**1. Introduction**

In recent years, there has been considerable interest in the statistical literature related to flexible families of distributions able of modeling data that present high degree of asymmetry, with kurtosis index greater or smaller than the captured by normal model. In this context, two proposals that have shown a promising behavior in this type of situations are the skew-normal (SN) distribution of Azzalini [1] and the power-normal (PN) distribution of Durrans [2]. The SN distribution has been widely studied by many authors, and its main drawback is that it presents singular Fisher information matrix, implying the inference is useless from the theory of large samples using the maximum likelihood (ML) approach. Although the PN model has a shorter asymmetry range than SN distribution, it presents non-singular information matrix and can easily be extended to censored scenarios, as it has a simple distribution function, see, for example, in Martínez-Flórez et al. [3].

The PN model is part of a wide family of distributions known as alpha-power, which has been widely studied by many authors. In addition to the normal distribution, the Birnbaum–Saunders (BS) distribution [4] has also been considered, see, for example, in Martínez-Flórez et al. [5], who propose an extension of the BS distribution based on the asymmetric alpha-power family of distributions to illustrate the applicability of the new proposal with a data set is related to the lifetimes in cycles ×10−<sup>3</sup> *n* = 101 aluminum 6061 − *T*6 pieces cut in parallel angle to the rotation direction of rolling at the rate of 18 cycles per second and maximum stress of 21.000 psi. More details of the PN distribution can be found in Gupta and Gupta [6] and Pewsey et al. [7].

An alternative propose for modeling asymmetric data that unifies the two previous approaches was introduced by Martínez-Flórez et al. [8]. The proposed model, which is called alpha-power skew-normal (APSN), has non-singular Fisher information matrix, and it can fit data with much more asymmetry than PN models it can handle. In addition, symmetry can be tested by using the likelihood ratio statistic, as the properties of large samples are satisfied for the ML estimator.

Another set of distributions with non-singular information matrices, useful for modeling asymmetric and heavy-tailed data, are based on generalizations of the Student-*t* distribution, see, for example, in [9–13]. Azzalini and Capitanio [9] for example, introduced a skew-*t* (ST) distribution as

an extension of the SN model for modeling asymmetric and heavy-tailed data as follows; The random variable *X* is said to have the ST distribution with parameter *λ* and degrees of freedom *ν*, if *X* has the probability density function (PDF) given by

$$f\_{ST}(\mathbf{x};\lambda,\nu) = 2f\_T(\mathbf{x};\nu) \, \mathcal{F}\_T\left(\lambda\sqrt{\frac{\nu+1}{\mathbf{x}^2+\nu}}\mathbf{x};\nu+1\right), \quad \mathbf{x} \in \mathbb{R} \tag{1}$$

where *λ* ∈ R is a parameter that controls the skewness of the distribution, and *fT* (·; *ν*) and F*T* (·; *ν*) denote the PDF and the cumulative distribution function (CDF) of a standard Student-*t* distribution with *ν* degree of freedom, respectively. The ST distribution, like an extension of the SN model, inherits the problem of the singularity of the information matrix and before this inconvenience Zhao and Kim [14] proposed the power Student-*t* (PT) distribution, whose information matrix is non-singular and for a given degree of freedom, the kurtosis range surpasses the kurtosis range of the skew-*t* model at all times. The PT distribution is defined as follows. The random variable *X* is said to have the PT distribution with parameter *α*, and degrees of freedom *ν*, if *X* has PDF given by

$$f\_{PT}(\mathbf{x}; \boldsymbol{\alpha}, \nu) = \mathfrak{a} f\_T(\mathbf{x}; \nu) \left[ \mathcal{F}\_T \left( \mathbf{x}; \nu \right) \right]^{\mathfrak{a} - 1}, \quad \mathbf{x} \in \mathbb{R} \tag{2}$$

where *α* > 0 is a parameter that controls the form of the distribution, and, again, *fT* (·; *ν*) and F*T* (·; *ν*) denote the PDF and the CDF of a standard Student-*t* distribution, respectively.

Based on the properties of the ST model, to fit data with high degree of asymmetry and the characteristic of the PN model to capture kurtosis larger than the normal model, in this paper, we introduce a new distribution for modeling asymmetric and heavy-tailed data. The proposed model possess non-singular information matrix, and it is able to fit data with far more asymmetry than ST and PT models can handle and with large sample properties satisfied for the ML estimator. The model introduced in this paper is named as alpha-power skew-*t* (APST) model and it extends both, ST and PT models. The APSN model by Martínez-Flórez et al. [8] is also a particular case when *ν* tends to infinite. Note that symmetry can be tested using the likelihood ratio statistics with its large sample chi-square distribution.

The rest of this paper is organized as follows. Section 2 introduces the APST model and some of its properties like moments are studied. In particular, skewness and kurtosis indices are computed showing that their ranges surpass those of the ST and PT models. Section 3 deals with the ML estimation for the location-scale situation and its observed information matrix is derived. The extension to censored data is also presented. Finally, two applications are shown in Section 4, revealing that the model proposed can present much improvement over competitors.

#### **2. The Alpha-Power Skew-t Distribution**

**Definition 1.** *The random variable X is said to have an alpha-power skew-t (APST) distribution, if X has PDF given by*

$$f\_{\rm APST}(\mathbf{x}; \lambda, \mathbf{a}, \nu) = \mathfrak{a} f\_{\rm ST}(\mathbf{x}; \lambda, \nu) \left[ \mathcal{F}\_{\rm ST}(\mathbf{x}; \lambda, \nu) \right]^{\mathfrak{a} - 1},\tag{3}$$

*for x* ∈ R*, λ* ∈ R*, and α*, *ν* ∈ R+*. Functions fST*(·) *and* F*ST*(·) *denote the PDF and the CDF of the standard ST distribution. A random variable having fAPST*(*x*; *λ*, *α*, *ν*) *distribution is denoted shortly by X* ∼ APST(*<sup>λ</sup>*, *α*, *<sup>ν</sup>*)*.*

Figure 1 displays the form of the APST distribution for some selected values of the parameters *λ* and *α* for *ν* = 6. Note from the figure that the asymmetry and kurtosis of the APST distribution are affected by the parameters *α* and *λ*; therefore, the APST model is more flexible to model data that can be highly skewed, as well as heavier tails than ST and PT models.

The following result provides some special cases of the model (3), which occur for different values of *λ*, *α*, and *ν*.

**Figure 1.** Probability density function of APST(*<sup>λ</sup>*, *α*, 10) for some values of *λ* and *α*.

**Proposition 1.** *Let X* ∼ APST(*<sup>λ</sup>*, *α*, *<sup>ν</sup>*)*,*


**Proof.** The proof of (i)–(vii) is immediate from the definition of APST distribution.
