The distribution defined by (1) is obtained by means of a change of variable from a two-parameter Lindley distribution defined in Shanker et al. (2013) (see also Zakerzadeh and Dolati, 2009).
Gómez-Déniz et al. (2014) studied some statistical properties of the Log-Lindley distribution as well as nice applications in insurance and inventory management. Furthermore, taking into account that this distribution has a bounded support, they gave a reparametrization of (1) to be used as a regression model for bounded responses, which provides an appealing alternative to the beta regression model.