**1. Introduction**

For many purposes, statistical distributions are used in a plethora of science fields. They are regularly useful tools to describe natural and social phenomena, providing suitable models which can help dealing with real problems, such as for instance, those concerning the prediction of an event of interest. Recent works have focused attention at formulating and describing new classes of probability distributions, which are defined generally as extensions of widely known models by adding a single or more parameters to the cumulative distribution function (cdf). Hopefully, the new models will provide more flexibility and better fitting to real data. Some examples are [1,2] where a shape parameter is added to the model by exponentiating the cdf. A general method of introducing a parameter to expand a family of distributions was presented by [3]; they applied the method to create a new two-parameter extension of the exponential distribution and a new three-parameter Weibull distribution.

A natural generalization of the Normal pdf was proposed by [4] and perhaps it is the most widely known generalized Normal distribution. The power 2 appearing in the original pdf was replaced by a shape parameter *s* > 0. Therewith, the new pdf becomes:

$$f(\mathbf{x}|\mu, \sigma, s) = K \exp\left\{-\left|\frac{\mathbf{x} - \mu}{\sigma}\right|^s\right\}, s$$

where *K* is a normalizing constant, which depends on *σ* and *s*. One can see that the Laplace distribution is a particular case of the generalized Normal of Nadarajah [4] when *s* = 1.

Azzalini [5] defined a mathematically tractable class that includes strictly (not just asymptotically) the Normal distribution. The general pdf of the class is <sup>2</sup>*<sup>G</sup>*(*<sup>λ</sup>y*)*f*(*y*) for −∞ < *y* < <sup>∞</sup>, where *λ* ∈ R, *G* is an absolutely continuous cdf, dd*y G* and *f* are pdfs symmetric about 0. Making *G* = Φ and *f* = *φ*, namely the standard normal cdf and pdf respectively, one gets to the well-known skew-normal distribution, whose pdf is *φ*(*y*; *λ*) = <sup>2</sup>*φ*(*y*)Φ(*<sup>λ</sup>y*). It is easy to see that *φ*(*y*; 0) = *φ*(*y*), but when *λ* = 0, the distribution is asymmetric and its coefficient of skewness has the same sign as *λ*.

A generalization denoted by compressed normal distribution was introduced by [6], whose objective was dealing with negatively skewed data (specifically with human longevity data); in this way, they induced a skew by adding *kx* to the denominator of the location-scale transformation, that is,

$$t(\mathbf{x}) = \frac{\mathbf{x} - \mu}{\sigma + k\mathbf{x}}$$

and when *k* < 0, the curve presents a negative skew; for *k* > 0, a positive skew occurs.

Classes with one or more additional parameters usually generalize existing classes as particular cases. The McDonald-Weibull distribution [7] is an important sub-model of the McDonald class; it has three extra parameters and includes the Beta-Weibull [8] and the Kumaraswamy-Weibull [9] as special cases.

A technique to derive families of continuous distributions using a pdf as a generator was introduced by [10] and the models emerged from such method are called members of the T-X family. In other words, if *r*(*t*) is the pdf of a random variable *T* ∈ [*a*, *b*], for −∞ ≤ *a* < *b* ≤ ∞ and *<sup>W</sup>*(*G*(*x*)) is a function of the cdf *<sup>G</sup>*(*x*) of a random variable *X* so that:


then *<sup>F</sup>*(*x*) = *<sup>W</sup>*(*G*(*x*)) *a r*(*t*)d*<sup>t</sup>* is the cdf of a new family of distributions.

An example of a T-X family member is the Gompertz-*G* class [11]; to define its cdf, the chosen functions were *<sup>W</sup>*[*G*(*x*)] = − log[1 − *<sup>G</sup>*(*x*)] and *r*(*t*) = *θe<sup>γ</sup>te*<sup>−</sup> *θγ* (*eγt*−<sup>1</sup>) for *t* > 0, given that *θ* > 0, *γ* > 0. Varying *<sup>G</sup>*(*x*), one can ge<sup>t</sup> different sub-models of the class.

The procedure to define a T-X family member is indeed capable to generalize a large number of distributions. Even though it can be regarded as a particular case described by the method of generating classes of probability distributions presented in the recent work of [12]. This new method has a high power of generalization. It consists of creating distribution classes by integrating a cdf, such that the limits of the integration are special functions that satisfy some conditions. Thus, the cdf of the general class is given by:

$$F(\mathbf{x}) = \zeta(\mathbf{x}) \sum\_{j=1}^{n} \int\_{L\_j(\mathbf{x})}^{l l\_j(\mathbf{x})} \mathbf{d}H(t) - \nu(\mathbf{x}) \sum\_{j=1}^{n} \int\_{M\_j(\mathbf{x})}^{V\_j(\mathbf{x})} \mathbf{d}H(t) \tag{1}$$

where *H* is a cdf, *n* ∈ N, *ζ*, *ν* : R → R and *Lj*, *Uj*, *Mj*, *Vj* : R → R ∪ {±∞} are the aforementioned special functions that will be discussed in the next section.

Based on this innovative method, we introduce the Normal-*G* class of distributions. We consider that this extension will yield good submodels. This paper aims to investigate and compare some of them with other competitive extended probability distributions.

### **2. The Normal-***G* **Class and Some Mathematical Properties**

The method established by [12] states that if *H*, *ζ*, *ν* : R → R and *Lj*, *Uj*, *Mj*, *Vj* : R → R ∪ {±∞} for *j* = 1, 2, 3, . . . , *n* are monotonic and right continuous functions such that:

(c1) *H* is a cdf and *ζ* and *ν* are non-negative;


$$\text{(c6)}\qquad\lim\_{\mathbf{x}\to+\infty}\mathcal{U}\_{\mathbf{n}}(\mathbf{x}) \ge \sup\{\mathbf{x}\in\mathbb{R}:H(\mathbf{x})<1\} \text{ and } \lim\_{\mathbf{x}\to+\infty}\mathcal{L}\_{1}(\mathbf{x}) \le \inf\{\mathbf{x}\in\mathbb{R}:H(\mathbf{x})>0\};$$

$$(\mathfrak{C}) \qquad \lim\_{x \to +\infty} \zeta(x) = 1;$$

(c8) lim *x*→+∞ *ν*(*x*) = 0 or lim *x*→+∞ *Mj*(*x*) = lim *x*→+∞ *Vj*(*x*) ∀*j* = 1, 2, 3, . . . , *n* and *n* ≥ 1;

$$\text{(c9)}\qquad\lim\_{\mathbf{x}\to+\infty}l l\_j(\mathbf{x}) = \lim\_{\mathbf{x}\to+\infty}L\_{j+1}(\mathbf{x})\,\forall j = 1,2,3,\dots,n-1 \text{ and } n \ge 2;$$

(c10) *H* is a cdf without points of discontinuity or all functions *Lj*(*x*) and *Vj*(*x*) are constant at the right of the vicinity of points whose image are points of discontinuity of *H*, being also continuous in that points. Moreover, *H* does not have any point of discontinuity in the set lim *x*→±∞ *Lj*(*x*), lim *x*→±∞ *Uj*(*x*), lim *x*→±∞ *Mj*(*x*), lim *x*→±∞ *Vj*(*x*) for some *j* = 1, 2, 3, . . . , *n*;

then Equation (1) is a cdf.

Let *n* = 1, *ζ*(*x*) = 1, *ν*(*x*) = 0, *<sup>L</sup>*1(*x*) = <sup>−</sup>∞, *<sup>U</sup>*1(*x*)=[2*<sup>G</sup>*(*x*) − 1]/ (*G*(*x*)[<sup>1</sup> − *<sup>G</sup>*(*x*)]) and *H*(*t*) = <sup>Φ</sup>(*t*); the function in Equation (1) turns into:

$$F\_G(x) = \int\_{-\infty}^{\frac{2G(x)-1}{G(x)(1-G(x))}} \mathbf{d}\Phi(t) \,, \tag{2}$$

where *<sup>G</sup>*(*x*) is a cdf. Since *ν*(*x*) = 0, there is no need to specify *<sup>M</sup>*1(*x*) and *<sup>V</sup>*1(*x*). The conditions (c1), (c7), (c8) and (c10) are straightforward; clearly (c4), (c5) and (c9) do not need to be verified in this case. Given that *<sup>G</sup>*(*x*) is non-decreasing:

$$\begin{aligned} \mathbf{x}\_1 < \mathbf{x}\_2 &\Rightarrow \quad \mathbf{G}(\mathbf{x}\_1) \le \mathbf{G}(\mathbf{x}\_2) \Rightarrow \frac{1}{1 - \mathbf{G}(\mathbf{x}\_1)} \le \frac{1}{1 - \mathbf{G}(\mathbf{x}\_2)}\\ &\Rightarrow \quad \frac{1}{1 - \mathbf{G}(\mathbf{x}\_1)} - \frac{1}{\mathbf{G}(\mathbf{x}\_1)} \le \frac{1}{1 - \mathbf{G}(\mathbf{x}\_2)} - \frac{1}{\mathbf{G}(\mathbf{x}\_2)}\\ &\Rightarrow \quad \frac{2\mathbf{G}(\mathbf{x}\_1) - 1}{\mathbf{G}(\mathbf{x}\_1)(1 - \mathbf{G}(\mathbf{x}\_1))} \le \frac{2\mathbf{G}(\mathbf{x}\_2) - 1}{\mathbf{G}(\mathbf{x}\_2)(1 - \mathbf{G}(\mathbf{x}\_2))} \Rightarrow \mathbf{U}\_1(\mathbf{x}\_1) \le \mathbf{U}\_1(\mathbf{x}\_2), \end{aligned}$$

so *<sup>U</sup>*1(*x*) is non-decreasing, as well as *ζ*(*x*); and since *<sup>L</sup>*1(*x*) is non-increasing, (c2) is true. Considering that *<sup>U</sup>*1(*x*) = 1/[1 − *<sup>G</sup>*(*x*)] − 1/*G*(*x*), it is easy to verify that lim *x*→−∞ *<sup>U</sup>*1(*x*) = −∞ = lim *x*→−∞ *<sup>L</sup>*1(*x*); and since lim *x*→−∞ *ν*(*x*) = 0, (c3) is satisfied. The condition (c6) is also true because lim *x*→+∞ *<sup>U</sup>*1(*x*)=+<sup>∞</sup> = sup{*x* ∈ R : *<sup>H</sup>*(*x*) < 1} and lim *x*→+∞ *<sup>L</sup>*1(*x*) = −∞ = inf{*x* ∈ R : *<sup>H</sup>*(*x*) > <sup>0</sup>}.

Therefore, according to the method exposed above, Equation (2) is a cdf and, from now on, we will denote it by Normal-*G* class of probability distributions. The new cdf can be viewed as a composed function of *<sup>G</sup>*(*x*), which will be referred as parent distribution or baseline; in agreemen<sup>t</sup> with [12], if the baseline is continuous (discrete), then the Normal-*G* will generate a continuous (discrete) distribution, whose support will be the same as *<sup>G</sup>*(*x*). It is worth remarking that the proposed class demands no additional parameters other than the ones of the parent distribution.

Although the Normal-*G* class has been defined as a composed function of a single *<sup>G</sup>*(*x*), it is possible to formulate classes that depend on more than one baseline; see [12] for further details.

*Symmetry* **2019**, *11*, 1407

> We can rewrite Equation (2) as:

$$F\_G(\mathbf{x}) = \int\_{-\infty}^{\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})(1 - G(\mathbf{x}))}} \frac{1}{\sqrt{2\pi}} e^{-t^2/2} \mathrm{d}t \,\mathrm{d}t \,\tag{3}$$

and since *φ*(*t*) = √12*π e*<sup>−</sup>*<sup>t</sup>*2/2, and <sup>Φ</sup>(*x*) = *x*−∞ *φ*(*t*)d*t*, we ge<sup>t</sup> to:

$$F\_G(\mathbf{x}) = \Phi\left(\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]}\right). \tag{4}$$

In case of continuous *<sup>G</sup>*(*x*), we can take the derivative of Equation (4) with respect to *x*:

$$f\_{\mathcal{G}}(\mathbf{x}) = \phi\left(\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]}\right) \frac{1 - 2G(\mathbf{x})[1 - G(\mathbf{x})]}{G(\mathbf{x})^2[1 - G(\mathbf{x})]^2} \mathcal{g}(\mathbf{x})\,. \tag{5}$$

The expression in Equation (5) is the pdf of the class Normal-*G*, whose hazard rate function (hrf) is given by:

$$\pi\_G(\mathbf{x}) = \frac{\Phi\left(\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]}\right)}{1 - \Phi\left(\frac{2G(\mathbf{x}) - 1}{G(\mathbf{x})[1 - G(\mathbf{x})]}\right)} \left[\frac{1 - 2G(\mathbf{x})[1 - G(\mathbf{x})]}{G(\mathbf{x})^2[1 - G(\mathbf{x})]^2} g(\mathbf{x})\right].$$

Many distributions presented in the statistical literature undergo the problem of non-identifiability. One cannot assume that the parameters of a non-identifiable model will be uniquely determined from a set of observed random variables; in other words, inferences on the parameters may not be reliable. As the Theorem 1 states, the Normal-*G* class is exempt from this problem, whenever the parent distribution *G* satisfies the property of identifiability.

**Theorem 1.** *If the cdf FG belongs to the Normal-G class and the cdf G is identifiable, then FG is identifiable.*

**Proof of Theorem 1.** Given that 0 < *<sup>G</sup>*(*x*|*ξj*) < 1 for *j* = 1, 2, where *ξj* is a parametric vector and assuming that *FG*(*x*|*ξ*1) = *FG*(*x*|*ξ*2), we have:

$$\Phi\left(\frac{2G(\mathbf{x}|\mathfrak{f}\_1) - 1}{G(\mathbf{x}|\mathfrak{f}\_1)[1 - G(\mathbf{x}|\mathfrak{f}\_1)]}\right) = \Phi\left(\frac{2G(\mathbf{x}|\mathfrak{f}\_2) - 1}{G(\mathbf{x}|\mathfrak{f}\_2)[1 - G(\mathbf{x}|\mathfrak{f}\_2)]}\right) \dots$$

Since the function Φ is injective, we can write:

$$\frac{1}{1 - G(\boldsymbol{x}|\boldsymbol{\xi}\_{1})} - \frac{1}{G(\boldsymbol{x}|\boldsymbol{\xi}\_{1})} = \frac{1}{1 - G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})} - \frac{1}{G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})}$$

$$\frac{G(\boldsymbol{x}|\boldsymbol{\xi}\_{1}) - G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})}{[1 - G(\boldsymbol{x}|\boldsymbol{\xi}\_{1})][1 - G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})]} = \frac{G(\boldsymbol{x}|\boldsymbol{\xi}\_{1}) - G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})}{-G(\boldsymbol{x}|\boldsymbol{\xi}\_{1})G(\boldsymbol{x}|\boldsymbol{\xi}\_{2})}$$

If *<sup>G</sup>*(*x*|*ξ*1) = *<sup>G</sup>*(*x*|*ξ*2), then:

$$[1 - G(x|\mathfrak{f}\_1)][1 - G(x|\mathfrak{f}\_2)] = -G(x|\mathfrak{f}\_1)G(x|\mathfrak{f}\_2) \tag{6}$$

The left-hand side of Equation (6) is necessarily positive for almost all *x* ∈ R, whereas the right-hand side is negative, a contradiction. Thereby, *<sup>G</sup>*(*x*|*ξ*1) = *<sup>G</sup>*(*x*|*ξ*2) ⇒ *ξ*1 = *ξ*2.

### *2.1. Special Normal-G Sub-Models*

Here we present two distributions from the Normal-*G* class.

### 2.1.1. The Normal-Weibull Distribution

Weibull is one of the most used models to describe natural phenomena and failure of several kinds of components. It is extensively used in survival analysis and reliability. In recent times, many authors have focused on new extensions for it, such as [13,14]. The two-parameter Weibull cdf is given by *GW*(*x*|*k*, *λ*) = 1 − *e*<sup>−</sup>(*x*/*λ*)*<sup>k</sup>* for *x* ≥ 0, where *k*, *λ* > 0. Replacing the baseline *G* in Equation (4) by *GW*, we ge<sup>t</sup> to the Normal-Weibull cdf, namely:

$$F\_{NW}(\mathbf{x}) = \Phi\left[\frac{\mathbf{c}^{(\mathbf{x}/\lambda)^k} - 2}{1 - \mathbf{c}^{-(\mathbf{x}/\lambda)^k}}\right] \; \; \; \; \tag{7}$$

x

for *x* ≥ 0. Using Equation (5) to write the corresponding pdf, we have:

x

$$f\_{\rm NW}(\mathbf{x}) = \phi \left[ \frac{e^{\mathbf{(x/\lambda)^k}} - 2}{1 - e^{-(\mathbf{x/\lambda})^k}} \right] \left( \frac{k \mathbf{x}^{k-1}}{\lambda^k} \right) \frac{1 - 2 \left[ 1 - e^{-(\mathbf{x/\lambda})^k} \right] e^{-(\mathbf{x/\lambda})^k}}{e^{-(\mathbf{x/\lambda})^k} \left[ 1 - e^{-(\mathbf{x/\lambda})^k} \right]^2}. \tag{8}$$

Plots of pdf and hrf of the Normal-Weibull distribution for different values of the parameters are portrayed in Figure 1. The different shapes of the hrf curve evince the flexibility of the model. Particularly for *k* = 1, the Weibull distribution is equivalent to an Exponential distribution, so the hrf is constant; in contrast, the Normal-Exponential model has an increasing hrf in some left-bounded interval.

**Figure 1.** Plots of pdf and hrf for the Normal-Weibull distribution.

In Figure 2, the vertical axis shows the range of values of Pearson's moment coefficient of skewness, which depends on the parameters *k* and *λ*. We can see in the graph that the Normal-Weibull distribution is also able to fit data with either positive or negative skew.

**Figure 2.** Skewness of the Normal-Weibull distribution.
