*3.2. GBS-ACD Models*

We now extend the class of BS-ACD(*<sup>r</sup>*,*<sup>s</sup>*) models by using the GBS distributions. As explained earlier, this family of distributions possesses either lighter or heavier tails than the BS density, thus providing more flexibility. From the density given in (6), the GBS-ACD(*<sup>r</sup>*,*<sup>s</sup>*) model can be obtained in a way analogous to that provided for the BS-ACD(*<sup>r</sup>*,*<sup>s</sup>*) model in (13) with an associated PDF expressed as

$$f\_{\rm GBS}(\mathbf{x}\_{i\prime};\boldsymbol{\theta}\_{i\prime}\boldsymbol{g}) = \boldsymbol{\varepsilon}\,\mathrm{g}\left(\boldsymbol{a}^{2}(\mathbf{x}\_{i\prime};\boldsymbol{\theta}\_{i})\right)\boldsymbol{a}^{\prime}(\mathbf{x}\_{i\prime};\boldsymbol{\theta}\_{i}), \quad i = 1,\ldots,n,\tag{15}$$

where *c* and *g* are as given in (6), *θi* = (*<sup>κ</sup>*, *<sup>σ</sup>i*) for *i* = 1, . . . , *n*, with

$$\ln \sigma\_i = \alpha + \sum\_{j=1}^{r} \beta\_j \ln \sigma\_{i-j} + \sum\_{j=1}^{s} \gamma\_j \lceil \frac{x\_{i-j}}{\sigma\_{i-j}} \rceil,\tag{16}$$

where *ξ* = (*<sup>κ</sup>*, *α*, *β*1, ... , *β<sup>r</sup>*, *γ*1, ... , *<sup>γ</sup>s*) and *ζ* = (*ζ*1, ... , *ζk*) denotes the additional parameters required by the density function in (6).

> Note that model (15) can be written as

$$X\_{\bar{i}} = \sigma\_{\bar{i}} \sigma\_{\bar{i}\prime} \tag{17}$$

where *ϕi* = exp(*<sup>ε</sup>i*) with *εi* being positively supported independent and identically distributed RVs following the SHS(*<sup>κ</sup>*, 0, 2, *g*) distribution, with density given by (11). Note that if *εi* ∼ SHS(*<sup>κ</sup>*, 0, 2, *g*), then exp(*<sup>ε</sup>i*) ∼ GBS(*<sup>κ</sup>*, 1, *g*) (see Proposition 1) with *Xi* ∼ GBS(*<sup>θ</sup>i*, *g*).
