Estimation and Influence Diagnostics for the Multivariate Linear Regression Models with Skew Scale Mixtures of Normal Distributions

Abstract

In this paper, we present recent results in the context of multivariate linear regression models considering that random errors follow multivariate skew scale mixtures of normal distributions. This class of distributions includes the scale mixtures of multivariate normal distributions, as special cases, and provides flexibility in capturing a wide variety of asymmetric behaviors. We implemented the algorithm ECM (Expectation/Conditional Maximization) and we obtained closed-form expressions for all the estimators of the parameters of the proposed model. Inspired by the ECM algorithm, we have developed an influence diagnostics for detecting influential observations to investigate the sensitivity of the maximum likelihood estimators. To examine the performance and the usefulness of the proposed methodology, we present simulation studies and analyze a real dataset.

Appendices

Appendix A. E-and M-steps in the ECM Algorithm

Firstly, we present the result of the exponential family whose corollaries allowed the complete establishment of the E-step in the STN-RM and the SSN-RM. After, we cite the main approximation result by weighted least squares from Powell (1981), which is useful to estimate the parameter τ in the STN-RM.

Proposition 2.

Given a random variable X in the exponential family, that is, with pdf $g(x|{\boldsymbol {\vartheta }})=\eta (x)\kappa ({\boldsymbol {\vartheta }})e^{{\sum }_{i=1}^{k}\omega _{i}({\boldsymbol {\vartheta }})t_{i}(x)}$ depending on 𝜗 = (𝜗₁,…,𝜗_m). Then,

$$ E\left( \sum\limits_{i=1}^{k}\frac{\partial \omega_{i}({\boldsymbol{\vartheta}})}{\partial\vartheta_{j}}T_{i}(X)\right)=-\frac{\partial}{\partial\vartheta_{j}}[\ln{\kappa({\boldsymbol{\vartheta}})}], \ \forall j\in\{1,\ldots,m\}. $$

Corollary 1.

Let $X\sim \text {Gamma}(\alpha ,\beta )$. Then, by Proposition 2,

$$E(\ln{X})=-\frac{\partial}{\partial\alpha}\left[\ln{\left( \frac{\beta^{\alpha}}{{\Gamma}(\alpha)}\right)}\right]={\Psi}(\alpha)-\ln{\beta}.$$

Corollary 2.

Let $X{\sim }\text {TGamma}(\alpha ,\beta ;(0,1)),$ where TGamma(r,s;(a,b)) denotes the univariate Gamma distribution (Gamma(r,s)), truncated on the interval (a,b). Then, by Proposition 2,

$$ E(\ln{X})=-\frac{\partial}{\partial\alpha}\left[\ln{\left( \frac{1}{{{\int}_{0}^{1}}\ x^{\alpha-1}e^{-\beta x}dx}\right)}\right]=\frac{\beta^{\alpha}{{\int}_{0}^{1}} x^{\alpha-1}\ln{(x)}e^{-\beta x}dx}{{\Gamma}(\alpha)P_{1}(\alpha,\beta)}.$$

Proposition 3.

Given $q\in {\mathscr{B}}=\mathcal {C}([a, b];\mathbb {R})$, i.e., a continuous real function on interval [a,b] and $w\in \mathcal {C}([a, b];(0,\infty ))$ a weight function, $p^{*}\in \mathcal {A}$ — a function in a finite-dimensional subspace of ${\mathscr{B}}$ — minimizes ${{\int \limits }_{a}^{b}} w(x)[q(x)-p(x)]^{2}dx=\left \langle q-p,q-p \right \rangle \text {with} p\in \mathcal {A}$, being called the best weighted least squares approximation for q within $\mathcal {A}$ if, and only if, the approximation error e^∗ = q − p^∗ satisfies $\left \langle e^{*},p \right \rangle =0 ,\ \forall p\in \mathcal {A}$.

Corollary 3.

If $\mathcal {U}=\{p_{0},p_{1},\ldots ,p_{n}\}$ is an orthogonal basis for a subspace $\mathcal {A}\subset {\mathscr{B}}=\mathcal {C}([a, b];\mathbb {R})$, then $p_{n}^{*}=\sum \limits _{j=0}^{n} c_{j}p_{j}$ with $c_{j}=\frac {\left \langle p_{j},q \right \rangle }{\left \|p_{j}\right \|^{2}}$ for each j ∈{0,1,…,n} is the best weighted least squares approximation for the function $q\in {\mathscr{B}}$ within $\mathcal {A}$.

Finally, we describe briefly how the above results have been used in each considered model.

• For the STN-RM $(U \sim Gamma(\tau /2,\tau /2)),$ we obtain the Q₂ function given in (3.5), where $\widehat {s}^{(k)}$ is defined in the expression (3.6) by applying Corollary 1. The derivative of this function relative to ν contains the function $e^{\eta (\nu )}=\ln {\left (\frac {\nu }{2}\right )}-{\Psi }\left (\frac {\nu }{2}\right )$. We apply a technique that was inspired by a method found in Powell (1981): approximation of functions by least squares weighted; see Proposition 3. This method has been used to approximate the function $\eta (x)=\ln {\left (\ln {x}-{\Psi }(x)\right )}$ through a function with the form $\xi (x)=c_{0}+c_{1}\ln {x}+c_{2}(\ln {x})^{2},$ with x = ν/2; see Corollary 3, considering the basis $\mathcal {U}$ as the orthogonalization of $\{1,\ln ,\ln ^{2}\}$. For that, we use the weight function $w(\nu )=\frac {2}{\nu }e^{-\nu /2}$ to obtain, with an accuracy of 10^− 4 on the norm constructed in Proposition 3, the approximation

  $$ \eta(\nu) \approx c_{0}+c_{1}\ln{\left( \frac{\nu}{2}\right)}+c_{2}\ln{\left( \frac{\nu}{2}\right)}^{2}, $$

  where $e^{c_{0}}\approx 0.5768854, c_{1}\approx -1.112673, c_{2}\approx 0.02783412$. Thus, by replacing the new function in $\frac {\partial Q_{2}}{\partial \nu }$ and solving its stationary equation, we find the expression (3.9).

• For the SSN-RM $(U\sim Beta(\tau ,1))$, we obtain

  $$Q_{2}\left( \nu|{\boldsymbol{\widehat{\theta}}}^{(k)}\right)=\sum\limits_{i=1}^{n}Q_{2i}\left( \nu|{\boldsymbol{\widehat{\theta}}}^{(k)}\right)=n\ln{\nu}+(\nu-1)\sum\limits_{i=1}^{n}\widehat{lu_{i}}^{(k)},$$

  where $\widehat {lu_{i}}^{(k)}$ is defined in the expression (3.7) by applying Corollary 2. Then a simple calculation shows that the solution of the stationary equation involving the derivative $\frac {\partial Q_{2}}{\partial \nu }$ is (3.10).

• For the SCN-RM, where τ = (ν,γ)^⊤ (U is a discrete variable assuming two states, such that h(u|τ) = νI_{u=γ} + (1 − ν)I_{u= 1}, the i th observation is drawn from of two populations: $\mathbf {Y}_{1} \sim ~ \text {SN}_{p}\left (\boldsymbol {\mu },\frac {\boldsymbol {\Sigma }}{\gamma },\frac {\boldsymbol {\lambda }}{\sqrt {\gamma }}\right )$ and $\mathbf {Y}_{2} \sim ~ \text {SN}_{p}\left (\boldsymbol {\mu },\boldsymbol {\Sigma },\boldsymbol {\lambda }\right )$. The complete data can be represented as $\mathbf {y}_{c}=\left (\mathbf {y}^{\top },\mathbf {v}_{1}^{\top },\mathbf {v}_{2}^{\top },\mathbf {t}^{\top }\right )^{\top },$ where $\mathbf {v}_{j}=\left (v_{j1},\ldots ,v_{jn}\right )^{\top },$ with V_ji = 1 if the i th observation is from the j th population and 0 otherwise, for i = 1,…,n and j = 1,2. It follows that the complete data likelihood for the i th observation is

  $$ \begin{array}{@{}rcl@{}} f_{c}\left( \mathbf{y}_{i},v_{1i},v_{2i},t_{i}\right)&=&2\left[\nu\phi_{p}\left( \mathbf{Y}_{i}|\mathbf{X}_{i}\boldsymbol{\beta},\boldsymbol{\Sigma}/\gamma\right)\right]^{v_{1i}}\left[(1-\nu)\phi_{p}(\mathbf{Y}_{i}|\mathbf{X}_{i}\boldsymbol{\beta},\boldsymbol{\Sigma})\right]^{v_{2i}}\\ &&\times \phi_{1}\left( t_{i}|\boldsymbol{\lambda}^{\top}\boldsymbol{\Sigma}^{-1/2}(\mathbf{Y}_{i}-\mathbf{X}_{i}\boldsymbol{\beta}),1\right). \end{array} $$

  In this context, the Q₂ function is given by

  $$ {Q}_{2}\left( \nu,\gamma|{\boldsymbol{\widehat{\theta}}}^{(k)}\right)=\left[\ln{(\nu)}+\frac{p}{2}\ln{(\gamma)}\right]\sum\limits_{i=1}^{n}\widehat{v_{1i}}^{(k)}+\ln{(1-\nu)}\sum\limits_{i=1}^{n}\widehat{v_{2i}}^{(k)}-\frac{\gamma}{2}\sum\limits_{i=1}^{n}\widehat{v_{1i}}^{(k)}\widehat{d_{i}}^{(k)}, $$

  where $\widehat {v_{1i}}^{(k)}$ and $\widehat {v_{2i}}^{(k)}$ are defined in the expression (3.8). Then, by solving the stationary conditions of Q₂ concerning ν and γ, we find the expressions given in (3.11).

Appendix B. Auxiliary Calculus of the Score Vector and the Information Matrix

Suppose that we have observations of n independent individuals, Y₁,…, Y_n, where $\mathbf {Y}_{i}\sim {SSMN}_{p}\left (\mathbf {X}_{i}\boldsymbol {\beta },\boldsymbol {\Sigma }({\boldsymbol {\alpha }}),\boldsymbol {\lambda },\boldsymbol {\tau }\right )$ for i = 1,…,n. Then, the log-likelihood function for $\boldsymbol {\theta }=\left (\boldsymbol {\beta }^{\top },{\boldsymbol {\alpha }}^{\top },\boldsymbol {\lambda }^{\top },\boldsymbol {\tau }^{\top }\right )^{\top }\in \mathbb {R}^{r},$ given the observed sample $\mathbf {y}=\left (\mathbf {y}^{\top }_{1},\ldots ,\mathbf {y}^{\top }_{n}\right )^{\top },$ is of the form

$$ \ell(\boldsymbol{\theta})=\sum\limits_{i=1}^{n} \ell_{i}(\boldsymbol{\theta}), $$

(B.1)

where $\ell _{i}(\boldsymbol {\theta })=\ln {2}-\frac {p}{2}\ln {2\pi }-\frac {1}{2}{\Lambda }+\ln {K_{i}}+\ln {\Phi }(A_{i})$ with $K_{i}=K_{i}(\boldsymbol {\theta })={\int \limits }^{\infty }_{0}u_{i}^{p/2}e^{-\frac {u_{i} d_{i}}{2}}h(u_{i}|\boldsymbol {\tau })du_{i}$. The score vector is given by $\frac {\partial \ell (\boldsymbol {\theta })}{\partial \boldsymbol {\theta }}=\sum \limits _{i=1}^{n}\frac {\partial \ell _{i}(\boldsymbol {\theta })}{\partial \boldsymbol {\theta }},$ where $\frac {\partial \ell _{i}(\boldsymbol {\theta })}{\partial \boldsymbol {\theta }}=\frac {\partial \ell _{i}}{\partial \boldsymbol {\theta }}=\left (\frac {\partial \ell _{i}}{\partial \boldsymbol {\beta }^{\top }},\frac {\partial \ell _{i}}{\partial {\boldsymbol {\alpha }}^{\top }},\frac {\partial \ell _{i}}{\partial \boldsymbol {\lambda }^{\top }},\frac {\partial \ell _{i}}{\partial \boldsymbol {\tau }^{\top }}\right )^{\top }$ with $\frac {\partial \ell _{i}}{\partial \boldsymbol {\beta }}=-\frac {1}{2}EU_{\mathbf {y}_{i}}\frac {\partial d_{i}}{\partial \boldsymbol {\beta }}+W_{\Phi }(A_{i})\frac {\partial A_{i}}{\partial \boldsymbol {\beta }}, \frac {\partial \ell _{i}}{\partial \boldsymbol {\alpha }}=-\frac {1}{2}\frac {\partial {\Lambda }}{\partial \boldsymbol {\alpha }}-\frac {1}{2}EU_{\mathbf {y}_{i}}\frac {\partial d_{i}}{\partial \boldsymbol {\alpha }}+W_{\Phi }(A_{i})\frac {\partial A_{i}}{\partial \boldsymbol {\alpha }},$ $\frac {\partial \ell _{i}}{\partial \boldsymbol {\lambda }}=W_{\Phi }(A_{i})\frac {\partial A_{i}}{\partial \boldsymbol {\lambda }}\quad \text {and}$ $\quad \frac {\partial \ell _{i}}{\partial \boldsymbol {\tau }}=\frac {1}{K_{i}}\frac {\partial K_{i}}{\partial \boldsymbol {\tau }},$ such that W_Φ(A_i) = ϕ₁(A_i)/Φ(A_i) and $EU_{\mathbf {y}_{i}}=E\left (U_{i}|\mathbf {Y}=\mathbf {y}_{i}\right )$. It follows that the matrix of the second derivatives concerning 𝜃 is given by

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\top}}= \begin{bmatrix} \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\beta}\partial\boldsymbol{\alpha}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\beta}\partial\boldsymbol{\tau}^{\top}} \\ \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\beta}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\lambda}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\tau}^{\top}} \\ \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\beta}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\alpha}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\lambda}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\tau}^{\top}} \\ \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\tau}\partial\boldsymbol{\beta}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\tau}\partial\boldsymbol{\alpha}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\tau}\partial\boldsymbol{\lambda}^{\top}} & \frac{\partial^{2}\ell_{i}(\boldsymbol{\theta})}{\partial\boldsymbol{\tau}\partial\boldsymbol{\tau}^{\top}} \end{bmatrix}, \end{array} $$

with

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}&=&-\frac{1}{2}EU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}+\frac{1}{4}VU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}^{\top}}+W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\beta}^{\top}},\\ \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\beta}\partial{\boldsymbol{\alpha}}^{\top}}&=&-\frac{1}{2}EU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\alpha}^{\top}}+\frac{1}{4}VU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}^{\top}}+W_{\Phi}(A_{i})\frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\alpha}^{\top}}+W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}^{\top}},\\ \frac{\partial^{2}\ell_{i}}{\partial{\boldsymbol{\alpha}}\partial{\boldsymbol{\alpha}}^{\top}}&=&-\frac{1}{2}\frac{\partial^{2}{\Lambda}}{\partial{\boldsymbol{\alpha}}\partial{\boldsymbol{\alpha}}^{\top}}-\frac{1}{2}EU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}+\frac{1}{4}VU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}^{\top}}+W_{\Phi}(A_{i})\frac{\partial^{2}A_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}\\&&+W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}}\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}^{\top}},\\ \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}&=&W_{\Phi}(A_{i})\frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}+W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}},\quad \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\lambda}^{\top}}=W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}},\\ \frac{\partial^{2}\ell_{i}}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\lambda}^{\top}}&=&W_{\Phi}(A_{i})\frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}+W^{\prime}_{\Phi}(A_{i})\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}},\quad \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\tau}^{\top}}=\mathbf{0},\\ \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\tau}\partial\boldsymbol{\tau}^{\top}}&=&\frac{1}{K_{i}}\frac{\partial^{2}K_{i}}{\partial\boldsymbol{\tau}\partial\boldsymbol{\tau}^{\top}}-\frac{1}{{K_{i}^{2}}}\frac{\partial K_{i}}{\partial\boldsymbol{\tau}}\frac{\partial K_{i}}{\partial\boldsymbol{\tau}^{\top}},\quad \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\tau}^{\top}}=\frac{1}{K_{i}}\frac{\partial^{2}K_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\tau}^{\top}}-\frac{1}{{K_{i}^{2}}}EU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}}\frac{\partial K_{i}}{\partial\boldsymbol{\tau}^{\top}},\\ \frac{\partial^{2}\ell_{i}}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\tau}^{\top}}&=&\frac{1}{K_{i}}\frac{\partial^{2}K_{i}}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\tau}^{\top}}-\frac{1}{{K_{i}^{2}}}EU_{\mathbf{y}_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}}\frac{\partial K_{i}}{\partial\boldsymbol{\tau}^{\top}}, \end{array} $$

where $W_{\Phi }^{\prime }(A_{i})=-W_{\Phi }(A_{i})(A_{i}+W_{\Phi }(A_{i}))$ and $VU_{\mathbf {y}_{i}}=Var\left (U_{i}|\mathbf {Y}=\mathbf {y}_{i}\right )$. The first and second derivatives of K_i(𝜃) concerning 𝜃 are given by $\frac {\partial K_{i}(\boldsymbol {\theta })}{\partial \boldsymbol {\theta }}=-\frac {1}{2} K_{i}(\boldsymbol {\theta }) \frac {\partial d_{i}}{\partial \boldsymbol {\theta }} EU_{\mathbf {y}_{i}}$ and $\frac {\partial ^{2} K_{i}(\boldsymbol {\theta })}{\partial \boldsymbol {\theta } \partial \boldsymbol {\theta }^{\top }}=-\frac {K_{i}(\boldsymbol {\theta })}{2}\left [\frac {\partial ^{2} d_{i}}{\partial \boldsymbol {\theta } \partial \boldsymbol {\theta }^{\top }} EU_{\mathbf {y}_{i}}-\frac {1}{2}\frac {\partial d_{i}}{\partial \boldsymbol {\theta }}\frac {\partial d_{i}}{\partial \boldsymbol {\theta }^{\top }}EU^{2}_{\mathbf {y}_{i}} \right ]$, such that $EU^{2}_{\mathbf {y}_{i}}=E\left ({U_{i}^{2}}|\mathbf {Y}=\mathbf {y}_{i}\right )$. The expected values $E\left (U^{l}|\mathbf {Y}=\mathbf {y}\right )$ for the SN, STN, SSN and SCN distributions can be found in Ferreira et al. (2016); see Section 2.1. According to Harville (1997), consider α = vech(B), where Σ^1/2 = B = B(α). Thus, the first and second derivatives of Λ, A_i and d_i can be obtained.

• Λ

  $$ \begin{array}{@{}rcl@{}} \frac{\partial{\Lambda}}{\partial\boldsymbol{\theta}}&=&\left( \mathbf{0},\frac{\partial{\Lambda}}{\partial{\boldsymbol{\alpha}}^{\top}},\mathbf{0},\mathbf{0}\right)^{\top}, \frac{\partial{\Lambda}}{\partial\boldsymbol{\alpha}}= \left[\frac{\partial{\Lambda}}{\partial\alpha_{j}}\right]_{j\in\{1,\ldots,p_{0}\}}=\left[2\text{tr}\left( \mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\right)\right]_{j\in\{1,\ldots,p_{0}\}},\\ \frac{\partial^{2}{\Lambda}}{\partial\boldsymbol{\theta}\partial\boldsymbol{\theta}^{\top}}&=&\begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \frac{\partial^{2}{\Lambda}}{\partial{\boldsymbol{\alpha}}\partial{\boldsymbol{\alpha}}^{\top}} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{bmatrix},\\ \frac{\partial^{2}{\Lambda}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}&=&\left[\frac{\partial^{2}{\Lambda}}{\partial\alpha_{j}\partial\alpha_{k}}\right]_{j,k\in\{1,\ldots,p_{0}\}}=\left[-\text{tr}\left( \mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\right)\right]_{j,k\in\{1,\ldots,p_{0}\}}, \end{array} $$

  where $\dot {\mathbf {B}}_{s}=\partial {\mathbf {B}(\boldsymbol {\alpha })}/\partial {\alpha _{s}}$ with s = 1,2,…,p₀ = p(p + 1)/2 and e_i = y_i −X_iβ.

• A_i

  $$ \begin{array}{@{}rcl@{}} \frac{\partial A_{i}}{\partial\boldsymbol{\beta}}&=&-\mathbf{X}_{i}^{\top}\mathbf{B}^{-1}\boldsymbol{\lambda}, \frac{\partial A_{i}}{\partial\alpha_{j}}=-\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}\left( \mathbf{Y}_{i}-\mathbf{X}_{i}\boldsymbol{\beta}\right), \frac{\partial A_{i}}{\partial\boldsymbol{\lambda}}=\mathbf{B}^{-1}\mathbf{e}_{i}, \frac{\partial A_{i}}{\partial\boldsymbol{\tau}}=\mathbf{0},\\ \frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}&=&\mathbf{0}, \frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\alpha_{j}}=\mathbf{X}_{i}^{\top}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}\boldsymbol{\lambda}, \frac{\partial^{2}A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}=-\mathbf{X}_{i}^{\top}\mathbf{B}^{-1},\quad \frac{\partial^{2}A_{i}}{\partial\boldsymbol{\theta}\partial\boldsymbol{\tau}^{\top}}=\mathbf{0},\\ \frac{\partial^{2}A_{i}}{\partial\alpha_{j}\partial\alpha_{k}}&=&-\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}+\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\right)\mathbf{B}^{-1}\mathbf{e}_{i}\\ \frac{\partial^{2}A_{i}}{\partial\alpha_{j}\partial\boldsymbol{\lambda}^{\top}}&=&-\mathbf{e}_{i}^{\top}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}, \frac{\partial^{2}A_{i}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\lambda}^{\top}}=\mathbf{0}. \end{array} $$

• d_i

  $$ \begin{array}{@{}rcl@{}} \frac{\partial d_{i}}{\partial\boldsymbol{\beta}}&=&-2\mathbf{X}_{i}^{\top}\mathbf{B}^{-2}\mathbf{e}_{i}, \frac{\partial d_{i}}{\partial\alpha_{j}}=-\mathbf{e}_{i}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{j}\mathbf{B}^{-1}+\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\right)\mathbf{B}^{-1}\mathbf{e}_{i}, \frac{\partial d_{i}}{\partial\boldsymbol{\lambda}}=\mathbf{0}, \frac{\partial d_{i}}{\partial\boldsymbol{\tau}}=\mathbf{0},\\ \frac{\partial^{2}d_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}&=&2\mathbf{X}_{i}^{\top}\mathbf{B}^{-2}\mathbf{X}_{i}, \frac{\partial^{2}d_{i}}{\partial\boldsymbol{\beta}\partial\alpha_{j}}=2\mathbf{X}_{i}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{j}\mathbf{B}^{-1}+\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\right)\mathbf{B}^{-1}\mathbf{e}_{i},\\ \frac{\partial^{2}d_{i}}{\partial\boldsymbol{\theta}\partial\boldsymbol{\lambda}^{\top}}&=&\mathbf{0}, \frac{\partial^{2}d_{i}}{\partial\boldsymbol{\theta}\partial\boldsymbol{\tau}^{\top}}=\mathbf{0},\\ \frac{\partial^{2}d_{i}}{\partial\alpha_{j}\partial\alpha_{k}}&=&\mathbf{e}_{i}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}+\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\mathbf{B}^{-1}+\dot{\mathbf{B}}_{j}\mathbf{B}^{-2}\dot{\mathbf{B}}_{k}+\dot{\mathbf{B}}_{k}\mathbf{B}^{-2}\dot{\mathbf{B}}_{j}\right.\\ &&\left.+\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}+\mathbf{B}^{-1}\dot{\mathbf{B}}_{j}\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\right)\mathbf{B}^{-1}\mathbf{e}_{i}, \end{array} $$

and the derivatives of ℓ_i can be obtained for the STN-RM, the SSN-RM and the SCN-RM as follows.

• STN-RM

  $$ \begin{array}{@{}rcl@{}} \frac{\partial \ell_{i}}{\partial\tau}&=&\frac{1}{2}\left[\ln{\left( \frac{\tau}{\tau+d_{i}}\right)}+{\Psi}\left( \frac{\tau+p}{2}\right)-{\Psi}\left( \frac{\tau}{2}\right)+\frac{d_{i}-p}{\tau+d_{i}}\right],\\\frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\theta}_{1}\partial\tau}&=&\frac{1}{2}\frac{p-d_{i}}{(\tau+d_{i})^{2}}\frac{\partial d_{i}}{\partial\boldsymbol{\theta}_{1}}\quad\text{and}\\\frac{\partial^{2}\ell_{i}}{\partial\tau^{2}}&=&\frac{1}{4}\left[{\Psi}_{1}\left( \frac{\tau+p}{2}\right)-{\Psi}_{1}\left( \frac{\tau}{2}\right)+\frac{2(d_{i}+p\tau)}{\tau(\tau+d_{i})^{2}}\right], \end{array} $$

  where Ψ₁(.) is the trigamma function.

• SSN-RM

  $$ \begin{array}{@{}rcl@{}} \frac{\partial \ell_{i}}{\partial\tau}&=&\frac{1}{\tau}+\frac{I(1,1)}{I(1,0)}, \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\theta}_{1}\partial\tau}=-\frac{1}{2}\left[\frac{I(0,1)}{I(1,0)}-\frac{I(1,1)I(0,0)}{I(1,0)^{2}}\right]\frac{\partial d_{i}}{\partial\boldsymbol{\theta}_{1}} \text{and} \\ \frac{\partial^{2}\ell_{i}}{\partial\tau^{2}}&=&\frac{I(1,2)}{I(1,0)}-\left[\frac{I(1,1)}{I(1,0)}\right]^{2}-\frac{1}{\tau^{2}}, \end{array} $$

  where $I(a,b)={{\int \limits }_{0}^{1}}\ u^{\nu +p/2-a}(\ln {u})^{b} e^{-ud_{i}/2}du$.

• SCN-RM, where τ = (ν,γ)^⊤

  $$ \begin{array}{@{}rcl@{}} \frac{\partial\ell_{i}}{\partial\nu}&=&\frac{\gamma^{p/2}e^{(1-\gamma) d_{i}/2}-1}{1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}},\quad \frac{\partial\ell_{i}}{\partial\gamma}=\frac{\nu}{2}\frac{(p-\gamma d_{i})\left( \gamma^{p/2-1}e^{(1-\gamma) d_{i}/2}-1\right)}{1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}},\\ \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\theta}_{1}\partial\nu}&=&\frac{1}{2}\frac{(1-\gamma)\gamma^{p/2}e^{(1-\gamma)d_{i}/2}}{\left[1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}\right]^{2}}\frac{\partial d_{i}}{\partial\boldsymbol{\theta}_{1}},\\ \frac{\partial^{2}\ell_{i}}{\partial\boldsymbol{\theta}_{1}\partial\gamma}&=&\frac{\nu}{4}\frac{(1-\nu)\gamma^{p/2-1}[p-\gamma(2+p+d_{i})+\gamma^{2}d_{i}]e^{(1-\gamma)d_{i}/2}-2\nu\gamma^{p} e^{(1-\gamma) d_{i}}}{\left[1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}\right]^{2}}\frac{\partial d_{i}}{\partial\boldsymbol{\theta}_{1}},\\ \frac{\partial^{2}\ell_{i}}{\partial\nu^{2}}&=&-\frac{\left[\gamma^{p/2}e^{(1-\gamma) d_{i}/2}-1\right]^{2}}{\left[1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}\right]^{2}},\quad \frac{\partial^{2}\ell_{i}}{\partial\nu\partial\gamma}=\frac{1}{2}\frac{(p-\gamma d_{i})\gamma^{p/2-1}e^{(1-\gamma) d_{i}/2}}{\left[1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}\right]^{2}},\\ \frac{\partial^{2}\ell_{i}}{\partial\gamma^{2}}&=&\frac{\nu}{4}\frac{(1-\nu)\gamma^{p/2-2}[(p-\gamma d_{i})^{2}-2p]e^{(1-\gamma)d_{i}/2}-2p\nu\gamma^{p-2}e^{(1-\gamma) d_{i}}}{\left[1-\nu+\nu\gamma^{p/2}e^{(1-\gamma)d_{i}/2}\right]^{2}}. \end{array} $$

Appendix C. Auxiliary Calculus for the Influence Measures

To obtain the diagnostic measures, it is necessary to compute $\frac {\partial ^{2}Q}{\partial \boldsymbol {\theta }\partial \boldsymbol {\theta }^{\top }}\left (\!{\boldsymbol {\theta }}|{\boldsymbol {\widehat {\theta }}}\right )$ $={\sum }_{i=1}^{n} \frac {\partial ^{2}Q_{i}}{\partial \boldsymbol {\theta }\partial \boldsymbol {\theta }^{\top }}\left ({\boldsymbol {\theta }}|{\boldsymbol {\widehat {\theta }}}\right )$. Hence, the Hessian matrix has elements given by

$$ \begin{array}{@{}rcl@{}} \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}&=&-\frac{1}{2}\widehat{u_{i}}\frac{\partial^{2} d_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\beta}^{\top}}-\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\beta}^{\top}},\\ \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\beta}\partial{\boldsymbol{\alpha}}^{\top}}&=&-\frac{1}{2}\widehat{u_{i}}\frac{\partial^{2} d_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\alpha}^{\top}}+\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial^{2} A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\alpha}^{\top}}-\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}^{\top}},\\ \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}&=&-\frac{1}{2}\frac{\partial^{2}{\Lambda}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}-\frac{1}{2}\widehat{u_{i}}\frac{\partial^{2} d_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}+\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial^{2} A_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\alpha}^{\top}}-\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}}\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}^{\top}},\\ \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}&=&\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial^{2} A_{i}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\lambda}^{\top}}-\frac{\partial A_{i}}{\partial\boldsymbol{\beta}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}},\\ \frac{\partial^{2}Q_{i}}{\partial{\boldsymbol{\alpha}}\partial\boldsymbol{\lambda}^{\top}}&=&\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial^{2} A_{i}}{\partial\boldsymbol{\alpha}\partial\boldsymbol{\lambda}^{\top}}-\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}},\\ \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\lambda}^{\top}}&=&-\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}}\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}^{\top}}, \quad \frac{\partial^{2} Q_{i}}{\partial\boldsymbol{\theta}_{1}\partial\boldsymbol{\tau}^{\top}}=\mathbf{0}, \end{array} $$

where $A_{i}=A_{i}(\boldsymbol {\theta })=\boldsymbol {\lambda }^{\top }\boldsymbol {\Sigma }^{-1/2}\left (\mathbf {y}_{i}-\mathbf {X}_{i}\boldsymbol {\beta }\right )$. The terms $\frac {\partial ^{2} Q_{i}}{\partial \boldsymbol {\tau }\partial \boldsymbol {\tau }^{\top }}$ can be readily evaluated for the STN-RM, the SSN-RM and the SCN-RM as follows.

• STN-RM

  $$ \frac{\partial^{2}Q_{i}}{\partial\tau^{2}}=\frac{1}{4}\left[\frac{2}{\tau}-{\varPsi}_{1}\left( \frac{\tau}{2}\right)\right],$$

  where Ψ₁(⋅) is the trigamma function.

• SSN-RM

  $$\frac{\partial^{2}Q_{i}}{\partial\tau^{2}}=-\frac{1}{\tau^{2}}.$$

• SCN-RM, where τ = (ν,γ)^⊤

  $$\frac{\partial^{2}Q_{i}}{\partial\nu^{2}}=-\left[\frac{\widehat{v_{1i}}}{\nu^{2}}+\frac{\widehat{v_{2i}}}{(1-\nu)^{2}}\right],\quad \frac{\partial^{2}Q_{i}}{\partial\nu\partial\gamma}=0 \quad\text{and} \quad \frac{\partial^{2}Q_{i}}{\partial\gamma^{2}}=-\frac{p\widehat{v_{1i}}}{2\gamma^{2}}.$$

In addition, the derivatives of Λ, d_i and A_i are given in Appendix B.

• (i) Case weight perturbation

  $$ \begin{array}{@{}rcl@{}} \frac{\partial Q_{i}}{\partial\boldsymbol{\beta}}&=&-\frac{1}{2}\widehat{u_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\beta}}+\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial A_{i}}{\partial\boldsymbol{\beta}},\quad \frac{\partial Q_{i}}{\partial\boldsymbol{\alpha}}=-\frac{1}{2}\frac{\partial{\Lambda}}{\partial\boldsymbol{\alpha}}-\frac{1}{2}\widehat{u_{i}}\frac{\partial d_{i}}{\partial\boldsymbol{\alpha}}+\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial A_{i}}{\partial\boldsymbol{\alpha}},\\ \frac{\partial Q_{i}}{\partial\boldsymbol{\lambda}}&=&\left( \widehat{t_{i}}-A_{i}\right)\frac{\partial A_{i}}{\partial\boldsymbol{\lambda}} \quad \text{and} \quad \frac{\partial Q_{i}}{\partial\boldsymbol{\tau}} \end{array} $$

can be readily evaluated for the STN-RM, the SSN-RM and the SCN-RM as follows.

• STN-RM

  $$ \begin{array}{@{}rcl@{}} \frac{\partial Q_{i}}{\partial\tau}=\frac{1}{2}\left[\ln{\left( \frac{\tau}{2}\right)}-{\Psi}\left( \frac{\tau}{2}\right)+1-\widehat{u_{i}}+\widehat{lu_{i}}\right].\end{array} $$

• SSN-RM

  $$ \begin{array}{@{}rcl@{}} \frac{\partial Q_{i}}{\partial\tau}=\frac{1}{\tau}+\widehat{lu_{i}}. \end{array} $$

• SCN-RM, where τ = (ν,γ)^⊤

  $$ \begin{array}{@{}rcl@{}}\frac{\partial Q_{i}}{\partial\nu}=\frac{\widehat{v_{1i}}}{\nu}-\frac{\widehat{v_{2i}}}{1-\nu}\quad \text{and} \quad \frac{\partial Q_{i}}{\partial\gamma}=\frac{1}{2\gamma}\widehat{v_{1i}}(p-\gamma d_{i}). \end{array} $$

• (ii) Response perturbation

  $$ \begin{array}{@{}rcl@{}} \frac{\partial A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\omega}^{\top}}&=&\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\mathbf{D}\mathbf{S}_{\mathbf{y}}\mathbf{c}_{i}^{\top}, \qquad \frac{\partial^{2}A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\omega}^{\top}}=\mathbf{B}^{-1}\mathbf{D}\mathbf{S}_{\mathbf{y}}\mathbf{c}_{i}^{\top},\\ \frac{\partial^{2}d_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\omega}^{\top}}&=&-2\mathbf{X}_{i}^{\top}\mathbf{B}^{-2}\mathbf{D}\mathbf{S}_{\mathbf{y}}\mathbf{c}_{i}^{\top}, \qquad \frac{\partial^{2}A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\alpha_{k}\partial\boldsymbol{\omega}^{\top}}=-\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\mathbf{D}\mathbf{S}_{\mathbf{y}}\mathbf{c}_{i}^{\top},\\ \frac{\partial^{2}d_{i}^{\boldsymbol{\omega}_{0}}}{\partial\alpha_{k}\partial\boldsymbol{\omega}^{\top}}&=&-2\mathbf{e}_{i}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{k}\mathbf{B}^{-1}+\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\right)\mathbf{B}^{-1}\mathbf{D}\mathbf{S}_{\mathbf{y}}\mathbf{c}_{i}^{\top}. \end{array} $$

• (iii) Explanatory perturbation

  $$ \begin{array}{@{}rcl@{}} \frac{\partial A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\omega}^{\top}}&=&-\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\mathbf{E}\mathbf{M}_{\mathbf{x}}\boldsymbol{\beta}\mathbf{c}_{i}^{\top}, \qquad \frac{\partial^{2}A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\omega}^{\top}}=-\mathbf{E}\mathbf{M}_{\mathbf{x}}^{\top}\mathbf{B}^{-1}\boldsymbol{\lambda}\mathbf{c}_{i}^{\top},\\ \frac{\partial^{2}A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\alpha_{k}\partial\boldsymbol{\omega}^{\top}}&=&\boldsymbol{\lambda}^{\top}\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\mathbf{B}^{-1}\mathbf{E}\mathbf{M}_{\mathbf{x}}\boldsymbol{\beta}\mathbf{c}_{i}^{\top}, \qquad \frac{\partial^{2}A_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\lambda}\partial\boldsymbol{\omega}^{\top}}=-\mathbf{B}^{-1}\mathbf{E}\mathbf{M}_{\mathbf{x}}\boldsymbol{\beta}\mathbf{c}_{i}^{\top},\\ \frac{\partial^{2}d_{i}^{\boldsymbol{\omega}_{0}}}{\partial\boldsymbol{\beta}\partial\boldsymbol{\omega}^{\top}}&=&-2\left( \mathbf{E}\mathbf{M}_{\mathbf{x}}^{\top}\mathbf{B}^{-2}\mathbf{e}_{i}-\mathbf{X}_{i}^{\top}\mathbf{B}^{-2}\mathbf{E}\mathbf{M}_{\mathbf{x}}\boldsymbol{\beta}\right)\mathbf{c}_{i}^{\top},\\ \frac{\partial^{2}d_{i}^{\boldsymbol{\omega}_{0}}}{\partial\alpha_{k}\partial\boldsymbol{\omega}^{\top}}&=&2\mathbf{e}_{i}^{\top}\mathbf{B}^{-1}\left( \dot{\mathbf{B}}_{k}\mathbf{B}^{-1}+\mathbf{B}^{-1}\dot{\mathbf{B}}_{k}\right)\mathbf{B}^{-1}\mathbf{E}\mathbf{M}_{\mathbf{x}}\boldsymbol{\beta}\mathbf{c}_{i}^{\top}. \end{array} $$