Influence diagnostics for Grubbs’s model with asymmetric heavy-tailed distributions

Abstract

Grubbs’s model (Grubbs, Encycl Stat Sci 3:42–549, 1983) is used for comparing several measuring devices, and it is common to assume that the random terms have a normal (or symmetric) distribution. In this paper, we discuss the extension of this model to the class of scale mixtures of skew-normal distributions. Our results provide a useful generalization of the symmetric Grubbs’s model (Osorio et al., Comput Stat Data Anal, 53:1249–1263, 2009) and the asymmetric skew-normal model (Montenegro et al., Stat Pap 51:701–715, 2010). We discuss the EM algorithm for parameter estimation and the local influence method (Cook, J Royal Stat Soc Ser B, 48:133–169, 1986) for assessing the robustness of these parameter estimates under some usual perturbation schemes. The results and methods developed in this paper are illustrated with a numerical example.

Appendices

Appendix A: Elements of the Hessian matrix $\ddot{Q}(\widehat{\varvec{\theta }})$

$$\begin{aligned} \frac{\partial ^2 Q_{1i}\left( \varvec{\theta }_1,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \varvec{\alpha }\partial {\varvec{\alpha }}^{\top }}&= -\widehat{u}_i\mathbb{I }_{(p)} D^{-1}(\varvec{\phi })\mathbb{I }_{(p)}^{\top },\, \displaystyle \frac{\partial ^2 Q_{1i}\left( \varvec{\theta }_1,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \varvec{\phi }\partial {\varvec{\alpha }}^{\top }}\\&= \left( -\widehat{u}_iD(\mathbf y _i-\mathbf a )+\widehat{ux}_i\mathbf I _p\right) D^{-2}(\varvec{\phi })\mathbb{I }^{\top }_{(p)},\\ \frac{\partial ^2 Q_{1i}\left( \varvec{\theta }_1,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \varvec{\phi }\partial {\varvec{\phi }}^{\top }}&= \left[ \frac{1}{2}D(\varvec{\phi })-\widehat{u}_iD^2(\mathbf y _i-\mathbf a ) +2\widehat{ux}_iD(\mathbf y _i-\mathbf a )-\widehat{ux^2}_i\mathbf I _p\right] D^{-3}(\varvec{\phi }),\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \mu _x \partial {\mu _x}}&= -\frac{1}{\nu _x^2}\widehat{u}_i,\,\, \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \mu _x \partial {\phi _x}} =\frac{\lambda _x\left( 1+\lambda _x^2\right) ^{1/2}}{2\phi _x^{3/2}}\widehat{ut}_i +\frac{\left( 1+{\lambda _x^2}\right) }{\phi _x^2}B_{1i},\\ \displaystyle \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \mu _x \partial {\lambda _x}}&= -\frac{2\lambda _x}{\phi _x}B_{1i}-\frac{\left( 1+2\lambda _x^2\right) }{\phi _x^{1/2}\left( 1+\lambda _x^2\right) ^{1/2}}\widehat{ut}_i,\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \phi _x \partial {\phi _x}}&= \frac{1}{2\phi _x^2}-\frac{3\lambda _x \left( 1+\lambda _x^2\right) ^{1/2}}{4\phi _x^{5/2}} B_{2i} -\frac{\left( 1+\lambda _x^2\right) }{\phi _x^3}B_{3i},\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \lambda _x \partial {\phi _x}}&= \frac{\left( 1+2\lambda _x^2\right) }{2\phi _x^{3/2}\left( 1+\lambda _x^2\right) ^{1/2}}B_{2i} +\frac{\lambda _x}{\phi _x^2}B_{3i},\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \lambda _x \partial {\lambda _x}}&= \frac{\left( 1-\lambda _x^2\right) }{\left( 1+\lambda _x^2\right) ^2}-\frac{1}{\phi _x}B_{3i}-\widehat{ut^2}_i -\frac{\lambda _x\left( 3+2\lambda _x^2\right) }{\phi _x^{1/2}\left( 1+\lambda _x^2\right) ^{3/2}}B_{2i}, \end{aligned}$$

where $B_{1i}=\widehat{u}_i\mu _x-\widehat{ux}_i$, $B_{2i}=\widehat{ut}_i\mu _x-\widehat{utx}_i$ and $B_{3i}=\widehat{ux^2}_i +\widehat{u}_i\mu _x^2-2\widehat{ux}_i\mu _x$.

Appendix B: Case weights perturbation

$$\begin{aligned} \frac{\partial ^2 Q_{1i}\left( \varvec{\theta }_1,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \varvec{\alpha }\partial {\omega _i}}&= \widehat{u}_i\mathbb{I }_{(p)}D^{-1}(\varvec{\phi })(\mathbf y _i-\mathbf a )-\widehat{ux}_i\mathbb{I }_{(p)}D^{-1}(\varvec{\phi })\mathbf 1 _p,\\ \frac{\partial ^2 Q_{1i}\left( \varvec{\theta }_1,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \varvec{\phi }\partial {\omega _i}}&= \frac{1}{2}\left[ \!-\!D(\varvec{\phi })\!+\!\widehat{u}_i D^2\left( \mathbf y _i\!-\!\mathbf a \right) \!-\!2\widehat{ux}_i D\left( \mathbf y _i\!-\!\mathbf a \right) \!+\!\widehat{ux^2}_i \mathbf I _p\right] D^{-2}(\varvec{\phi })\mathbf 1 _p,\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \mu _x \partial {\omega _i}}&= -\frac{1}{\nu _x^2}\left( \widehat{u}_i\mu _x-\widehat{ux}_i+\widehat{ut}_i\tau _x\right) ,\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \phi _x \partial {\omega _i}}&= -\frac{1}{2\phi _x}+\frac{\lambda _x\left( 1+\lambda _x^2\right) ^{1/2}}{2\phi _x^{3/2}}B_{2i}+\frac{1+\lambda _x^2}{2\phi _x^{2}}B_{3i},\\ \frac{\partial ^2 Q_{2i}\left( \varvec{\theta }_2,\varvec{\omega }_0|\widehat{\varvec{\theta }}\right) }{\partial \lambda _x \partial {\omega _i}}&= \frac{\lambda _x}{\left( 1+\lambda _x^2\right) }- \frac{\lambda _x}{\phi _x} B_{3i}-\lambda _x\widehat{ut^2}_i-\frac{\left( 1+2\lambda _x^2\right) }{\phi _x^{1/2}\left( 1+\lambda _x^2\right) ^{1/2}}B_{2i}. \end{aligned}$$

Appendix C: Moments of the latent variable $x_i$

For $x_i \sim \text{ SMSN }(\mu _x,\phi _x,\lambda _x;H)$. Let $\kappa _m=E\left[ \kappa ^{m/2}(U_i)\right]$, $\tau _x=\phi _x^{1/2}\delta _x$ and $\delta _x={\lambda _x}/{\sqrt{1+\lambda _x^2}}$. Then: (a) If $\kappa _1<\infty$, $E[x_i]= \mu _x+\sqrt{\frac{2}{\pi }} \kappa _1\tau _x$; (b) If $\kappa _2<\infty$, $Var[x_i]=\kappa _2\phi _x-\frac{2}{\pi }\kappa _1^2 \tau _x^2$. For special cases: (i) Normal and SN; $\kappa _1=\kappa _2=1$; (ii) ST; $\kappa _1=\frac{\Gamma \left( \frac{\nu -1}{2}\right) }{\Gamma (\frac{\nu }{2})},\nu >1\;\text{ and }\;\kappa _2=\frac{\nu }{\nu -2},\nu >2$; (iii) SSL: $\kappa _1=\frac{2\nu }{2\nu -1}, \nu >1/2\;\text{ and }\;\kappa _2=\frac{\nu }{\nu -1},\nu >1$; (iv) SCN: $\kappa _1=\frac{\nu }{\sqrt{\gamma }}+1-\nu \;\text{ and }\;\kappa _2=\frac{\nu }{\gamma }+1-\nu$.