**1. Introduction**

The half-normal (HN) distribution is a very important model in the statistical literature. Its density function has a closed-form and its cumulative distribution function (cdf) depends on the cdf of the standard normal model (or the error function), which is implemented in practically all mathematical and statistical software. Pewsey [1,2] provides the maximum likelihood (ML) estimation for the general location-scale HN distribution and its asymptotic properties. Wiper et al. [3] and Khan and Islam [4] perform analysis and applications for the HN model from a Bayesian framework. Moral et al. [5] also present the hnp R package, which produces half-normal plots with simulated envelopes using different diagnostics from a range of different fitted models. The HN model is also presented in the stochastic representation of the skew-normal distribution in Azzalini [6,7] and Henze [8]. In recent years this distribution has been used to model positive data, and it is becoming an important model in reliability theory despite the fact that it accommodates only decreasing hazard rates. Some of the generalizations of this distribution can be found in Cooray and Ananda [9], Olmos et al. [10], Cordeiro et al. [11], Gómez and Bolfarine [12], among others.

In particular, we focused on the extension of Cooray and Ananda [9]. The authors provided a motivation related to static fatigue life to consider the transformation *Z* = *<sup>σ</sup>Y*1/*<sup>α</sup>*, where *Y* ∼ *HN*(1). This model was named the generalized half-normal (GHN) distribution. An alternative way to extend the HN model was introduced by Gómez et al. [13] considering a normal distribution with mean and standard deviation *μ* and *σ*, respectively, truncated to the interval (0, <sup>+</sup>∞) and considering the reparametrization *λ* = *μ*/*<sup>σ</sup>*. This model was named the truncated positive normal (TPN) distribution with density function given by

$$f(z; \sigma, \lambda) = \frac{1}{\sigma \Phi(\lambda)} \phi \left(\frac{z}{\sigma} - \lambda\right), \quad z, \sigma > 0, \lambda \in \mathbb{R}, \tag{1}$$

where *φ*(·) and <sup>Φ</sup>(·) denote the density and cdf of the standard normal models, respectively. We use TPN(*<sup>σ</sup>*, *λ*) to refer to a random variable (r.v.) with density function as in Equation (1). Note that TPN(*<sup>σ</sup>*, 0) ≡ *HN*(*σ*).

In this work we consider a similar idea to that used in Cooray and Ananda [9] to extend the TPN model including the transformation *Y* = *σ <sup>X</sup>*1/*<sup>α</sup>*, where *X* ∼ *TPN*(1, *<sup>λ</sup>*). We will refer to this distribution as the generalized truncation positive normal (GTPN).

The rest of the manuscript is organized as follows. Section 2 is devoted to study of some important properties of the model, as well as its moments, quantile and hazard functions and its entropy. In Section 3 we perform an inference and present the Fisher information matrix for the proposed model. Section 4 discusses the selection model in nested and non-nested models for the GTPN distribution. In Section 5 we carry out a simulation study in order to study properties of the ML estimators in finite samples for the proposed distribution. Section 6 presents two applications to real data-sets to illustrate that the proposed model is competitive versus other common models for positive data in the literature. Finally, in Section 7, we present some concluding remarks.
