**1. Introduction**

Over the last few years, the search for flexible probabilistic families capable of modeling different levels of bias and kurtosis has been an issue of grea<sup>t</sup> interest in the field of distributions theory. This was mainly motivated by the seminal work of Azzalini [1]. In that paper, the probability density function (pdf) of a skew-symmetric distribution was introduced. The expression of this density is given by

$$\log(z;\lambda) = 2f(z)G(\lambda z), \quad z, \lambda \in \mathbb{R},\tag{1}$$

where *f*(·) is a symmetric pdf about zero; *<sup>G</sup>*(·) is an absolutely continuous distribution function, which is also symmetric about zero; and *λ* is a parameter of asymmetry. For the case where *f*(·) is the standard normal density (from now on, we reserve the symbol *φ* for this function), and *<sup>G</sup>*(·) is the standard normal cumulative distribution function (henceforth, denoted by Φ), the so-called skew-normal (SN ) distribution with density

$$\phi\_{\mathbb{Z}}(z;\lambda) = 2\phi(z)\Phi(\lambda z), \quad z,\lambda \in \mathbb{R}, \tag{2}$$

is obtained. We use the notation *Z* ∼ SN (*λ*) to denote the random variable *Z* with pdf given by Equation (2). A generalization of the SN distribution is introduced by Arellano-Valle et al. [2] and Arellano-Valle et al. [3]; they study Fisher's information matrix of this generalization. For further details about the SN distribution, the reader is referred to Azzalini [4]. Martínez-Flórez et al. [5] used generalizations of the SN distribution to extend the Birnbaum-Saunders model, and Contreras-Reyes and Arellano-Valle [6] utilized the Kullback–Leibler divergence measure to compare the multivariate normal distribution with the skew-multivariate normal.

One of the main limitations of working with the family given by Equation (1) is that the information matrix could be singular for some of its particular models (see Azzalini [1]). This might lead to some difficulties in the estimation, due to the asymptotic convergence of the maximum likelihood (ML) estimators. To overcome this issue, some authors (see Chiogna [7] or Arellano-Valle and Azzalini [8]) have used a reparametrization of the SN model to obtain a nonsingular information matrix. However, this methodology cannot be extended to all type of skew-symmetric models which suffers of this convergence problem. On the other hand, the family of power-symmetric (PS) distributions does not have this problem of singularity in the information matrix (see, Pewsey et al. [9]). The pdf of this family of distribution is given by

$$\mathfrak{a}\,\varphi\_{\mathbb{F}}(z;a) = \mathfrak{a}f(z)\{F(z)\}^{a-1}, \qquad z \in \mathbb{R}, \; a \in \mathbb{R}^+.\tag{3}$$

where *<sup>F</sup>*(·) is itself a cumulative distribution function (cdf) and *α* is the shape parameter. For the particular case that *<sup>F</sup>*(·) = <sup>Φ</sup>(·), the power-normal (PN ) distribution is obtained, with density given by

$$f(z;a) = a\phi(z)\{\Phi(z)\}^{a-1}, \quad z \in \mathbb{R}, \ a \in \mathbb{R}^+.\tag{4}$$

For some references where this family is discussed, the reader is referred to Lehmann [10], Durrans [11], Gupta and Gupta [12], and Pewsey et al. [9], among other papers. Other extensions of this model are given in Martínez-Flórez et al. [13], where a multivariate version from the model is introduced; also, Martínez-Flórez et al. [14] carried out applications by using regression models; finally, Martínez-Flórez et al. [15] examined the exponential transformation of the model , and Martínez-Flórez et al. [16] examined a version of the model doubly censored with inflation in a regression context. Truncations of the PN distribution were considered by Castillo el al. [17].

In this paper, a modification in the pdf of the PS probabilistic family is implemented to increase the degree of kurtosis. This methodology is later used to explain datasets that include atypical observations. Usually, this methodology is accomplished by increasing the number of parameters in the model.

The paper is organized as follows. In Section 2, first, we introduce the modified power symmetric distribution. Then, the particular case of the modified power normal distribution is derived. Some of the most relevant statistical properties of this model, including moments and kurtosis coefficient, are presented. Next, in Section 3, some methods of estimation are discussed. Later, a simulation study is provided to illustrate the behavior of the shape parameter. A numerical application where the modified power normal distribution is compared to the SN and PN distributions is given in Section 4. Finally, Section 5 concludes the paper.

### **2. Genesis and Properties of Modified Power-Normal Distribution**

In this section, we introduce a new family of probability distributions. The idea is to make a transformation to a given probability density, as the skew-symmetric or power-symmetric distributions does. As there exists a certain resemblance between our formula (Equation (6)) and the formula for the power-symmetric distributions (Equation (3)), we agree to name these new distributions as modified power-symmetric (MPS) distributions. From the standard normal distribution, we obtain the so-called Modified Power-Normal (MPN ) distribution. The main parameters and properties of this particular distribution will be studied throughout this work.
