A New Family of Continuous Distributions: Properties and Estimation

Abstract

We introduce a new flexible modified alpha power (MAP) family of distributions by adding two parameters to a baseline model. Some of its mathematical properties are addressed. We show empirically that the new family is a good competitor to the Beta-F and Kumaraswamy-F classes, which have been widely applied in several areas. A new extension of the exponential distribution, called the modified alpha power exponential (MAPE) distribution, is defined by applying the MAP transformation to the exponential distribution. Some properties and maximum likelihood estimates are provided for this distribution. We analyze three real datasets to compare the flexibility of the MAPE distribution to the exponential, Weibull, Marshall–Olkin exponential and alpha power exponential distributions.

1. Introduction

The characteristics of classical distributions are limited and cannot represent all situations found in applications or may not provide a good fit in many practical situations. Adding parameters to a parent distribution is an effective way to make it more flexible and to improve the goodness of fit to real data. However, the properties of the generating distribution will need further investigation, as they are, generally, different from those of the parent distribution. There are different methods to include parameters to a distribution. Ref. [1] seems to have been the first to raise an exponential cumulative distribution function (cdf) to a positive power. Exponentiated distributions are also known as proportional reversed hazard rate models with a constant of proportionality $\alpha$ [2] or Lehman alternatives [3]. Ref. [4] added a shape parameter to the normal distribution to obtain a skew-normal distribution. Ref. [5] included an extra parameter $\alpha$ via the transformation $G\left(x\right)=F\left(x\right)/[1-(1-\alpha )\overline{F}\left(x\right)]$, which transforms any cdf $F(·)$ to a new cdf $G(·)$, where $\overline{F}\left(x\right)=1-F\left(x\right)$. Ref. [6] added two parameters $(r,p)$ to the survival function by considering a countable mixture of positive integer powers of general survival functions, where the mixing proportions are Pascal (r,p). Ref. [7] included two parameters to any probability density function (pdf), based on the form of a mixture, to construct a new family. Ref. [8] defined a method of adding a parameter to a parent distribution called the alpha power (AP) transformation. Henceforth, the pdf and cdf of any continuous baseline random variable W are denoted by $f(·)$ and $F(·)$, respectively. Then, the cdf of the AP family (for $x\in \mathbb{R}$) has the form

$$F_{\mathrm{AP}}\left(x\right)=\frac{{\alpha }^{F\left(x\right)}-1}{\alpha -1},\mathrm{if}\alpha >0,\alpha \ne 1,$$

and the corresponding pdf has the form

$$f_{\mathrm{AP}}\left(x\right)=\frac{log\alpha }{\alpha -1}{\alpha }^{F\left(x\right)}f\left(x\right).$$

The AP transformation has been used by many authors to construct new distributions from baseline models. Among others, Refs. [8,9,10,11,12,13,14] derived the alpha power exponential (APE), AP Weibull, AP inverse Weibull, AP Lindley, AP Pareto, AP Kumaraswamy and AP Gompertz distributions from the exponential, Weibull, inverse Weibull, Lindley, Pareto, Kumaraswamy and Gompertz distributions, respectively. Ref. [15] introduced the three-parameter AP transformed extended exponential, which includes the AP transformed exponential and AP transformed Lindley distributions. Ref. [16] defined the cdf of an extended AP class (say “EAP”) (for $x\in \mathbb{R}$)

$$F_{\mathrm{EAP}}\left(x\right)=\frac{{\alpha }^{F\left(x\right)}-{\mathrm{e}}^{F\left(x\right)}}{\alpha -\mathrm{e}},\mathrm{if}\alpha >0,\alpha \ne \mathrm{e}.$$

Recently, Ref. [17] included a new parameter by combining the type II half-logistic-G family with the inverted Topp–Leone distribution to define the half-logistic inverted Topp–Leone family. In this paper, we propose a modified alpha power (MAP) family by adding two parameters to a baseline model. Therefore, the properties of the generated distributions require further investigation, as they are different from those of the baseline models. We apply the new transformation to the exponential distribution to construct the modified alpha power exponential (MAPE) distribution.

The paper is organized as follows: Section 2.1 defines the MAP family and provides a linear representation for its density function to derive some of its properties. Section 2.2 determines the MAPE distribution, addresses some of its properties and estimates the unknown parameters. A statistical analysis of two real datasets to verify the effectiveness of the proposed model is reported in Section 3. Some conclusions are offered in Section 4.

2. Materials and Methods

2.1. The MAP Transformation

Definition 1.

For every continuous cdf $F\left(x\right)$, the cdf of the new MAP family is defined by the monotonic increasing cdf $G:\mathbb{R}\to [0,1]$ (for $x\in \mathbb{R}$)

$$G\left(x\right)=\frac{{\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}-1}{\alpha \beta -1},\mathrm{if}\alpha ,\beta \ge 1,\alpha \beta \ne 1.$$

(1)

The main motivation for (1) follows from the fact that the new family succeeds in fitting real data as a competitive model to widely applied classes. Clearly, we can adopt ${\beta }^{F^{\gamma }\left(x\right)}$ for an unknown parameter $\gamma >0$, but this will lead to a model with three extra parameters. This proposal can be investigated following the same lines of the MAP family.

It is clear that

• ${lim}_{x\to -\infty }G\left(x\right)=0$ and ${lim}_{x\to \infty }G\left(x\right)=1.$
• For $\beta =1$ and $\alpha \to 1$, $G\left(x\right)\to F\left(x\right)$, which means that the baseline distribution is a sub-model of the new family.
• $G(·)$ is not defined for $\alpha \beta =1(\alpha =\beta =1)$.

The pdf corresponding to (1) can be expressed as

$$\begin{array}{c}\hfill g\left(x\right)=\frac{1}{\alpha \beta -1}{\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}\left[ln\alpha +2F\left(x\right)ln\beta \right]f\left(x\right).\end{array}$$

(2)

Equation (2) is a weighted version of the baseline density $f\left(x\right)$, where the weight function has the form

$$w\left(x\right)={\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}\left[ln\alpha +2F\left(x\right)ln\beta \right].$$

Thus,

$$E\left[w\left(X\right)\right]={\int }_{-\infty }^{\infty }{\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}\left[ln\alpha +2F\left(x\right)ln\beta \right]f\left(x\right)dx=\alpha \beta -1.$$

Therefore, Equation (2) can be expressed as

$$g\left(x\right)=\frac{1}{\alpha \beta -1}w\left(x\right)f\left(x\right).$$

(3)

Such weighted distributions play an important role in distribution theory [18].

Figure 1 and Figure 2 show the effects of the extra parameters on the cdf and pdf of some parent distributions. Clearly, we can control the skewness and kurtosis of the generated distributions through the extra parameters. For an exponential distribution with decreasing pdf, the new transformed exponential pdf is a concave function. The plots in Figure 2 reveal that the parameters $\alpha$ and $\beta$ can produce asymmetrical densities to the right and left and change tail lengths.

2.1.1. Linear Representation

The two power series hold

$${\beta }^{F^2\left(x\right)}={\mathrm{e}}^{F^2\left(x\right)ln\beta }=\sum _{i=0}^{\infty }\frac{{\left(ln\beta \right)}^i}{i!}F{\left(x\right)}^{2i},$$

and

$${\alpha }^{F\left(x\right)}={\mathrm{e}}^{F\left(x\right)ln\alpha }=\sum _{j=0}^{\infty }\frac{{\left(ln\alpha \right)}^j}{j!}F{\left(x\right)}^j.$$

Therefore, the cdf of the MAP distribution (1) can be expressed as

$$G\left(x\right)=\frac{1}{\alpha \beta -1}\left[\sum _{i,j=0}^{\infty }{\psi }_{i,j}\left(\alpha ,\beta \right)F{\left(x\right)}^{2i+j}-1\right],$$

where ${\psi }_{i,j}\left(\alpha ,\beta \right)={\left(ln\alpha \right)}^j{\left(ln\beta \right)}^i/(i!j!)$ (for $i,j\ge 0$).

Then, the MAP family density (2) can be expressed as

$$g\left(x\right)=\frac{1}{\alpha \beta -1}\left[\sum _{i,j=0;2i+j\ge 1}^{\infty }\left(2i+j\right){\psi }_{i,j}\left(\alpha ,\beta \right)F{\left(x\right)}^{2i+j-1}f\left(x\right)\right],$$

(4)

and then

$$g\left(x\right)=\frac{1}{\alpha \beta -1}\left[\sum _{i,j=0;2i+j\ge 1}^{\infty }{\psi }_{i,j}\left(\alpha ,\beta \right){\pi }_{2i+j}\left(x\right)\right],$$

(5)

where ${\pi }_{2i+j}\left(x\right)=\left(2i+j\right)F{\left(x\right)}^{2i+j-1}f\left(x\right)$ is the exponentiated-F (exp-F) density with power parameter $2i+j$ (for $2i+j\ge 1$).

Hence, based on the linear representation (5), we can obtain directly some mathematical properties (ordinary and incomplete moments, generating function and mean deviations, among others) of the MAP family from those properties of the exp-F distributions, which were reported in several papers; see Table 1 of [19]. Some other properties, such as different types of entropy, reliability and order statistics, and their computations may be addressed in future research.

For a specific $F\left(x\right)$, these properties can also be calculated approximately from (4) by well-known methods of numerical integration; see, for example, [20].

2.1.2. Mode

The mode $x^{*}$ of the MAP family can be found by maximizing the log-pdf from (2)

$$lng\left(x\right)=-ln\left(\alpha \beta -1\right)+F^2\left(x\right)ln\beta +F\left(x\right)ln\alpha +lnf\left(x\right)+ln\left[ln\alpha +2F\left(x\right)ln\beta \right].$$

At $x=x^{*}$, the derivative of $lng\left(x\right)$ with respect to x vanishes. We can obtain the mode of the MAP family by solving the following equation

$$2F\left(x^{*}\right)f\left(x^{*}\right)ln\beta +f\left(x^{*}\right)ln\alpha +\frac{f^{\prime }\left(x^{*}\right)}{f\left(x^{*}\right)}+\frac{2f\left(x^{*}\right)ln\beta }{ln\alpha +2F\left(x^{*}\right)ln\beta }=0,$$

(6)

which has no explicit solution for $x^{*}$. Therefore, the mode of the MAP family does not have a closed form. Given $F(·)$ and $f(·)$, direct maximization of $lng\left(x\right)$ using a numerical optimization algorithm or a one-dimensional root-finding algorithm of (6) can give an approximation for the mode.

For simplicity, let $a=ln\alpha \ge 0$ and $b=ln\beta \ge 0$, since $\alpha \ge 1$ and $\beta \ge 1$. The first derivative of the log-density is

$${(lng)}^{\prime }\left(x\right)=2bf\left(x\right)F\left(x\right)+af\left(x\right)+{(lnf\left(x\right))}^{\prime }+{[ln(a+2bF\left(x\right))]}^{\prime }.$$

Let $K\left(x\right)=f\left(x\right)[a+2bF(x\left)\right]$. Then,

$$\begin{array}{cc}\hfill {(lng)}^{\prime }\left(x\right)& =f\left(x\right)(2bF\left(x\right)+a)+{(lnf\left(x\right)+ln(a+2bF\left(x\right)))}^{\prime }\hfill \\ \hfill & =f\left(x\right)(2bF\left(x\right)+a)+{(ln\left[f\left(x\right)(a+2bF\left(x\right))\right])}^{\prime }=K\left(x\right)+{(lnK\left(x\right))}^{\prime }.\hfill \end{array}$$

Let $J\left(x\right)=lnK\left(x\right)$. Then, ${(lng)}^{\prime }\left(x\right)=K\left(x\right)+J^{\prime }\left(x\right)$ and ${(lng)}^{″}\left(x\right)=K^{\prime }\left(x\right)+J^{″}\left(x\right)$.

Hence, a sufficient condition for the unimodality of the MAP density is $K^{\prime }\left(x\right)+J^{″}\left(x\right)\le 0$, which gives

$$K^{\prime }\left(x\right)[K{\left(x\right)}^2-K^{\prime }\left(x\right)]+K\left(x\right)K^{″}\left(x\right)\le 0.$$

2.1.3. Quantiles

There is no explicit form for the quantile function of the MAP distribution, and its qth quantile ($x_q$) can be approximated using a one-dimensional root-finding algorithm in $G\left(x_q\right)=q$.

2.1.4. Survival and Hazard Functions

Let X be a random variable with the MAP family, pdf $g(·)$ and cdf $G(·)$. Then, the survival function of X, say $S_G\left(x\right)=P(X>x)$, is

$$\begin{array}{ccc}\hfill S_G\left(x\right)& =& \frac{\alpha \beta }{\alpha \beta -1}\left[1-{\beta }^{F^2\left(x\right)-1}{\alpha }^{F\left(x\right)-1}\right]\hfill \\ & =& \frac{\alpha \beta }{\alpha \beta -1}\left[1-{\beta }^{S_F\left(x\right)(S_F\left(x\right)-2)}{\alpha }^{{-S}_F\left(x\right)}\right].\hfill \end{array}$$

The hazard rate function (hrf) $h_G\left(x\right)$ and the proportional hrf of X are

$$h_G\left(x\right)=\frac{{\beta }^{F^2\left(x\right)-1}{\alpha }^{F\left(x\right)-1}\left[ln\alpha +2F\left(x\right)ln\beta \right]f\left(x\right)}{1-{\beta }^{F^2\left(x\right)-1}{\alpha }^{F\left(x\right)-1}},$$

(7)

$$h_G^{*}\left(x\right)=\frac{{\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}\left[ln\alpha +2F\left(x\right)ln\beta \right]f\left(x\right)}{{\beta }^{F^2\left(x\right)}{\alpha }^{F\left(x\right)}-1}.$$

(8)

The cdf of the MAP family (and also for any cdf) can be expressed in terms of the hrf and proportional hrf from [21] as

$$G\left(x\right)=\frac{h_G\left(x\right)}{h_G\left(x\right)+h_G^{*}\left(x\right)}.$$

Equation (7) can be rewritten as

$$h_G\left(x\right)=\frac{{\beta }^{F^2\left(x\right)-1}{\alpha }^{F\left(x\right)-1}\left[ln\alpha +2F\left(x\right)ln\beta \right]\left(1-F\left(x\right)\right)}{\left[1-{\beta }^{F^2\left(x\right)-1}{\alpha }^{F\left(x\right)-1}\right]}{h}_F\left(x\right),$$

where $h_F\left(x\right)=f\left(x\right)/[1-F\left(x\right)]$ is the baseline hrf. Hence, the hrf of the MAP class satisfies the following properties:

$$\underset{x\to -\infty }{lim}{h}_G\left(x\right)=\frac{ln\alpha }{\alpha \beta -1}\underset{x\to -\infty }{lim}{h}_F\left(x\right),$$

and

$$\underset{x\to \infty }{lim}{h}_G\left(x\right)=\underset{x\to \infty }{lim}{h}_F\left(x\right).$$

Figure 3 illustrates, for some parameter values, the effects of the additional parameters on the hrf for some parent distributions (solid line). For example, the exponential distribution has a constant hrf, whereas the MAPE distribution has an increasing hrf.

2.1.5. Random Number Generator

There is no explicit form for the quantile function of the MAP family, say $G^{-1}(·)$. We can use acceptance–rejection criteria [22] to generate a random variate from the MAP class. Since, for $\alpha ,\beta \ge 1,\alpha \beta \ne 1$, we have

$$\frac{g\left(x\right)}{f\left(x\right)}\le \frac{\alpha \beta \left[ln\alpha +2ln\beta \right]}{\alpha \beta -1},$$

and then the acceptance–rejection algorithm can be used as follows:

Step 1: Generate a random variable W from the baseline distribution F.

Step 2: Generate a uniform (0, 1) random variable U (independent of W).

Step 3: If

$$U\le \frac{g\left(W\right)}{cf\left(W\right)},$$

where $c=\alpha \beta \left[ln\alpha +2ln\beta \right]/(\alpha \beta -1)$, then set $X=W$ (accept); otherwise, go back to step 1 (reject).

2.1.6. Estimation

Let $X_1,\dots ,X_n$ be a random sample of size n from the MAP model with parameters ($\alpha ,\beta ,\theta$), and observed values $x_1,\dots ,x_n$, where $\theta$ is the parameter vector of the parent distribution. The log-likelihood function for the full vector of parameters of X has the form

$$\begin{array}{c}\hfill \ell \left(\alpha ,\beta ,\theta \right)=-nln\left(\alpha \beta -1\right)+ln\beta \sum _{i=1}^n{F}^2\left(x_i;\theta \right)+ln\alpha \sum _{i=1}^{n}F\left(x_i;\theta \right)\\ \hfill +\sum _{i=1}^{n}lnf\left(x_i;\theta \right)+\sum _{i=1}^{n}ln\left[ln\alpha +2F\left(x_i;\theta \right)ln\beta \right].\end{array}$$

By differentiating $\ell \left(\alpha ,\beta ,\theta \right)$ with respect to the parameters, we obtain the score components:

$$\begin{array}{c}\hfill U_{\alpha }=-\frac{n\beta }{\alpha \beta -1}+\frac{1}{\alpha }\sum _{i=1}^{n}F\left(x_i;\theta \right)+\frac{1}{\alpha }\sum _{i=1}^n\frac{1}{ln\alpha +2F\left(x_i;\theta \right)ln\beta },\end{array}$$

$$\begin{array}{c}\hfill U_{\beta }=-\frac{n\alpha }{\alpha \beta -1}+\frac{1}{\beta }\sum _{i=1}^n{F}^2\left(x_i;\theta \right)+\frac{2}{\beta }\sum _{i=1}^n\frac{F\left(x_i;\theta \right)}{ln\alpha +2F\left(x_i;\theta \right)ln\beta },\end{array}$$

and

$$\begin{array}{c}\hfill U_{\theta }=2ln\beta \sum _{i=1}^n\left[F\left(x_i;\theta \right)+\frac{1}{ln\alpha +2F\left(x_i;\theta \right)ln\beta }\right]\frac{\partial F\left(x_i;\theta \right)}{\partial \theta }\\ \hfill +ln\alpha \sum _{i=1}^n\frac{\partial F\left(x_i;\theta \right)}{\partial \theta }+\sum _{i=1}^n\frac{1}{f\left(x_i;\theta \right)}\frac{\partial f\left(x_i;\theta \right)}{\partial \theta }.\end{array}$$

The maximum-likelihood estimates (MLEs) $(\widehat{\alpha },\widehat{\beta },\widehat{\theta })$ are calculated by solving the nonlinear equations $U_{\alpha }=0$, $U_{\beta }=0$ and $U_{\theta }=0$. These equations cannot be solved analytically, but we can use iterative techniques such as a Newton–Raphson type algorithm to calculate these estimates. The Broyden–Fletcher–Goldfarb–Shanno method with analytical derivatives can be used for maximizing $\ell \left(\alpha ,\beta ,\theta \right)$. Some optimization algorithms for maximizing log-likelihood functions based on swarm intelligence have been proposed over the last decades. One very popular swarm intelligence method is the Particle Swarm Optimization (PSO) [23].

2.2. The MAPE Distribution

Exponential distributions have an important role in the analysis of product reliability and lifetime data. In this section, we define the MAPE distribution by applying the MAP transformation to the exponential distribution with rate parameter $\lambda$.

Definition 2.

The random variable X is said to have a three-parameter MAPE distribution, say MAPE$\left(\alpha ,\beta ,\lambda \right)$, if its cdf is given by

$$G_E\left(x\right)=\frac{{\beta }^{{\left(1-{\mathrm{e}}^{-\lambda x}\right)}^2}{\alpha }^{\left(1-{\mathrm{e}}^{-\lambda x}\right)}-1}{\alpha \beta -1},\mathrm{if}\alpha ,\beta \ge 1,\alpha \beta \ne 1,x>0.$$

The pdf of X (under the same conditions on the parameters) reduces to

$$\begin{array}{c}\hfill g_E\left(x\right)=\frac{\lambda }{\alpha \beta -1}{\beta }^{{\left(1-{\mathrm{e}}^{-\lambda x}\right)}^2}{\alpha }^{\left(1-{\mathrm{e}}^{-\lambda x}\right)}{\mathrm{e}}^{-\lambda x}\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x}\right)ln\beta \right].\end{array}$$

Further, the survival function and hrf of X are

$$S_E\left(x\right)=\frac{\alpha \beta }{\alpha \beta -1}\left[1-{\beta }^{{{\mathrm{e}}^{-\lambda x}(\mathrm{e}}^{-\lambda x}-2)}{\alpha }^{-{\mathrm{e}}^{-\lambda x}}\right]$$

and

$$h_E\left(x\right)=\frac{\lambda {\mathrm{e}}^{-\lambda x}{\beta }^{{{\mathrm{e}}^{-\lambda x}(\mathrm{e}}^{-\lambda x}-2)}{\alpha }^{-{\mathrm{e}}^{-\lambda x}}\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x}\right)ln\beta \right]}{1-{\beta }^{{{\mathrm{e}}^{-\lambda x}(\mathrm{e}}^{-\lambda x}-2)}{\alpha }^{-{\mathrm{e}}^{-\lambda x}}},$$

respectively.

2.2.1. Moments

For a random variable X following a MAPE distribution with parameters $\theta =(\alpha ,\beta ,\lambda )$, the rth ordinary moment of X (for $r=1,2,\dots$) can be determined from (4) and the well-known moments of the exp-exponential model.

Alternatively, the moments can be expressed as

$${\mu }_r=\frac{1}{{\lambda }^r\left(\alpha \beta -1\right)}{\int }_0^1{\beta }^{y^2}{\alpha }^y\left[ln\alpha +2yln\beta \right]{\left[ln\left(\frac{1}{1-y}\right)\right]}^{r}dy.$$

By expanding ${\beta }^{y^2}$ and ${\alpha }^y$ in power series, we can write

$${\mu }_r=\frac{1}{{\lambda }^r\left(\alpha \beta -1\right)}\sum _{i,j=0}^{\infty }\left(2i+j\right){\psi }_{i,j}\left(\alpha ,\beta \right){\int }_0^1{\left[ln\left(\frac{1}{1-y}\right)\right]}^r{y}^{2i+j}dy,$$

where ${\psi }_{i,j}$ was defined in Section 2.1.1.

Alternatively, the rth ordinary moment of X can be expressed as

$${\mu }_r=\frac{r!\alpha \beta }{{\lambda }^r(\alpha \beta -1)}\left[ln\alpha \sum _{i,j,k=0}^{\infty }{\psi }_{i,j,k}{\eta }_{i,j,k}+2ln\beta \sum _{i,j,k=0}^{\infty }{\psi }_{i,j,k}\left[{\eta }_{i,j,k}-{\eta }_{i,j,k+1}\right]\right],$$

where

$${\psi }_{i,j,k}={\left(-2\right)}^j{\left(ln\beta \right)}^{i+j}{\left(-ln\alpha \right)}^k/i!j!k!$$

and

$${\eta }_{i,j,k}=\frac{1}{{\left(2i+j+k+1\right)}^{r+1}}.$$

For more details, see the Appendix A.

Table 1. Statistics for the MAPE distribution for some parameter values.

$\mathit{\lambda }$	$\mathit{\alpha }$	$\mathit{\beta }$	${\mathit{\mu }}_1$	${\mathit{\mu }}_2$	$\mathit{Var}\left(\mathit{X}\right)$	Skewness	Kurtosis	Mode	${\mathit{Q}}_{\mathbf{1}}$	Median	${\mathit{Q}}_{\mathbf{3}}$
0.1	1.1	1	10.2395	207.219	102.372	1.96475	8.79095	0	2.99858	7.17254	14.221
0.2	2	3	9.17665	123.628	39.4172	1.26739	5.63906	5.81714	4.57863	7.99777	12.4161
0.5	3	1.5	3.11442	15.4874	5.78776	1.42055	6.15969	1.43523	1.32826	2.59054	4.30670
1	4	5	2.12560	6.19867	1.68051	1.14173	5.29179	1.55745	1.19497	1.91373	2.81104
2	2	2	0.82705	1.05722	0.37321	1.36477	5.96256	0.44342	0.37569	0.70074	1.13450

reports the approximate values of some statistics for the MAPE distribution. The mode of X was approximated via direct maximization of $ln{g}_E\left(x\right)$ using the R optim function and the unit root-finding function for the function

$$\frac{g_E^{\prime }\left(x\right)}{g_E\left(x\right)}=-\lambda +2\lambda {\mathrm{e}}^{-\lambda x}\left(1-{\mathrm{e}}^{-\lambda x}\right)+\lambda {\mathrm{e}}^{-\lambda x}ln\alpha +\frac{2\lambda {\mathrm{e}}^{-\lambda x}ln\beta }{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x}\right)ln\beta }.$$

The quartiles were found using the unit root-finding function for $G_E\left(x\right)-q=0$ ($q=0.25,0.5,0.75)$. These functions are available in Ref. [24].

2.2.2. Estimation

Let $X_1,\dots ,X_n$ be a random sample of size n from the MAPE distribution with parameters ($\alpha ,\beta ,\lambda$) and observed values $x_1,\dots ,x_n$. The log-likelihood function for the parameters can be expressed as

$$\begin{array}{ccc}\hfill \ell \left(\alpha ,\beta ,\lambda \right)& =& n\left(ln\alpha +ln\lambda \right)-nln\left(\alpha \beta -1\right)+ln\beta \sum _{i=1}^n{\left(1-{\mathrm{e}}^{-\lambda x_i}\right)}^2\hfill \\ & -& ln\alpha \sum _{i=1}^n{\mathrm{e}}^{-\lambda x_i}-\lambda \sum _{i=1}^n{x}_i+\sum _{i=1}^{n}ln\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right].\hfill \end{array}$$

The maximum likelihood equations are

$$\frac{n}{\alpha }-\frac{n\beta }{\alpha \beta -1}-\frac{1}{\alpha }\sum _{i=1}^n{\mathrm{e}}^{-\lambda x_i}+\frac{1}{\alpha }\sum _{i=1}^n\frac{1}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta }=0,$$

$$-\frac{n\alpha }{\alpha \beta -1}+\frac{1}{\beta }\sum _{i=1}^n{\left(1-{\mathrm{e}}^{-\lambda x_i}\right)}^2+\frac{2}{\beta }\sum _{i=1}^n\frac{1-{\mathrm{e}}^{-\lambda x_i}}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta }=0,$$

$$\begin{array}{ccc}\hfill \frac{n}{\lambda }& +& 2ln\beta \sum _{i=1}^n{x}_i{\mathrm{e}}^{-\lambda x_i}\left(1-{\mathrm{e}}^{-\lambda x_i}\right)+ln\alpha \sum _{i=1}^n{x}_i{\mathrm{e}}^{-\lambda x_i}-\sum _{i=1}^n{x}_i\hfill \\ & +& 2ln\beta \sum _{i=1}^n\frac{x_i{\mathrm{e}}^{-\lambda x_i}}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta }=0.\hfill \end{array}$$

Like most of the extensions of the exponential distribution, the MLEs of these parameters cannot be obtained in closed form. Therefore, the maximum likelihood equations should be solved numerically. Alternatively, a numerical maximization of the log-likelihood function is a good technique for calculating the estimates of the unknown parameters.

Using large sample approximation, $\sqrt{n}\left(\widehat{\alpha },\widehat{\beta },\widehat{\lambda }\right)$ are asymptotically normally distributed with mean vector $\mathbf{0}$ and covariance matrix estimated by the inverse of the observed information matrix

$$I^{-1}\left(\widehat{\alpha },\widehat{\beta },\widehat{\lambda }\right)={\left[\begin{array}{ccc}-\frac{{\partial }^2\ell }{\partial {\alpha }^2}& -\frac{{\partial }^2\ell }{\partial \alpha \partial \beta }& -\frac{{\partial }^2\ell }{\partial \alpha \partial \lambda }\\ -\frac{{\partial }^2\ell }{\partial \beta \partial \alpha }& -\frac{{\partial }^2\ell }{\partial {\beta }^2}& -\frac{{\partial }^2\ell }{\partial \beta \partial \lambda }\\ -\frac{{\partial }^2\ell }{\partial \lambda \partial \alpha }& -\frac{{\partial }^2\ell }{\partial \lambda \partial \beta }& -\frac{{\partial }^2\ell }{\partial {\lambda }^2}& \end{array}\right]}_{\alpha =\widehat{\alpha },\beta =\widehat{\beta },\lambda =\widehat{\lambda }}^{-1},$$

where

$$\begin{array}{ccc}\hfill \frac{{\partial }^2\ell }{\partial {\alpha }^2}& =& -\frac{n}{{\alpha }^2}+\frac{n{\beta }^2}{{\left(\alpha \beta -1\right)}^2}+\frac{1}{{\alpha }^2}\sum _{i=1}^n{\mathrm{e}}^{-\lambda x_i}-\frac{1}{{\alpha }^2}\sum _{i=1}^n\frac{1}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2}\hfill \\ & & -\frac{1}{{\alpha }^2}\sum _{i=1}^n\frac{1}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta },\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \alpha \partial \beta }& =& \frac{n}{{\left(\alpha \beta -1\right)}^2}-\frac{2}{\alpha \beta }\sum _{i=1}^n\frac{1-{\mathrm{e}}^{-\lambda x_i}}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \alpha \partial \lambda }& =& \frac{1}{\alpha }\sum _{i=1}^n{x}_i{\mathrm{e}}^{-\lambda x_i}-\frac{2ln\beta }{\alpha }\sum _{i=1}^n\frac{x_i{\mathrm{e}}^{-\lambda x_i}}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2},\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial {\beta }^2}& =& \frac{n{\alpha }^2}{{\left(\alpha \beta -1\right)}^2}-\frac{1}{{\beta }^2}\sum _i^n{\left(1-{\mathrm{e}}^{-\lambda x_i}\right)}^2-\frac{4}{{\beta }^2}\sum _{i=1}^n\frac{{\left(1-{\mathrm{e}}^{-\lambda x_i}\right)}^2}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2}\hfill \\ & & -\frac{2}{{\beta }^2}\sum _{i=1}^n\frac{1-{\mathrm{e}}^{-\lambda x_i}}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta },\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial \beta \partial \lambda }& =& \frac{2}{\beta }\sum _{i=1}^n{x}_i{\mathrm{e}}^{-\lambda x_i}\left(1-{\mathrm{e}}^{-\lambda x_i}\right)-\frac{4ln\beta }{\beta }\sum _{i=1}^n\frac{x_i{\mathrm{e}}^{-\lambda x_i}\left(1-{\mathrm{e}}^{-\lambda x_i}\right)}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2}\hfill \\ & & +\frac{2}{\beta }\sum _{i=1}^n\frac{x_i{\mathrm{e}}^{-\lambda x_i}}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta },\hfill \\ \hfill \frac{{\partial }^2\ell }{\partial {\lambda }^2}& =& -\frac{n}{{\lambda }^2}+2ln\beta \sum _{i=1}^n{x}_i^2{\mathrm{e}}^{-2\lambda x_i}-ln\alpha \sum _{i=1}^n{x}_i^2{\mathrm{e}}^{-\lambda x_i}-2ln\beta \sum _{i=1}^n{x}_i^2{\mathrm{e}}^{-\lambda x_i}\left(1-{\mathrm{e}}^{-\lambda x_i}\right)\hfill \\ & & -4{\left(ln\beta \right)}^2\sum _{i=1}^n\frac{x_i^2{\mathrm{e}}^{-2\lambda x_i}}{{\left[ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta \right]}^2}-2ln\beta \sum _{i=1}^n\frac{x_i^2{\mathrm{e}}^{-\lambda x_i}}{ln\alpha +2\left(1-{\mathrm{e}}^{-\lambda x_i}\right)ln\beta }.\hfill \end{array}$$

Thus, the $100(1-\tau )\%$ approximate confidence intervals for $\alpha ,\beta$ and $\lambda$ can be constructed from the estimated diagonal elements of the above matrix.

2.3. MAP-Normal (MAPN) Model

Henceforth, we work with a random variable Z having the standard MAPN$(\alpha ,\beta ,0,1)$ distribution because of the transformation $X=\mu +\sigma Z$. Then, the density of Z reduces to ($\alpha ,\beta \ge 1,\alpha \beta \ne 1$)

$$G\left(z\right)=\frac{{\beta }^{{\mathsf{\Phi }}^2\left(z\right)}{\alpha }^{\mathsf{\Phi }\left(z\right)}-1}{\alpha \beta -1},$$

(9)

where $\mathsf{\Phi }\left(z\right)$ is the standard normal cdf.

Some properties of Z can be determined from (5) as

$$g\left(z\right)=\frac{1}{\alpha \beta -1}\left[\sum _{i,j=0;2i+j\ge 1}^{\infty }\left(2i+j\right){\psi }_{i,j}\left(\alpha ,\beta \right)\mathsf{\Phi }{\left(z\right)}^{2i+j-1}\varphi \left(z\right)\right],$$

(10)

where $\varphi \left(z\right)$ is the standard normal density.

Setting $u=\mathsf{\Phi }\left(z\right)$, we can write ${\mu }_r=E\left(Z^r\right)$ in terms of the standard normal qf $Q_{SN}\left(u\right)={\mathsf{\Phi }}^{-1}\left(u\right)$:

$$\begin{array}{c}\hfill {\mu }_r=\frac{1}{\alpha \beta -1}\left[\sum _{i,j=0;2i+j\ge 1}^{\infty }\left(2i+j\right){\psi }_{i,j}\left(\alpha ,\beta \right){\int }_0^1{Q}_{SN}{\left(u\right)}^r{u}^{2i+j-1}du\right].\end{array}$$

(11)

A power series for the standard normal qf holds [25]

$$\begin{array}{c}\hfill Q_{SN}\left(u\right)=\sum _{k=0}^{\infty }{b}_k{\left[\sqrt{2\pi }(u-1/2)\right]}^{2k+1},\end{array}$$

where the coefficient $b_k$ is calculated recursively from

$$b_{k+1}=\frac{1}{2(2k+3)}\sum _{r=0}^k\frac{(2r+1)(2k-2r+1)b_r{b}_{k-r}}{(r+1)(2r+1)}.$$

Here, $b_0=1$, $b_1=1/6$, $b_2=7/120$, $b_3=127/7560,\dots$ Therefore, we can evaluate numerically the integral

$$I_{r,i,j}=\underset{0}{\overset{1}{\int }}{\left\{\sum _{k=0}^{\infty }{b}_k{\left[\sqrt{2\pi }(u-1/2)\right]}^{2k+1}\right\}}^r{u}^{2i+j-1}du,$$

and determine the ordinary moments of the MAPN distribution.

The skewness and kurtosis measures of Z can be calculated from the ordinary moments using well-known relationships. The rth incomplete moment of Z also follows from an integral, as above, by changing one for a convenient constant. The first incomplete moment immediately gives the mean deviations of Z about the mean or any other location.

3. Results

3.1. Simulation Study

A simulation is performed to study the behavior of the MLEs of the parameters $\alpha$, $\beta$ and $\lambda$ of the MAPE distribution in terms of their averages, absolute biases (ABs) and mean square errors (MSEs) of the estimates obtained from 1000 samples under different scenarios for sample sizes $n=50,100,150,200,250$. The results are in

Table 2. Simulation findings.

			$\widehat{\mathit{\alpha }}$			$\widehat{\mathit{\beta }}$			$\widehat{\mathit{\lambda }}$
	$\mathit{n}$	Average	AB	MSE	Average	AB	MSE	Average	AB	MSE
	50	2.6519	0.3481	0.5395	1.9405	0.0595	0.0177	1.6455	0.3545	0.0427
$\alpha =3$	100	2.6913	0.3087	0.2258	1.9521	0.0479	0.0151	1.8823	0.1177	0.0311
$\beta =2$	150	3.1520	0.1520	0.0926	1.9567	0.0433	0.0144	1.8939	0.1061	0.0217
$\lambda =2$	200	3.0472	0.0472	0.0912	1.9582	0.0418	0.0133	1.8952	0.1048	0.0205
	250	3.0408	0.0408	0.0892	1.9667	0.0333	0.0104	1.8973	0.1027	0.0195
	50	3.2085	0.2085	0.4249	1.2341	0.7659	0.5996	0.2772	0.2228	0.0549
$\alpha =3$	100	2.8072	0.1928	0.2484	2.3468	0.3468	0.5218	0.3231	0.1769	0.0360
$\beta =2$	150	2.8204	0.1796	0.2041	2.1426	0.1426	0.4321	0.3296	0.1704	0.0329
$\lambda =0.5$	200	2.8767	0.1233	0.1267	1.8923	0.1077	0.4070	0.4059	0.0941	0.0122
	250	3.0134	0.0134	0.0794	1.9269	0.0731	0.3913	0.4081	0.0919	0.0092
	50	2.0467	0.9533	0.9876	1.9624	0.9624	0.9883	0.7984	0.2016	0.0675
$\alpha =3$	100	2.1031	0.8969	0.9265	1.9503	0.9503	0.9417	1.1421	0.1421	0.0362
$\beta =1$	150	3.4661	0.4661	0.2615	1.2780	0.2780	0.1018	1.1070	0.1070	0.0312
$\lambda =1$	200	3.1602	0.1602	0.1323	1.2060	0.2060	0.0613	1.0574	0.0574	0.0101
	250	3.0999	0.0999	0.1152	1.1093	0.1093	0.0176	1.0360	0.0360	0.0088
	50	3.0915	0.5915	0.9428	2.2471	0.2529	0.5766	0.5467	0.1533	0.0260
$\alpha =2.5$	100	2.8179	0.3179	0.4582	2.2679	0.2321	0.5339	0.5570	0.1430	0.0255
$\beta =0.5$	150	2.7816	0.2816	0.4437	2.3269	0.1731	0.3822	0.5802	0.1198	0.0176
$\lambda =0.7$	200	2.3223	0.1777	0.2118	2.3927	0.1073	0.2378	0.5835	0.1165	0.0146
	250	2.4409	0.0591	0.0340	2.5391	0.0391	0.0384	0.6859	0.0141	0.0143
	50	1.9513	1.5487	2.4017	2.0566	0.9566	0.9181	0.8239	0.0761	0.0180
$\alpha =3.5$	100	3.2120	0.2880	0.4656	1.2540	0.1540	0.1661	0.9397	0.0397	0.0174
$\beta =1.1$	150	3.2751	0.2249	0.1472	1.1974	0.0974	0.0243	0.8694	0.0306	0.0144
$\lambda =0.9$	200	3.4461	0.0539	0.0617	1.0803	0.0197	0.0061	0.9217	0.0217	0.0046
	250	3.5330	0.0330	0.0324	1.0965	0.0035	0.0036	0.8825	0.0175	0.0032
	50	2.0278	0.1722	0.0463	1.9497	0.2503	0.0737	1.8585	0.1415	0.0437
$\alpha =2.2$	100	2.0292	0.1708	0.0403	1.9588	0.2412	0.0682	1.8934	0.1066	0.0217
$\beta =2.2$	150	2.0419	0.1581	0.0388	1.9647	0.2353	0.0677	1.8934	0.1066	0.0217
$\lambda =2$	200	2.0445	0.1555	0.0358	1.9735	0.2265	0.0585	1.8960	0.1040	0.0202
	250	2.0539	0.1461	0.0356	1.9752	0.2248	0.0577	1.8968	0.1032	0.0197

.

We compare the MAPE distribution with some other extensions of the exponential distribution, specifically the Beta exponential (Beta-E), Kumaraswamy exponential (Kw-E) and Marshall–Olkin exponential (MOE), using a random sample of size 100 generated from the MAPE distribution with $\alpha =2$, $\beta =2$ and $\lambda =0.5$. The sample values are:

1.885041737, 4.986048585, 3.066673918, 0.959772040, 2.825368368, 0.824435649, 1.294124159, 1.012800439, 5.167502802, 1.068702611, 9.623282708, 2.156111659, 5.752932638, 3.670298312, 1.832103403, 8.127591949, 3.185589749, 0.068173249, 3.384436805, 1.746315020, 2.993498932, 12.453617304, 4.934371758, 7.789793514, 2.734155022, 2.642853318, 3.251226831, 7.176859866, 0.203007024, 5.584507972, 1.169154800, 0.437596269, 2.547156770, 1.606106547, 4.204954374, 4.496538512, 1.295321912, 3.177033049, 1.044723861, 1.646249159, 0.645593200, 2.402715247, 6.392833545, 7.284959565, 0.001960174, 4.312072308, 2.283924227, 0.462808553, 6.438231855, 4.652210157, 0.370191192, 3.436350472, 2.670331310, 3.058598962, 0.850709308, 6.201166734, 4.893258106, 3.137402020, 5.488975387, 6.466631233, 4.510068949, 2.098932480, 1.210160010, 0.350602649, 6.132940426, 1.215480205, 3.524247143, 1.599821076, 2.080543331, 1.868623797, 7.776530949, 1.203310307, 6.869488403, 5.233742062, 0.889446491, 4.337990211, 8.693465053, 1.555651987, 0.927514150, 4.653270469, 2.223475490, 2.077584863, 5.174423467, 10.262362345, 3.124613475, 3.487773970, 3.801199648, 2.102226125, 3.247546812, 2.429156739, 4.481736907, 1.595783483, 0.693145920, 6.019549617, 5.427266512, 2.496379854, 1.445541279, 2.085743446, 1.159817418 and 6.736977612.

The measures adopted to compare the models are the Akaike information criterion (AIC), Bayesian information criterion (BIC) and Hannan–Quinn information criterion (HQIC), along with the Kolmogorov–Smirnov (KS) goodness of fit and its p-value. The Kw-E, MOE and Beta-E distributions are fitted with the R script Newdistns [26].

Table 3. Results for the simulated MAPE data.

	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\lambda }}$	Minus Log-Likelihood	AIC	BIC	HQIC	KS Statistic	KS p-Value
MAPE	1.6161	1.8750	0.4688	216.4997	438.9995	446.8150	442.1625	0.0400	0.999998
Beta-E	1.4770	6.0754	0.0685	218.9210	443.8421	451.6576	447.0051	0.0537	0.935500
Kw-E	1.3599	16.9271	0.0355	217.7373	441.4745	449.2901	444.6376	0.0489	0.970400
MOE	3.4943	0.5118	-	218.9981	441.9962	447.2065	443.3049	0.0406	0.996526

reports the MLEs of the model parameters and the statistics mentioned before. It is clear that the MAPE model fitted the sample with the largest p-value among all models and smaller KS statistics, as well as minimum values of the maximized log-likelihood, AIC, BIC and HQIC.

We compare the MAPN distribution with the (i) normal distribution and (ii) sinh-arcsinh distribution (SAS) with parameters $ϵ$ and $\delta$ (defined in [27]) using a random sample of size 100 generated from the standard MAPN$(20,20,0,1)$ model. The sample values are:

−0.08363545, −0.18607357, 1.86273702, 2.25638263, 0.73660602, 0.43807657, 0.70991234, 1.49379206, 1.22935919, 1.04610514, 0.06031000, 1.95055682, 2.36414619, 1.35644650, 2.18061418, 1.02696357, 1.14251296, 0.33830318, 1.43612497, 0.99086614, 1.05799729, 3.68323963, 1.56528289, 1.75877235, 1.01704320, 1.26556227, 1.35099527, −0.63300406, 2.20036565, 1.55478240, 0.80134930, 0.88848599, 1.45800153, 0.80765750, 0.10462775, 1.52265250, 1.09206144, 0.98159601, 2.16970680, 1.37697204, 1.53911634, 1.14643762, 1.46446260, 2.21276767, 1.73042459, 1.21417199, 1.20884293, 1.40146066, 0.92637279, 0.73564445, 0.22756120, 2.35066872, 2.67684033, 0.94056519, 1.04040666, 1.19934400, 1.00711623, 1.88954724, 3.57866792, 1.81844198, 0.90083021, 2.00987153, 1.77569782, 1.32619050, 1.42991050, 1.93232500, 0.76668545, 1.30735528, 0.97714293, 0.47458649, 0.85462981, 2.24507741, 3.08341050, 2.30797564, 0.69125516, 1.09488414, 2.19492667, 1.10441197, 1.32840119, 1.99843395, 2.51819041, 1.42571479, 2.98604819, 1.72972425, 1.18167532, 1.47923676, 0.75889338, 2.05254023, 1.27701296, 1.98314559, 0.07153798, 2.10196116, 2.10919545, 1.23909188, 1.06198103, 0.22265900, 1.03588769, 1.16160506, −1.52414087 and 0.70722877.

Table 4. Results for the simulated MAPN data.

			Minus Log-Likelihood	AIC	BIC	HQIC	KS Statistic	KS p-Value
MAPN	$\widehat{\alpha }=$ 8.9431	$\widehat{\beta }=$ 14.9992	118.7269	241.4538	246.6642	243.5626	0.05	0.9996
Normal	$\widehat{\mu }=$1.3407	$\widehat{\sigma }=$ 0.8069	120.4394	244.8789	250.0892	246.9876	0.09	0.8127
SAS	$\widehat{ϵ}=$−0.8774	$\widehat{\delta }=$1.1704	149.7595	303.5189	308.7293	305.6276	0.33	$3.73\times {10}^{-5}$

provides the MLEs of the model parameters and the statistics mentioned before. It is clear that the MAPN model fitted to the generated sample has the smallest KS, minus log-likelihood, AIC, BIC and HQIC values and largest p-value.

3.2. Real Datasets

The performance of the MAPE distribution is verified by comparing it with the baseline distribution (exponential with rate $\lambda$) and five other competitive models: (i) the Weibull distribution with shape parameter $\alpha$ and scale parameter $\lambda$, (ii) the Marshall–Olkin exponential (MOE) distribution with parameters $\alpha$ and $\lambda$ as given by [5], (iii) the APE distribution with parameters $\alpha$ and $\lambda$ as defined in [8], (iv) the Beta-E distribution with parameters $\alpha$, $\beta$ and $\lambda$ as defined in [28] and (v) the Kumaraswamy exponential (Kw-E) distribution with parameters $\alpha$, $\beta$ and $\lambda$ as defined in [29].

Three datasets were analyzed to demonstrate the performance of the proposed model. The measures used to compare the models are the Akaike information criterion (AIC), Bayesian information criterion (BIC) and Hannan–Quinn information criterion (HQIC), along with the Kolmogorov–Smirnov (KS) goodness of fit and its p-value. The exponential and Weibull distributions are fitted with the R script fitdistrplus [30], whereas the MOE, Kw-E and Beta-E distributions are fitted with the R script Newdistns [26].

3.2.1. Diagnostic Wisconsin Breast Cancer Data

The two datasets are part of the Diagnostic Wisconsin Breast Cancer Data, which describe some characteristics of the cell nuclei presented in a digitized image of a fine needle aspirate of a breast mass. The dataset was collected and made open by Wolberg et al. at the University of Wisconsin. These data were first used in [31], and they are available at the UC Irvine Machine Learning Repository [32].

The Diagnostic Wisconsin Breast Cancer Data contain 30 features (V3, V4,..., V32) for each cell nucleus. The MAPE model is applied to V8 and V14.

Table 5. MLEs and goodness of fit statistics for V8.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	Minus Log-Likelihood	AIC	BIC	HQIC	KS Statistic	p-Value
Exponential	9.58396	-	-	−716.9918	−1431.99	−1427.6	−1430.3	0.27944	2.2 × ${10}^{-16}$
Weibull	0.11833	2.1092	-	−914.7481	−1825.50	−1816.8	−1822.1	0.06678	0.158
APE	0.46448	$5.8\times {10}^{-10}$	-	−706.5671	−1409.13	−1400.4	−1405.7	0.28471	2.2 × ${10}^{-16}$
MOE	31.2859	20.2769	-	−902.5504	−1801.10	−1801.1	−1797.7	0.07475	0.0035
MAPE	26.0899	297.0363	3.9260	-937.1483	−1868.30	−1855.3	−1863.2	0.05097	0.4508

and

Table 6. MLEs and goodness of fit statistics for V14.

Distribution	$\widehat{\mathit{\lambda }}$	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	Minus Log-Likelihood	AIC	BIC	HQIC	KS Statistic	p-Value
Exponential	0.8218	-	-	680.6767	1363.353	1367.697	1365.048	0.31634	2.2 × ${10}^{-16}$
Weibull	1.3750	2.3027	-	433.8886	871.7772	880.465	875.1672	0.073814	0.09007
APE	2.1263	1609.99	-	401.5559	807.1118	815.800	810.5018	0.054482	0.3671
MOE	3.2667	42.5702	-	423.8381	851.6763	860.364	855.0663	0.059195	0.0371
MAPE	2.5742	136.6009	53.2925	387.8032	781.6064	794.638	786.6913	0.033392	0.9089

provide, for V8 and V14, respectively, the MLEs of the parameters of all models and the statistics: minus maximized log-likelihood function, AIC, BIC, HQIC, KS and p-values. It is clear that whereas the exponential distribution failed to fit the data, the MAPE distribution succeeded. It fitted the two datasets with the lowest values of the minus maximized log-likelihood, AIC, BIC, HQIC and KS statistics, as well as the largest p-values among all models. Thus, it is revealed to be the best model for these data. To compare the empirical distribution of the data with the MAPE distribution, Figure 4 and Figure 5 display: (a) a relative histogram and fitted MAPE distribution, (b) plots of the fitted MAPE survival function and the empirical survival and (c) a probability vs. probability (P-P) plot, respectively. These plots support the results reported in Table 5 and Table 6.

3.2.2. Failure Stress of Carbon Fibers

The dataset refers to the failure stresses (in GPa) of 64 bundles of carbon fibres [33] which were also analyzed in [34] using LET-F family. We transform the recorded failure stresses (x) using the transformation $y=x-min\left(x\right)$. The observations are:

0.000, 0.231, 0.302, 0.327, 0.356, 0.449, 0.460, 0.495, 0.496, 0.544, 0.553, 0.553, 0.573, 0.617, 0.621, 0.624, 0.631, 0.674, 0.713, 0.715, 0.717, 0.723, 0.758, 0.774, 0.837, 0.839, 0.955, 1.016, 1.027, 1.036, 1.036, 1.076, 1.095, 1.129, 1.224, 1.238, 1.244, 1.319, 1.322, 1.334, 1.342, 1.363, 1.371, 1.393, 1.431, 1.445, 1.476, 1.507, 1.534, 1.592, 1.600, 1.636, 1.653, 1.661, 1.727, 1.951, 1.970, 1.985, 2.070, 2.123, 2.126, 2.324, 2.494 and 3.119.

Table 7. MLEs and goodness of fit statistics for the failure stress data.

	$\widehat{\mathit{\alpha }}$	$\widehat{\mathit{\beta }}$	$\widehat{\mathit{\lambda }}$	Minus Log-Likelihood	AIC	BIC	HQIC	KS Statistic	KS p-Value
MAPE	7.4968	6.7698	2.0128	56.6248	119.2495	125.7262	121.8010	0.0781	0.9898
Beta-E	39.7249	4.1564	0.8100	57.2210	120.4420	126.9186	122.9935	0.0876	0.7093
Kw-E	67.6731	1.8710	1.3638	57.0680	120.1361	126.6127	122.6876	0.0861	0.7298

provides the MLEs of the parameters of the models and the above statistics. Clearly, the MAPE model fitted the data with the largest p-value among all fitted models and smaller KS Statistics, as well as minimum values of the maximized log-likelihood, AIC, BIC and HQIC. Figure 6 supports the results reported in Table 7.

4. Conclusions

Over the last few decades, many generators have been proposed in different areas of data analysis. We presented a new class of distributions with two extra shape parameters called the modified alpha power (MAP) family, which can be a competitive model to well-known classes such as beta-G in [28] and Kumaraswamy-G in [29], among others. Some properties of the new family can be easily found from a linear representation of its density function. We defined an extended exponential distribution. Some of its mathematical properties and the maximum likelihood estimates are reported. The flexibility of an extended exponential distribution in this family is proved empirically by means of three real datasets. Future research can be conducted for other properties and applications.