X}{\varvec{\beta }})^{\top }{\varvec{\varSigma }}^{-1}(\mathbf {Z}- \mathbf {X}{\varvec{\beta }})\).

The second derivatives matrix, is given by

$$\begin{aligned} \mathbf {L}({\varvec{\theta }})=\mathbf {L}=\left( \begin{array}{ll} L_{\beta \beta } &{}\quad L_{\beta \phi } \\ L_{\phi \beta } &{}\quad L_{\phi \phi } \end{array}\right) , \end{aligned}$$

where \(L_{\beta \beta }=\frac{\partial ^2 {\mathcal L}({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial {\varvec{\beta }}^{\top }}= 2\mathbf {X}^{\top }{\varvec{\varSigma }}^{-1}\{W_{g}(\delta ){\varvec{\varSigma }}+ 2 W^{'}_{g}(\delta ){\varvec{\epsilon }}{\varvec{\epsilon }}^{\top }\}{\varvec{\varSigma }}^{-1}\mathbf {X}\), \(L_{\beta \phi }=\frac{\partial ^2 {\mathcal L}({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial {\varvec{\phi }}^{\top }}\), with \(\frac{\partial ^2 {\mathcal L}({\varvec{\theta }})}{\partial {\varvec{\beta }}\partial \phi _{j}}=2\mathbf {X}^{\top }{\varvec{\varSigma }}^{-1}\Big \{W^{'}_{g}(\delta ) {\varvec{\epsilon }}{\varvec{\epsilon }}^{\top }{\varvec{\varSigma }}\frac{\partial {\varvec{\varSigma }}}{\partial \phi _j} + W_{g}(\delta )\frac{\partial {\varvec{\varSigma }}}{\partial \phi _j}\Big \}{\varvec{\varSigma }}^{-1}{\varvec{\epsilon }}\), as its \(j\)th column, for \(j=1, 2, 3\); \(L_{\phi \beta }=L_{\beta \phi }^{\top }\) and \(L_{\phi \phi }=\frac{\partial ^2 {\mathcal L}({\varvec{\theta }})}{\partial {\varvec{\phi }}\partial {\varvec{\phi }}^{\top }}\), with elements

$$\begin{aligned} \dfrac{\partial ^2 {\mathcal L}({\varvec{\theta }})}{\partial \phi _i \partial \phi _{j}}&= \dfrac{1}{2} \mathop {\mathrm{tr}}\nolimits \Big \{{\varvec{\varSigma }}^{-1}\left( \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _i}{\varvec{\varSigma }}^{-1}\dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _j} -\dfrac{\partial ^2{\varvec{\varSigma }}}{\partial \phi _i\partial \phi _j}\right) \Big \}\\&+{\varvec{\epsilon }}^{\top }{\varvec{\varSigma }}^{-1}\Big \{W^{'}_{g}(\delta ) \left( \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _i}{\varvec{\varSigma }}^{-1} {\varvec{\epsilon }}{\varvec{\epsilon }}^{\top }{\varvec{\varSigma }}^{-1}\dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _j}\right) \\&+W_{g}(\delta )\left( \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _i} {\varvec{\varSigma }}^{-1} \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _j} - \dfrac{\partial ^2 {\varvec{\varSigma }}}{\partial \phi _i \partial \phi _j} + \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _j} {\varvec{\varSigma }}^{-1} \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _i} \right) \Big \} {\varvec{\varSigma }}^{-1} {\varvec{\epsilon }}, \end{aligned}$$

for \(i, j=1, 2, 3\). The derivatives of first and second-order of the scale matrix \({\varvec{\varSigma }}\), (3), with respect to \(\phi _{j}\), for \(j=1, 2, 3\); for some covariance functions are presented in Uribe-Opazo et al. (2012).

Appendix B: \({\varvec{\varDelta }}\) matrix for perturbation scheme of the mean

In this case we have that \({\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})\) is given by

$$\begin{aligned} {\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})=-\dfrac{1}{2}\log |{\varvec{\varSigma }}| + \log g(\delta _{\omega }), \end{aligned}$$
(11)

where \(\delta _{\omega }=\{\mathbf {Z}- {\varvec{\mu }}({{\varvec{\omega }}})\}^{\top }{\varvec{\varSigma }}^{-1}\{\mathbf {Z}- {\varvec{\mu }}({{\varvec{\omega }}})\}={\varvec{\epsilon }}_{\omega }^{\top }{\varvec{\varSigma }}^{-1}{\varvec{\epsilon }}_{\omega }\), \({\varvec{\epsilon }}_{\omega }=\mathbf {Z}-{\varvec{\mu }}({{\varvec{\omega }}})\) and \({\varvec{\mu }}({{\varvec{\omega }}})=\mathbf {X}{\varvec{\beta }}+\mathbf {A}{{\varvec{\omega }}}\). Then

$$\begin{aligned} \dfrac{\partial {\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})}{\partial {{\varvec{\omega }}}^{\top }}=-2W_{g} (\delta _{\omega })\{\mathbf {Z}-{\varvec{\mu }}({{\varvec{\omega }}})\}^{\top }{\varvec{\varSigma }}^{-1}\mathbf {A}. \end{aligned}$$
(12)

Differentiating (12) with respect to \({\varvec{\beta }}\), see Nel (1980),

$$\begin{aligned} \dfrac{\partial ^{2}{\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})}{\partial {\varvec{\beta }}\partial {{\varvec{\omega }}}^{\top }}=2\mathbf {X}^{\top } {\varvec{\varSigma }}^{-1}\{W_{g}(\delta _{\omega }){\varvec{\varSigma }}+ 2W^{'}_{g} (\delta _{\omega }){\varvec{\epsilon }}_{\omega }{\varvec{\epsilon }}_{\omega }^{\top }\} {\varvec{\varSigma }}^{-1}\mathbf {A}. \end{aligned}$$
(13)

The derivative with respect to \(\phi _{j}\) is given by,

$$\begin{aligned} \dfrac{\partial ^{2}{\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})}{\partial \phi _{j}\partial {{\varvec{\omega }}}^{\top }}=-2{\varvec{\epsilon }}_{\omega }^{\top } \big \{W_{g}(\delta _{\omega })({\varvec{\varSigma }}^{-1}\dfrac{\partial \mathbf {A}}{\partial \phi _j}-\mathbf {D}_{j}\mathbf {A})-W^{'}_{g}(\delta _{\omega }) {\varvec{\epsilon }}_{\omega }^{\top }\mathbf {D}_{j}{\varvec{\epsilon }}_{\omega }{\varvec{\varSigma }}^{-1}\mathbf {A}\big \}, \end{aligned}$$
(14)

for \(j=1, 2, 3\). Evaluating (13) and (14) at \({{\varvec{\omega }}}={{\varvec{\omega }}}_{0}\), we obtain the \({\varvec{\varDelta }}=({\varvec{\varDelta }}^{\top }_{\beta }, {\varvec{\varDelta }}^{\top }_{\phi })^{\top }\) matrix.

Appendix C: The Fisher information matrix \(\mathbf {G}({{\varvec{\omega }}})\)

To select an adequate matrix \(\mathbf {A}\) we can use the methodology proposed by Zhu et al. (2007). In effect, the score function for \({{\varvec{\omega }}}\) in the perturbed log-likelihood function (11) is given by

$$\begin{aligned} U({{\varvec{\omega }}}) = \dfrac{\partial {\mathcal L}({\varvec{\theta }}, {{\varvec{\omega }}})}{\partial {{\varvec{\omega }}}} = -2W_{g}(\delta _{\omega }) \mathbf {A}^{\top }{\varvec{\varSigma }}^{-1}\{\mathbf {Z}- {\varvec{\mu }}({{\varvec{\omega }}})\}. \end{aligned}$$

Following Zhu et al. (2007) let \(\mathbf {G}({{\varvec{\omega }}})\), the Fisher information matrix with respect to the perturbation vector \({{\varvec{\omega }}}\). That is, \(\mathbf {G}({{\varvec{\omega }}})=E_{\omega }\{U({{\varvec{\omega }}})U^{\top }({{\varvec{\omega }}})\}\), where \(E_{\omega }\) denotes the expectation with respect to \(f(\mathbf {z}, {\varvec{\theta }},{{\varvec{\omega }}})\). A perturbation \({{\varvec{\omega }}}\) is appropriate if it satisfies \(\mathbf {G}({{\varvec{\omega }}}_{0})=c\mathbf {I}_n\), where \(c>0\). In our case we have

$$\begin{aligned} \mathbf {G}({{\varvec{\omega }}})=c({{\varvec{\omega }}})\mathbf {A}^{\top }{\varvec{\varSigma }}^{-1}\mathbf {A}. \end{aligned}$$

That is, \(\mathbf {G}({{\varvec{\omega }}}_{0})=c\mathbf {A}^{\top }{\varvec{\varSigma }}^{-1}\mathbf {A}\) with \(c=c({{\varvec{\omega }}}_{0})\) a positive constant, see Appendix . Notice that usually \(\mathbf {A}^{\top }{\varvec{\varSigma }}^{-1}\mathbf {A}\ne \mathbf {I}_{n}\). However, if \(\mathbf {A}={\varvec{\varSigma }}^{1/2}\), then \(\mathbf {G}({{\varvec{\omega }}}_{0})=c\mathbf {I}_{n}\) and so \({\varvec{\mu }}({{\varvec{\omega }}})=\mathbf {X}{\varvec{\beta }}+{\varvec{\varSigma }}^{1/2}{{\varvec{\omega }}}\) is a perturbation scheme appropriate. The derivatives \(\partial {\varvec{\varSigma }}^{1/2}/\partial \phi _{j}\) for \(j=1, 2, 3\), are given in Appendix .

Appendix D: Derivative of the square root \({\varvec{\varSigma }}^{1/2}\)

Corresponding to any matrix \({\varvec{\varSigma }}\) \(n \times n\) symmetric and nonnegative definite, there is a matrix symmetric nonnegative definite \({\varvec{\varSigma }}^{1/2}=\mathbf {W}\), such that \({\varvec{\varSigma }}={\varvec{\varSigma }}^{1/2}{\varvec{\varSigma }}^{1/2}={\mathbf {W}}^2\). Furthermore, \(\mathbf {W}\) is unique and can be expressed by

$$\begin{aligned} \mathbf {W}=\mathbf {P}\mathbf {A}^{1/2}\mathbf {P}^{\top }, \end{aligned}$$

where \(\mathbf {A}^{1/2}=\mathop {\mathrm{diag}}\nolimits (\sqrt{\alpha _1},\ldots ,\sqrt{\alpha _n})\), with \(\alpha _1,\ldots ,\alpha _n\) the eigenvalues of \({\varvec{\varSigma }}\) and \(\mathbf {P}\) is a matrix \(n \times n\) orthogonal \((\mathbf {P}\mathbf {P}^{\top } = \mathbf {I}_n)\) such that \(\mathbf {P}{\varvec{\varSigma }}\mathbf {P}^{\top }=\mathbf {A}\), with \(\mathbf {A}=\mathop {\mathrm{diag}}\nolimits (\alpha _1,\ldots ,\alpha _n)\). So, derivatives of \({\varvec{\varSigma }}\) with respect to \(\phi _j\) is given by

$$\begin{aligned} \dfrac{\partial {\varvec{\varSigma }}}{\partial \phi _j}= \mathbf {W}\dfrac{\partial \mathbf {W}}{\partial \phi _j}+ \dfrac{\partial \mathbf {W}}{\partial \phi _j}\mathbf {W}, \quad \hbox {for}\ j=1, 2, 3. \end{aligned}$$
(15)

This equation can be written as \(\dot{{\varvec{\varSigma }}}_j = \mathbf {W} \dot{\mathbf {W}}_j + \dot{\mathbf {W}}_j \mathbf {W}\), where \(\dot{{\varvec{\varSigma }}}_j=\frac{\partial {\varvec{\varSigma }}}{\partial \phi _j}\) and \(\frac{\partial \mathbf {W}}{\partial \phi _j}=\dot{\mathbf {W}}_j\), which has been extensively studied in the literature; see for instance Jameson (1968). Note that \(\dot{{\varvec{\varSigma }}}_j\), \(\mathbf {W}\) and \(\dot{\mathbf {W}}_j\) are symmetric matrices. Let \(\mathbf {J}_j=\mathbf {P}^{\top }\dot{{\varvec{\varSigma }}}_j\mathbf {P}\) and \(\mathbf {Q}=[(q_{rs})]\) symmetric matrices \(n\times n\), with \(q_{rs}=(\sqrt{\alpha _r}+\sqrt{\alpha _s})^{-1}\), for \(r,s=1,\ldots ,n\). Then, the solution to Eq. (15) is given by

$$\begin{aligned} \dfrac{\partial \mathbf {W}}{\partial \phi _j}=\dfrac{\partial {\varvec{\varSigma }}^{1/2}}{\partial \phi _j}=\mathbf {P}(\mathbf {J}_j\odot \mathbf {Q})\mathbf {P}^{\top }, \end{aligned}$$

where \(\odot \) denotes the Hadamard product for \(j=1, 2, 3\).

Appendix E: The likelihood function of the \({\varvec{t}}\) model is an increasing function of \(\nu \)

As noted by Zellner (1976), for the case of the usual linear regression model, “the necessary conditions on \({\varvec{\beta }}\), \({\varvec{\varSigma }}=\phi _{1}\mathbf {I}\) and \(\nu \) for a maximum of the likelihood function cannot be satisfied for \(\nu \ge 1\)”. In our case, the likelihood function is an increasing function of \(\nu \). For illustration, we consider the bivariate case, \({\varvec{t}}_{2}(0, \mathbf {I}, \nu )\) with density function given by

$$\begin{aligned} f(\mathbf {z}, \nu , \delta )=\frac{\varGamma ((\nu +2)/2)}{\varGamma (\nu /2)(\nu \pi )}\{1+\delta /\nu \}^{-(\nu +2)/2}, \end{aligned}$$

with \(\delta =\mathbf {z}^{\top }\mathbf {z}\). Clearly, from Fig. 5, the likelihood function is an increasing function of \(\nu \) and also of \(\delta \).

Fig. 5
figure 5

Likelihood function versus \(\nu \) and \(\delta \)

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De Bastiani, F., Mariz de Aquino Cysneiros, A.H., Uribe-Opazo, M.A. et al. Influence diagnostics in elliptical spatial linear models. TEST 24, 322–340 (2015). https://doi.org/10.1007/s11749-014-0409-z

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