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Variable dispersion beta regressions with parametric link functions

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Abstract

This paper presents a new class of regression models for continuous data restricted to the interval (0, 1), such as rates and proportions. The proposed class of models assumes a beta distribution for the variable of interest with regression structures for the mean and dispersion parameters. These structures consider covariates, unknown regression parameters, and parametric link functions. Link functions depend on parameters that model the relationship between the random component and the linear predictors. The symmetric and asymmetric Aranda-Ordaz link functions are considered in details. Depending on the parameter values, these link functions refer to particular cases of fixed links such as logit and complementary log–log functions. Joint estimation of the regression and link function parameters is performed by maximum likelihood. Closed-form expressions for the score function and Fishers information matrix are presented. Aspects of large sample inferences are discussed, and some diagnostic measures are proposed. A Monte Carlo simulation study is used to evaluate the finite sample performance of point estimators. Finally, a practical application that employs real data is presented and discussed.

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Notes

  1. When constant mean and dispersion are considered, no regression structures are considered; thus, there are no estimates for \(\lambda _\delta \).

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Acknowledgements

This research was partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Brazil.

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Authors and Affiliations

  1. Departamento de Estatística and LACESM, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil

    Diego Ramos Canterle & Fábio Mariano Bayer

Authors
  1. Diego Ramos Canterle
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  2. Fábio Mariano Bayer
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Correspondence to Fábio Mariano Bayer.

Appendix

Appendix

In this appendix we obtain the score function and the Fisher’s information matrix for all parameters (\(\varvec{\beta }\),\(\varvec{\gamma }\),\(\lambda _1\),\(\lambda _2\)).

The elements of the score vector are given by:

$$\begin{aligned} U_{\beta _i}(\varvec{\theta })=\frac{\partial \ell (\varvec{\theta })}{\partial \beta _i}=&\sum \limits _{t=1}^{n}\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\frac{\partial \mu _t}{\partial \eta _{1t}}\frac{\partial \eta _{1t}}{\partial \beta _{i}},\\ U_{\gamma _j}(\varvec{\theta })=\frac{\partial \ell (\varvec{\theta })}{\partial \gamma _j}=&\sum _{t=1}^{n}\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}},\\ U_{\lambda _1}(\varvec{\theta })=\frac{\partial \ell (\varvec{\theta })}{\partial \lambda _1} =&\sum _{t=1}^{n}\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \lambda _1},\\ U_{\lambda _2}(\varvec{\theta })=\frac{\partial \ell (\varvec{\theta })}{\partial \lambda _2} =&\sum _{t=1}^{n}\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \lambda _2}, \end{aligned}$$

for \(i=1,\ldots ,r\) and \(j=1, \ldots , s\), where \(\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t} = \dfrac{1-\sigma ^2_t}{\sigma ^2_t}(y^*_t-\mu ^*_t)\), \(\dfrac{\partial \mu _t}{\partial \eta _{1t}} = \left[ \dfrac{\partial g_1(\mu _{t},\lambda _1)}{\partial \mu _t}\right] ^{-1}\), \(\dfrac{\partial \eta _{1t}}{\partial \beta _{i}}=x_{ti}\), \(\dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}=a_t\), \(\dfrac{\partial \sigma _t}{\partial \eta _{2t}} = \left[ \dfrac{\partial g_2(\sigma _{t},\lambda _2)}{\partial \sigma _t}\right] ^{-1}\) and \(\dfrac{\partial \eta _{2t}}{\partial \gamma _{i}}=z_{tj}\).

The second order derivatives of the log-likelihood function are given by:

$$\begin{aligned} \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \beta _p}&= \sum _{t=1}^{n}\dfrac{\partial }{\partial \mu _t} \left( \dfrac{\partial \ell _t(\mu _t,\sigma _t)}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) \dfrac{\partial \mu _t}{\partial \eta _{1t}} \dfrac{\partial \eta _{1t}}{\partial \beta _p} \dfrac{\partial \eta _{1t}}{\partial \beta _i} \\&= \sum _{t=1}^{n} \left( \dfrac{\partial _2\ell _t(\mu _t,\sigma _t)}{\partial \mu _t^2}\dfrac{\partial \mu _t}{\partial \eta _{1t}} + \dfrac{\partial \ell _t(\mu _t,\sigma _t)}{\partial \mu _t} \dfrac{\partial }{\partial \mu _t} \left( \dfrac{\partial \mu _t}{\partial \eta _{1t}}\right) \right) \\&\quad \times \, \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti}x_{tp}, \; p=1,\ldots ,r,\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \gamma _j}&= \sum _{t=1}^{n} \left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}} \right) \frac{\partial \mu _t}{\partial \eta _{1t}} \frac{\partial \eta _{1t}}{\partial \beta _{i}} \\&=\sum _{t=1}^{n} \left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}\left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj} \right) \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _1}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _1} \left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) \frac{\partial \eta _{1t}}{\partial \beta _{i}}\\&=\sum _{t=1}^{n} \bigg [ \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t^2} \dfrac{\partial \mu _t}{\partial \lambda _1} \dfrac{\partial \mu _t}{\partial \eta _{1t}} + \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t} \dfrac{\partial }{\partial \lambda _1} \left( \dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) \bigg ] x_{ti},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _2}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _2} \left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) \frac{\partial \eta _{1t}}{\partial \beta _{i}}\\&=\sum _{t=1}^{n} \bigg [ \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t} \dfrac{\partial \sigma _t}{\partial \lambda _2} \dfrac{\partial \mu _t}{\partial \eta _{1t}} + \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t} \dfrac{\partial }{\partial \lambda _2} \left( \dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) \bigg ] x_{ti}\\&= \sum _{t=1}^{n} \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t} \varrho _t \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \gamma _l}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \gamma _l}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}} \right) \\&= \sum _{t=1}^{n} \bigg ( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2} \dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}} \dfrac{\partial \sigma _t}{\partial \eta _{2t}} + \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t} \dfrac{\partial }{\partial \gamma _l} \left( \dfrac{\partial \sigma _t}{\partial \eta _{2t}} \right) \bigg )z_{tj},\\ l=1,\ldots ,s,\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _1}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _1}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}} \right) \\&= \sum _{t=1}^{n} \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t\partial \mu _t}\dfrac{\partial \mu _t}{\partial \lambda _1} \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _2}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _2}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \eta _{2t}}\dfrac{\partial \eta _{2t}}{\partial \gamma _{j}} \right) \\&= \sum _{t=1}^{n} \bigg ( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2}\dfrac{\partial \sigma _t}{\partial \lambda _2} \dfrac{\partial \sigma _t}{\partial \eta _{2t}} + \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t} \dfrac{\partial }{\partial \lambda _2}\left( \dfrac{\partial \sigma _t}{\partial \eta _{2t}} \right) \bigg ) z_{tj},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1^2}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _1}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \lambda _1}\right) \\&= \sum _{t=1}^{n} \bigg ( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t^2}\dfrac{\partial \mu _t}{\partial \lambda _1} \dfrac{\partial \mu _t}{\partial \lambda _1} \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t} \dfrac{\partial ^2\mu }{\partial \lambda _1^2} \bigg ),\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1\partial \lambda _2}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _2}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t}\dfrac{\partial \mu _t}{\partial \lambda _1}\right) = \sum _{t=1}^{n} \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \lambda _2} \dfrac{\partial \mu _t}{\partial \lambda _1},\\ \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _2^2}&= \sum _{t=1}^{n} \dfrac{\partial }{\partial \lambda _2}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\dfrac{\partial \sigma _t}{\partial \lambda _2}\right) \\&= \sum _{t=1}^{n} \bigg ( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2}\dfrac{\partial \sigma _t}{\partial \lambda _2} \dfrac{\partial \sigma _t}{\partial \lambda _2} \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t} \dfrac{\partial ^2\sigma }{\partial \lambda _2^2} \bigg ), \end{aligned}$$

where \(\dfrac{\partial }{\partial \lambda _2} \left( \dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) =0\),

$$\begin{aligned} \dfrac{\partial ^2\ell _t(\mu _t,\sigma _t)}{\partial \mu _t^2}&= - \left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) ^2 \bigg [ \psi '\left( \mu _t \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) + \psi '\left( (1-\mu _t) \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \bigg ],\\ \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}&=-\dfrac{2}{\sigma _t^3}(y^*_t-\mu ^*_t) - \dfrac{1-\sigma ^2_t}{\sigma ^2_t}\dfrac{2}{\sigma _t^3} \bigg [ (1-\mu _t)\psi '\left( (1-\mu _t) \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \\&\quad -\,\mu _t\psi '\left( \mu _t \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \bigg ],\\ \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2}&= -\dfrac{4}{\sigma ^6_t}\bigg [-\psi '\left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) + \mu _t^2\psi '\left( \mu _t\dfrac{1-\sigma _t^2}{\sigma _t^2}\right) + (1-\mu _t)^2\\&\quad \times \,\psi '\left( (1-\mu _t)\dfrac{1-\sigma _t^2}{\sigma _t^2}\right) \bigg ] +\dfrac{3}{\sigma _t}\dfrac{2}{\sigma _t^3} \bigg [ \mu _t(y_t^*-\mu _t^*)+\psi \left( \frac{1-\sigma _t^2}{\sigma _t^2}\right) \\&\quad -\,\psi \left( (1-\mu _t)\frac{1-\sigma _t^2}{\sigma _t^2}\right) +\log (1-y_t) \bigg ]. \end{aligned}$$

Taking the expected value of the second order derivatives given above, since \(\mathbb {E}\left( \dfrac{\partial \ell _t(\mu _t,\sigma _t)}{\partial \mu _t} \right) = 0\), we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial \ell (\varvec{\theta })}{\partial \beta _i\partial \beta _p} \right)= & {} \sum _{t=1}^{n} \mathbb {E} \Bigg [ \left( \dfrac{\partial _2\ell _t(\mu _t,\sigma _t)}{\partial \mu _t^2}\left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} \right) \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti}x_{tp} \Bigg ]\\= & {} \sum _{t=1}^{n} \mathbb {E} \left[ \dfrac{\partial _2\ell _t(\mu _t,\sigma _t)}{\partial \mu _t^2}\left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-2} x_{ti}x_{tp} \right] \\= & {} - \sum _{t=1}^{n} \mathbb {E} \Bigg [ \left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) \left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) \bigg [ \psi '\left( \mu _t \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \\&+\, \psi '\left( (1-\mu _t) \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \bigg ]\left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-2} x_{ti}x_{tp} \Bigg ]\\= & {} - \sum _{t=1}^{n} \left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) w_t x_{ti} x_{tp}. \end{aligned}$$

Since

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}\right)&=- \dfrac{1-\sigma ^2_t}{\sigma ^2_t}\dfrac{2}{\sigma _t^3}\bigg [ (1-\mu _t)\psi '\bigg ( (1-\mu _t) \dfrac{1-\sigma _t^2}{\sigma _t^2} \bigg )\\&\quad -\,\mu _t\psi '\left( \mu _t \dfrac{1-\sigma _t^2}{\sigma _t^2} \right) \bigg ], \end{aligned}$$

we arrive at the conclusion that

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \gamma _j}\right)&=-\sum _{t=1}^{n}c_t \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} z_{tj} x_{ti}. \end{aligned}$$

In relation to \(\beta _i\) and \(\lambda _1\), we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _1}\right)&= \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t^2} \dfrac{\partial \mu _t}{\partial \lambda _1} \dfrac{\partial \mu _t}{\partial \eta _{1t}} \right) = \sum _{t=1}^{n} \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t^2} \dfrac{\partial \mu _t}{\partial \lambda _1} \dfrac{\partial \mu _t}{\partial \eta _{1t}} \\&= -\sum _{t=1}^{n} \nu _t \rho _t \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti}. \end{aligned}$$

The expected value of the second order derivative with respect to \(\beta _i\) and \(\lambda _2\) is given by:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _2}\right)&= \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \mu _t\partial \sigma _t}\right) \varrho _t \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti} \\&= \sum _{t=1}^{n} c_t \varrho _t \left( \dfrac{\partial g_1(\mu _t,\lambda _1)}{\partial \mu _t} \right) ^{-1} x_{ti}. \end{aligned}$$

Since \(\mathbb {E}\left( \dfrac{\partial \ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t}\right) =0\), we have

$$\begin{aligned}&\mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \gamma _l}\right) = \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2}\right) \left( \frac{g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-2} z_{tl} z_{tj}, \end{aligned}$$

where

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2}\right)&= -\dfrac{4}{\sigma ^6_t}\bigg [-\psi '\left( \dfrac{1-\sigma _t^2}{\sigma _t^2}\right) + \mu _t^2\psi '\left( \mu _t\dfrac{1-\sigma _t^2}{\sigma _t^2}\right) \\&\quad +\, (1-\mu _t)^2\psi '\left( (1-\mu _t)\dfrac{1-\sigma _t^2}{\sigma _t^2}\right) \bigg ]. \end{aligned}$$

With respect to \(\gamma _j\) and \(\lambda _1\), we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _1}\right)&= \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t\partial \mu _t} \right) \rho _t \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj} \\&= - \sum _{t=1}^{n} c_t \rho _t \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj}. \end{aligned}$$

For \(\gamma _j\) and \(\lambda _2\), we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _2}\right)&= \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _{t}(\mu _{t},\sigma _{t})}{\partial \sigma _t^2} \right) \varrho _t \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj} \\&= - \sum _{t=1}^{n} d_t^* \varrho _t \left( \dfrac{\partial g_2(\sigma _t,\lambda _2)}{\partial \sigma _t} \right) ^{-1} z_{tj}. \end{aligned}$$

Finally, we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1^2}\right)&= -\sum _{t=1}^{n}\nu _t\rho _t\rho _t,\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1\partial \lambda _2}\right)&= \sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _t(\mu _t\sigma _t)}{\partial \mu _t\partial \sigma _t}\right) \varrho _t\rho _t= -\sum _{t=1}^{n} c_t\varrho _t\rho _t, \end{aligned}$$

and

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _2^2}\right) =\sum _{t=1}^{n} \mathbb {E}\left( \dfrac{\partial ^2\ell _t(\mu _t,\sigma _t)}{\partial \sigma _t^2} \right) \varrho _t\varrho _t = -\sum _{t=1}^{n}d_t^*\varrho _t\varrho _t. \end{aligned}$$

In matrix form, we have:

$$\begin{aligned} \mathbb {E}\left( \dfrac{\partial \ell (\varvec{\theta })}{\partial \beta _i\partial \beta _p} \right)&= -\varvec{X}^\top \varvec{\Sigma } \varvec{W}\varvec{X},\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \gamma _j}\right)&=-\varvec{X}^\top \varvec{C}\varvec{T}\varvec{H}\varvec{Z},\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _1}\right)&= -\varvec{X}^\top \varvec{V}\varvec{T}\varvec{\rho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \beta _i\partial \lambda _2}\right)&= -\varvec{X}^\top \varvec{C}\varvec{T}\varvec{\varrho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \gamma _l}\right)&= -\varvec{Z}^\top \varvec{D}^*\varvec{H}\varvec{H}^\top \varvec{Z},\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _1}\right)&= -\varvec{Z}^\top \varvec{C}\varvec{H}\varvec{\rho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \gamma _j\partial \lambda _2}\right)&= -\varvec{Z}^\top \varvec{D}^*\varvec{H}\varvec{\varrho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1^2}\right)&= -\varvec{\rho }^\top \varvec{V}\varvec{\rho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _1\partial \lambda _2}\right)&= -\varvec{\rho }^\top \varvec{C}\varvec{\varrho },\\ \mathbb {E}\left( \dfrac{\partial ^2\ell (\varvec{\theta })}{\partial \lambda _2^2}\right)&= -\varvec{\varrho }^\top \varvec{D}^*\varvec{\varrho }. \end{aligned}$$

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Canterle, D.R., Bayer, F.M. Variable dispersion beta regressions with parametric link functions. Stat Papers 60, 1541–1567 (2019). https://doi.org/10.1007/s00362-017-0885-9

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