On a new extreme value distribution: characterization, parametric quantile regression, and application to extreme air pollution events

Abstract

Extreme-value distributions are important when modeling weather events, such as temperature and rainfall. These distributions are also important for modeling air pollution events. Particularly, the extreme-value Birnbaum-Saunders regression is a helpful tool in the modeling of extreme events. However, this model is implemented by adding covariates to the location parameter. Given the importance of quantile regression to estimate the effects of covariates along the wide spectrum of a response variable, we introduce a quantile extreme-value Birnbaum-Saunders distribution and its corresponding quantile regression model. We implement a likelihood-based approach for parameter estimation and consider two types of statistical residuals. A Monte Carlo simulation is performed to assess the behavior of the estimation method and the empirical distribution of the residuals. We illustrate the introduced methodology with unpublished real air pollution data.

Appendices

Appendices

1.1 Appendix A: Proofs

Proof of Proposition 2.1

This is immediate from a routine differentiation and hence the proof is omitted. $\square$

Proof of Theorem 2.2

Let us denote

$$\begin{aligned}&g(t) = \exp ( -\mathfrak {a}_t )-1 = \exp \Biggl ( -{1\over \alpha }\biggl (x-{1\over x}\biggr ) \Biggr )-1,\\&\quad h(t) = -{\mathfrak {a}''_t\over (\mathfrak {a}'_t)^2} = \dfrac{\displaystyle x+{3/x}}{ \displaystyle ({1/\alpha }) \left( x+{1/x}\right) ^2}, \end{aligned}$$

with $x=\sqrt{t/\beta }$ and $\beta ={4Q\Lambda _{\alpha ,q}}$. By Proposition 2.1, when $\xi =0$, all mode t satisfies $g(t)=h(t)$. Since

$$\begin{aligned}&g'(t)=-\dfrac{1}{2\alpha \beta x} \biggl (\dfrac{1}{x^2}+1\biggr ) \exp \Biggl (\displaystyle -{1\over \alpha }\biggl (x-{1\over x}\biggr ) \Biggr )<0,\\&g''(t)=\dfrac{1}{4\alpha ^2\beta ^2 x^6} \big (\alpha x(x^2+3)+(x^2+1)^2\big ) \\&\quad \exp \Biggl (\displaystyle -{1\over \alpha }\biggl (x-{1\over x}\biggr ) \Biggr )>0, \forall t>0, \end{aligned}$$

g is strictly decreasing and concave up. Moreover, $g(\beta )= 0$, $\lim _{t\rightarrow 0^+}g(t)=\infty$ and $\lim _{t\rightarrow \infty }g(t)=-1$. Consider

$$\begin{aligned} h'(t) = -\dfrac{\alpha (x^4+6x^2-3)}{2\beta x(x^2+1)^3},\quad h''(t) = \dfrac{3\alpha (x^6+9x^4-9x^2-1)}{4\beta ^2 x^3(x^2+1)^4}. \end{aligned}$$

Note that $h'(t)=0 \Longleftrightarrow x^4+6x^2-3=0 \Longleftrightarrow x=\sqrt{2\sqrt{3}-3} \Longleftrightarrow t=t_0=\beta (2\sqrt{3}-3),$ in which $h''(t_0)=3(1.75+\sqrt{3})(80-48\sqrt{3})\alpha /\big (64(2\sqrt{3}-3)^{3/2}\beta ^2\big )<0.$ This implies that $t_0$ is the unique maximum point of h given by $h(t_0)=\alpha (({9 + 6 \sqrt{3}})^{1/2})/4$. Furthermore, $h(t)> 0$ for all $t>0$, and $\lim _{t\rightarrow 0^+}h(t)=\lim _{t\rightarrow \infty }h(t)=0$. Notice also that $h''(t)=0 \Longleftrightarrow \alpha (x^2-1)/(\beta x)=0 \Longleftrightarrow x=1 \Longleftrightarrow t=\beta .$ That is, $t=\beta$ is an inflection point of h. With the descriptions of g and h mentioned, we expect g and h to intersect at a single point; see Fig. 6. This ensures the existence of a single positive root of $g(t)=h(t)$. Hence, the QEVBS PDF has a single critical point. Since $\lim _{t\rightarrow 0^+}f_T(t;\varvec{\theta }_0)=0$ and $\lim _{t\rightarrow \infty }f_T(t;\varvec{\theta }_0)=0$, the unimodality follows. $\square$

Fig. 6

Plots of functions g and h for $\xi =0$

Proof of Lemma 2.3

This is immediate by applying the L’Hôpital rule and omitted due to reasons of space. $\square$

Proof of Theorem 2.4

Let us consider r and h as in Lemma 2.3 and Theorem 2.2, respectively. By Proposition 2.1, when $\xi \ne 0$, all mode t is a positive root of the equation $r(t)=h(t)$. Differentiating r with respect to t gives $r'(t) = (\xi +1) (1+\xi \mathfrak {a}_t)^{(-1/\xi )-2} (\xi (1+\xi \mathfrak {a}_t)^{1/\xi }-1)\mathfrak {a}'_t.$ When $\xi >0$; $r'(t)=0 \Longleftrightarrow \mathfrak {a}_t=((1/\xi )^\xi -1)/\xi \Longleftrightarrow t_1=\mathfrak {a}_{(\xi ^{-\xi } -1)/\xi }^{-1},$ which belongs to the interval $(t_\xi ,\infty )$. Since $\lim _{t\rightarrow t_\xi ^+} r(t)=\infty$ and $\lim _{t\rightarrow \infty } r(t) = 0$ (Lemma 2.3), it is clear that $t_1$ is a minimum point of r with minimum value $r(t_1)=-\xi ^{\xi }$. Moreover, $r'(t)<0$ (respectively, $>0$) $\Longleftrightarrow$ $t<t_1$ (respectively, $t>t_1$), because $1+\xi \mathfrak {a}_t>0$ and $\mathfrak {a}'_t>0$. Hence, there is a single point such that $r(t)=h(t)$ (Fig. 7 (left-top)), and then the QEVBS PDF has a single critical point. Since $\lim _{t\rightarrow t_\xi ^+} f_T(t;\varvec{\theta }_\xi )=\lim _{t\rightarrow \infty } f_T(t;\varvec{\theta }_\xi ) = 0$, the unimodality stated in (i) follows.

When $\xi =-1$; $r(t)=1$ (Lemma 2.3). Since h is a positive function with maximum value $h(t_0)=\alpha (\sqrt{9 + 6 \sqrt{3}}\,)/4$ in $t_0=\beta (2\sqrt{3}-3)$, we have (Fig. 7 (right-top)): (a) r and h have no point in common for $h(t_0)<1$; (b) r and h have a single point in common for $h(t_0)=1$; and (c) r and h have two points in common for $h(t_0)>1$. The scenarios (i)-(iii) ensure that the QEVBS PDF has at most two critical points. However, $\lim _{t\rightarrow 0^+} f_T(t;\varvec{\theta }_\xi ) = 0$ and $\lim _{t\rightarrow t_\xi ^-} f_T(t;\varvec{\theta }_\xi ) = \mathfrak {a}'_{t_\xi } (>0)$. Thus, the proofs of (ii), (iii), and (iv) follow.

Now, let $\xi <-1$ (respectively, $-1<\xi <0$). In this case, by using an expression stated earlier, see that $r'(t)>0$ (respectively, $r'(t)<0$) for all $0<t<t_\xi$, because $1+\xi \mathfrak {a}_t>0$ and $\mathfrak {a}'_t>0$. When $\xi <-1$; r is positive, increasing on $(0,t_\xi )$, $\lim _{t\rightarrow 0^+} r(t)=0$ and $\lim _{t\rightarrow t_\xi ^-} r(t) = \infty$ (Lemma 2.3). Since h is also positive with maximum value $h(t_0)$, we have the following scenarios (Fig. 7(left-bottom)): (a) r and h have no point in common for $r(t_0)>h(t_0)$; (b) r and h have a single point in common for $r(t_0)\leqslant h(t_0)$; and (c) r and h have two points in common for $r(t_0)<h(t_0)$. We claim that scenario (b) cannot occur; otherwise, the QEVBS PDF would have a single critical point. Since $\lim _{t\rightarrow 0^+} f_T(t;\varvec{\theta }_\xi )=0$ and $\lim _{t\rightarrow t_\xi ^-} f_T(t;\varvec{\theta }_\xi )=\infty$, the QEVBS PDF is forced to have none or at least an even number of critical points, which is a contradiction. This proves the claim. In addition, scenarios (a) and (c) ensure that the QEVBS PDF has zero or two critical points. Nevertheless, $\lim _{t\rightarrow 0^+} f_T(t;\varvec{\theta }_\xi )=0$ and $\lim _{t\rightarrow t_\xi ^-} f_T(t;\varvec{\theta }_\xi )=\infty$. Then, the proofs of (v) and (vi) follows. In the remainder of the proof, we consider the last case $-1<\xi <0$. In this case, r is decreasing on $(0,t_\xi )$, crosses the abscissa at the point $t_*=\mathfrak {a}_{((\xi +1)^{-\xi }-1)/\xi }^{-1}$ and $\lim _{t\rightarrow 0^+} r(t)=\infty$ and $\lim _{t\rightarrow t_\xi ^-} r(t) = -\infty$ (Lemma 2.3). Now, we have the following scenarios (Fig. 7(right-bottom)): (d) r and h have a single point in common; and (e) r and h have three points in common. Both scenarios ensure that the QEVBS PDF has one or three critical points. Since $\lim _{t\rightarrow 0^+} f_T(t;\varvec{\theta }_\xi ) =0$ and $\lim _{t\rightarrow t_\xi ^-} f_T(t;\varvec{\theta }_\xi )=0$, the uni- or bimodality stated in (vii) follows. $\square$

Fig. 7

Plots of functions r and h for (left-top) $\alpha /\xi \geqslant (1/\sqrt{2\sqrt{3}-3})-\sqrt{2\sqrt{3}-3}$ and $\xi >0$ –in this case, $t_\xi \leqslant t_0$–; (right-top) $\xi =-1$ –in this case, it is always satisfied that, $t_\xi > t_0$–, with three scenarios being highlighted: (a) $h(t_0)<1$, (b) $h(t_0)=1$ and (c) $h(t_0)>1$; (left-bottom) $\xi <-1$ –in this case, it is always satisfied that, $t_\xi > t_0$–, with three scenarios being highlighted: (a) $r(t_0)>h(t_0)$, (b) $r(t_0)\leqslant h(t_0)=1$ and (c) $r(t_0)<h(t_0)$; and (right-bottom) $-1<\xi <0$ –in this case, it is always satisfied that, $t_\xi > t_0$–, with the scenarios (d) and (e) being highlighted

Proof of Proposition 2.5

If $T\sim \text {QEVBS}(\varvec{\mathbb {\theta }}_\xi )$, then $\mathbbm {P}(X\leqslant x) = \mathbbm {P}(T\leqslant \mathfrak {a}^{-1}_x) = F_T(\mathfrak {a}^{-1}_x;\varvec{\mathbb {\theta }}_\xi ) {\mathop {=}\limits ^{(2.2)}} F_\mathrm{GEV}(x;0,1,\xi ).$ This proves (i), with (ii) being proved similarly. The proof of (iii) follows by using the property $\mathfrak {a}_{t/c}(\alpha ,Q)=\mathfrak {a}_{t}(\alpha ,cQ)$. $\square$

Proof of Proposition 2.6

If $X\sim \mathrm{Weibull}(\sigma ,\mu )$, it is well-known that $\mu (1-\sigma \log (X/\sigma ))\sim \mathrm{GEV}(\mu ,\sigma ,0)$.

Equivalently, $\mu \log (X/\sigma )\sim \mathrm{GEV}(0,1,0)$. Then, by using (ii) of Proposition 2.5, the proof of (i) follows.

For $T\sim \text {QEVBS}(\varvec{\mathbb {\theta }}_0)$, by (i) of Proposition 2.5, $\mathfrak {a}_T\sim \mathrm{GEV}(0,1,0)$. By combining this with the known result given by $X\sim \mathrm{GEV}(\mu ,\sigma ,0)$ implies $\sigma \exp (-(X-\mu )/(\mu \sigma ))\sim \mathrm{Weibull}(\sigma ,\mu )$, the proof of (ii) follows.

Let assume that $T_1 \sim \text {QEVBS}(\varvec{\mathbb {\theta }}_0)$ and $T_2 \sim \text {QEVBS}(\varvec{\mathbb {\theta }}_0)$ are independent. By (i) of Proposition 2.5, $\mathfrak {a}_{T_1}\sim \mathrm{GEV}(0,1,0)$ and $\mathfrak {a}_{T_2}\sim \mathrm{GEV}(0,1,0)$. By combining this result with the known fact $X_1\sim \mathrm{GEV}(\mu _1,\sigma ,0)$ and $X_2\sim \mathrm{GEV}(\mu _2,\sigma ,0)$ are independent, then $X_1-X_2\sim \mathrm{Logistic} (\mu _1-\mu _2,\sigma )$ and the proof of (iii) follows. $\square$

1.2 Appendix B: Observed Fisher information matrix

By using the partial derivatives of Sect. 3.2, a simple calculus shows that the elements of the observed Fisher information matrix $\mathcal{J}(\varvec{\theta }_{\xi })=-\partial ^2\ell (\varvec{\theta }_{\xi };\varvec{t})/\partial \varvec{\theta }_{\xi }\partial \varvec{\theta }_{\xi }^{\top }$ are given as follows:

Case $\xi \ne 0$. For $r,s \in \{0,1,\dots,k\},$ we have that

$$\begin{aligned}&{\partial ^2 \ell (\varvec{\theta }_\xi ;\varvec{t})\over \partial \gamma _r\partial \gamma _s} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha ^2_i} \biggl ( {\xi -1\over 1+\xi \mathfrak {a}_{t_i}} + {1\over (1+\xi \mathfrak {a}_{t_i})^{(\xi +1)/\xi } } \biggr ) \\&\quad - \biggl ({\partial \mathfrak {a}_{t_i}\over \partial \alpha _i}\biggr )^2 \biggl ( {(\xi -1)\xi \over (1+\xi \mathfrak {a}_{t_i})^2} + { {(\xi +1)}(1+\xi \mathfrak {a}_{t_i})^{1/\xi } \over (1+\xi \mathfrak {a}_{t_i})^{2(\xi +1)/\xi } } \biggr ) \biggr ) {\partial \alpha _i\over \partial \gamma _r} {\partial \alpha _i\over \partial \gamma _s} \\&\quad - \sum _{i=1}^{n} {1\over \mathfrak {a}_{t_i}} \left( {1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} - {\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha ^2_i} \right) {\partial \alpha _i\over \partial \gamma _r} {\partial \alpha _i\over \partial \gamma _s} \\&\quad + \sum _{i=1}^{n} \biggl ( {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} \biggl ( {\xi -1\over 1+\xi \mathfrak {a}_{t_i}} + {1\over (1+\xi \mathfrak {a}_{t_i})^{(\xi +1)/\xi } } \biggr ) + {1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} \biggr ) {\partial ^2 \alpha _i\over \partial \gamma _r\partial \gamma _s}, \\&{\partial ^2 \ell (\varvec{\theta }_\xi ;\varvec{t})\over \partial \beta _r\partial \beta _s} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial Q_i^2} \biggl ( {\xi -1\over 1+\xi \mathfrak {a}_{t_i}} + {1\over (1+\xi \mathfrak {a}_{t_i})^{(\xi +1)/\xi }} \biggr )\\&\quad - \left( {\partial \mathfrak {a}_{t_i}\over \partial Q_i} \right) ^2 \biggl ( {(\xi -1)\xi \over (1+\xi \mathfrak {a}_{t_i})^2} + { {(\xi +1)}(1+\xi \mathfrak {a}_{t_i})^{1/\xi } \over (1+\xi \mathfrak {a}_{t_i})^{2(\xi +1)/\xi } } \biggr ) \biggr ) {\partial Q_i\over \partial \beta _r} {\partial Q_i\over \partial \beta _s} \\&\quad - \sum _{i=1}^{n} {1\over \mathfrak {a}_{t_i}} \left( {1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}_{t_i}\over \partial Q_i} {\partial \mathfrak {a}'_{t_i}\over \partial Q_i} - {\partial ^2 \mathfrak {a}'_{t_i}\over \partial Q_i^2} \right) {\partial Q_i\over \partial \beta _r} {\partial Q_i\over \partial \beta _s} \\&\quad + \sum _{i=1}^{n} \biggl ( {\partial \mathfrak {a}_{t_i}\over \partial Q_i} \biggl ( {\xi -1\over 1+\xi \mathfrak {a}_{t_i}} + {1\over (1+\xi \mathfrak {a}_{t_i})^{(\xi +1)/\xi }} \biggr ) + {1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}'_{t_i}\over \partial Q_i} \biggr ) {\partial ^2 Q_i\over \partial \beta _r\partial \beta _s}, \\&{\partial ^2 \ell (\varvec{\theta }_\xi ;\varvec{t})\over \partial \gamma _s\partial \beta _r} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha _i \partial Q_i} \biggl ( {\xi -1\over 1+\xi \mathfrak {a}_{t_i}} + {1\over (1+\xi \mathfrak {a}_{t_i})^{(\xi +1)/\xi } } \biggr )\\&\quad - {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}_{t_i}\over \partial Q_i} \biggl ( {(\xi -1)\xi \over (1+\xi \mathfrak {a}_{t_i})^2} + { {(\xi +1)}(1+\xi \mathfrak {a}_{t_i})^{1/\xi } \over (1+\xi \mathfrak {a}_{t_i})^{2(\xi +1)/\xi } } \biggr ) \biggr ) {\partial \alpha _i\over \partial \gamma _s} {\partial Q_i\over \partial \beta _r} \\&\quad - \sum _{i=1}^{n} {1\over \mathfrak {a}_{t_i}} \left( {1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}'_{t_i}\over \partial Q_i} - {\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha _i\partial Q_i} \right) {\partial \alpha _i\over \partial \gamma _s} {\partial Q_i\over \partial \beta _r}, \end{aligned}$$

with

$$\begin{aligned}&{\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha ^2_i} = {\mathfrak {a}_{t_i}\over \alpha _i^2} - {1\over \alpha _i}\, {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} - {t_i \over \phi _i}\, {\partial \phi _i \over \partial \alpha _i}\, {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} + {t_i \mathfrak {a}'_{t_i}\over \phi _i} \left( {1\over \phi _i}\, \left( {\partial \phi _i \over \partial \alpha _i}\right) ^2 - {\partial ^2\phi _i \over \partial \alpha _i^2} \right) , \\&{\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha _i\partial Q_i} = -{t_i \over \phi _i}\, {\partial \phi _i \over \partial Q_i}\, {\partial \mathfrak {a}'_{t_i}\over \alpha _i} + {t_i\mathfrak {a}'_{t_i}\over \phi _i} \left( {1\over \phi _i}\,{\partial \phi _i \over \partial \alpha _i}\,{\partial \phi _i \over \partial Q_i} - {\partial ^2\phi _i \over \partial \alpha _i\partial Q_i} \right) , \\&{\partial ^2 \mathfrak {a}_{t_i}\over \partial Q_i^2} = -{t_i \over \phi _i}\, {\partial \phi _i \over \partial Q_i}\, {\partial \mathfrak {a}'_{t_i}\over \partial Q_i} + {t_i \mathfrak {a}'_{t_i} \over \phi _i} \left( {1\over \phi _i}\, \left( {\partial \phi _i \over \partial Q_i}\right) ^2 - {\partial ^2\phi _i \over \partial Q_i^2} \right) , \\&{\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha ^2_i} = {\mathfrak {a}'_{t_i}\over \alpha _i^2} - {1\over \alpha _i}\, {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} - {1\over 4t_i \phi _i}\, {\partial \phi _i \over \partial \alpha _i}\, {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i}\\&\quad + {\mathfrak {a}_{t_i}\over 4t_i \phi _i^2}\, \left( {1\over \phi _i}\, \left( {\partial \phi _i \over \partial \alpha _i}\right) ^2 - {\partial ^2\phi _i \over \partial \alpha _i^2} \right) , \\&{\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha _i\partial Q_i} = -{1 \over 4t_i \phi _i}\, {\partial \phi _i \over \partial Q_i}\, {\partial \mathfrak {a}_{t_i}\over \alpha _i} + {\mathfrak {a}_{t_i}\over 4t_i \phi _i}\, \left( {1\over \phi _i}\,{\partial \phi _i \over \partial \alpha _i}\,{\partial \phi _i \over \partial Q_i} - {\partial ^2\phi _i \over \partial \alpha _i\partial Q_i} \right) , \\&{\partial ^2 \mathfrak {a}'_{t_i}\over \partial Q_i^2} = -{1 \over 4t_i \phi _i}\, {\partial \phi _i \over \partial Q_i}\, {\partial \mathfrak {a}_{t_i}\over \partial Q_i} + {\mathfrak {a}_{t_i} \over 4t_i\phi _i}\, \left( {1\over \phi _i}\, \left( {\partial \phi _i \over \partial Q_i}\right) ^2 - {\partial ^2\phi _i \over \partial Q_i^2} \right) . \end{aligned}$$

As in Sect. 3.2, we adopt the notation $\phi _i=4Q_i/\Lambda _{\alpha _i,q}$, where $\Lambda _{\alpha _i,q}=(\alpha _i z_q +({\alpha ^2_i z_q^2+4})^{1/2})^2$. Hence,

$$\begin{aligned} {\partial ^2\phi _i\over \partial \alpha _i^2} = -{2z_q\over \sqrt{\alpha ^2_i z_q^2+4}} \left( {\partial \phi _i\over \partial \alpha _i} - \phi _i\, {\alpha _i z_q^2\over \alpha _i^2z_q^2+4} \right) , \quad {\partial ^2\phi _i\over \partial \alpha _i\partial Q_i}={1\over Q_i}\,{\partial \phi _i\over \partial \alpha _i}, \quad {\partial ^2\phi _i\over \partial Q_i^2}=0. \end{aligned}$$

Furthermore, by using the relation stated in (3.1), we have

$$\begin{aligned}&{\partial ^2 Q_i\over \partial \beta _0^2} = -\frac{g''(Q_i){\textbf {}}}{(g'(Q_i))^3}, \quad {\partial ^2 Q_i\over \partial \beta _0 \partial \beta _r} = -\frac{g''(Q_i){\textbf {}}}{(g'(Q_i))^3}\, x_{ir}, \\&\quad {\partial ^2 Q_i\over \partial \beta _r\partial \beta _s} = -\frac{g''(Q_i){\textbf {}}}{(g'(Q_i))^3}\, x_{ir} x_{is}, \\&{\partial ^2 \alpha _i\over \partial \gamma _0^2} = -\frac{h''(\alpha _i)}{(h'(\alpha _i))^3}, \quad {\partial ^2 \alpha _i\over \partial \gamma _0 \partial \gamma _r} = -\frac{h''(\alpha _i)}{(h'(\alpha _i))^3}\, w_{ir}, \\&\quad {\partial ^2 \alpha _i\over \partial \gamma _r\partial \gamma _s} = -\frac{h''(\alpha _i)}{(h'(\alpha _i))^3}\, w_{ir} w_{is}. \end{aligned}$$

Case $\xi = 0$. For $r,s\in \{0,1,\dots ,k\}$, we get

$$\begin{aligned}&{\partial ^2 \ell (\varvec{\theta }_0;\varvec{t})\over \partial \gamma _r \partial \gamma _s} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha ^2_i}\, (\exp (-\mathfrak {a}_{t_i})-1) - \left( {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i}\right) ^2\\&\quad \exp (-\mathfrak {a}_{t_i}) - {1\over \mathfrak {a}^2_{t_i}}\, {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} + {1\over \mathfrak {a}_{t_i}}\, {\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha ^2_i} \biggr ) {\partial \alpha _i\over \partial \gamma _r} {\partial \alpha _i\over \partial \gamma _s} \\&\quad + \sum _{i=1}^{n} \biggl ({\partial \mathfrak {a}_{t_i}\over \partial \alpha _i}\, (\exp (-\mathfrak {a}_{t_i})-1) +{1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i}\biggr ) {\partial ^2 \alpha _i\over \partial \gamma _r\partial \gamma _s}, \\&{\partial ^2 \ell (\varvec{\theta }_0;\varvec{t})\over \partial \beta _r\partial \beta _s} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial Q_i^2} \, (\exp (-\mathfrak {a}_{t_i})-1) - \left( {\partial \mathfrak {a}_{t_i}\over \partial Q_i}\right) ^2\\&\quad \exp (-\mathfrak {a}_{t_i}) - {1\over \mathfrak {a}^2_{t_i}}\, {\partial \mathfrak {a}_{t_i}\over \partial Q_i} {\partial \mathfrak {a}'_{t_i}\over \partial Q_i} + {1\over \mathfrak {a}_{t_i}}\, {\partial ^2 \mathfrak {a}'_{t_i}\over \partial Q_i^2} \biggr ) {\partial Q_i\over \partial \beta _r} {\partial Q_i\over \partial \beta _s} \\&\quad + \sum _{i=1}^{n} \biggl ({\partial \mathfrak {a}_{t_i}\over \partial Q_i} \, (\exp (-\mathfrak {a}_{t_i})-1) +{1\over \mathfrak {a}_{t_i}}\, {\partial \mathfrak {a}'_{t_i}\over \partial Q_i}\biggr ) {\partial ^2 Q_i\over \partial \beta _r\partial \beta _s}, \\&{\partial ^2 \ell (\varvec{\theta }_0;\varvec{t})\over \partial \gamma _s\partial \beta _r} = \sum _{i=1}^{n} \biggl ( {\partial ^2 \mathfrak {a}_{t_i}\over \partial \alpha _i\partial Q_i}\, (\exp (-\mathfrak {a}_{t_i})-1) - {\partial \mathfrak {a}_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}_{t_i}\over \partial Q_i}\\&\quad \exp (-\mathfrak {a}_{t_i}) - {1\over \mathfrak {a}^2_{t_i}}\, {\partial \mathfrak {a}'_{t_i}\over \partial \alpha _i} {\partial \mathfrak {a}_{t_i}\over \partial Q_i} + {1\over \mathfrak {a}_{t_i}}\, {\partial ^2 \mathfrak {a}'_{t_i}\over \partial \alpha _i\partial Q_i} \biggr ) {\partial \alpha _i\over \partial \gamma _s} {\partial Q_i\over \partial \beta _r}, \end{aligned}$$

where the partial derivatives of $\mathfrak {a}_{t_i}$, $\mathfrak {a}'_{t_i}$, $\alpha _i$, and $Q_i$with respect to parameters are as in Case $\xi \ne 0$.

Notice that all first-order derivatives of the functions involved in computing the elements of the matrix $\mathcal{J}(\varvec{\theta }_{\xi })$ above are found in Sect. 3.2.