Cubic rank transmuted generalized Gompertz distribution: properties and applications

Abstract

In this paper, we introduce a new lifetime distribution as an alternative to generalized Gompertz, Gompertz distribution and its modified ones. This new distribution is a special case of the family of distributions introduced by Granzotto et al. [D.C.T. Granzotto, F. Louzada and N. Balakrishnan, Cubic rank transmuted distributions: inferential issues and applications., J. Stat. Comput. Simul. 87 (2017), pp. 2760–2778]. We obtain some characteristic properties of suggested distribution such as hazard function, ordinary moments, coefficient of skewness, coefficient of kurtosis, moment generating function, quantile function and median. We discuss three different methods of estimation to estimate the parameters of proposed distribution. A comprehensive Monte Carlo simulation study is performed in order to compare the performances of estimators according to mean square errors and biases. Finally, three real data applications are performed to illustrate usefulness of suggested distribution.

1. Introduction

Modeling real data and providing statistical inferences about these data is an enlightening indicator for the path to be followed in the area of relevant research. There are many statistical distributions in the literature used in modelling real data. Determining the best-fitted statistical distribution to obtained data in many fields such as economics, medicine, social sciences, engineering, physics, chemistry, biology, veterinary medicine has become a very important problem in applied science. However, it is necessary to obtain new statistical distributions that are more flexible than existing distributions due to the increasing variety of data. In recent years, many methods have been proposed to obtain new statistical distributions. These methods help to generate a new statistical distribution by transforming or combining existing statistical distributions. Most of the distributions obtained by using these methods have a more flexible structure than baseline ones in real-data analysis because they contain additional parameters. Statistical distributions having to bathtub-shaped hazard function has the potential to model a wider variety of real data than ones have increasing or decreasing hazard rates. Obtaining new statistical distributions that will allow modelling of a wide variety of real data groups. Analyzing the properties of the suggested distributions and providing statistical inferences about the distribution and applying these inferences to real-life data is very important in terms of proving the existence of the proposed distribution not only in statistical theory but also in real-life applications. Therefore, there has been an increased interest in developing more flexible distributions. El-Gohary et al. [12] suggested a new three-parameter lifetime distribution. This distribution is a generalization of Gompertz $(a,b)$ distribution. The cumulative distribution function (cdf) and probability distribution function (pdf) of generalized Gompertz ( $GG(a,b,\alpha )$,for short) distribution are

$$G\left(x\right)={[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha }$$ (1)

and

$$g\left(x\right)=\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha -1},$$ (2)

where $,a,b,\alpha >0$ and x>0. ( $GG(a,b,\alpha )$ distribution is one of the useful lifetime distributions in reliability and survival analyses. Some of the papers about ( $GG(a,b,\alpha )$ distribution are listed as follows: Tahmasebi and Jafari [28], Benkhelifa [5], Benkhelifa [6]. Khan et al. [18] transmuted generalized Gompertz (TGG) distribution by using Quadratic Rank Transmutation method (QRTM) proposed by Shaw and Buckley [26,27]. The cdf and pdf of TGG distribution are as follows.

$$\begin{array}{rl}F(x;a,b,\alpha ,\lambda )& ={[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha }\\ & \times [1+\lambda -\lambda {[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha }]\end{array}$$ (3)

$$\begin{array}{rl}f(x;a,b,\alpha ,\lambda )& =\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha -1}\\ & \times [1+\lambda -2\lambda {[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha }]\end{array}$$ (4)

where $a,b,\alpha >0,\lambda \in [-1,1]$ and x>0  [18]. In last decades many authors have studied on transmuted distributions such as Tian et al. [30], Saboor et al. [22], Shahzad and Asghar [25], Khan et al. [17], Bhatti et al. [7], Alizadeh et al. [1], Tanış et al. [29], Saraçoğlu and Tanış [24]. Granzotto et al. [13] have introduced a new method called cubic rank transmutation map (CRTM) to generate new distributions. CRTM can be summarized as follows:

Let $X_1$, $X_2$, $X_3$ be a random sample from the distribution with cdf $G(x)$ and pdf $g(x)$. $X_{1:n}$, $X_{2:n}$ and $X_{3:n}$ be the order statistics of this sample.

Let us define a random variable Y by

$$\begin{array}{ll}Y\stackrel{d}{=}{\text{ X}}_{1:3},& \text{with probability}{p}_1\\ Y\stackrel{d}{=}{\text{ X}}_{2:3},& \text{with probability}{p}_2\\ Y\stackrel{d}{=}{\text{ X}}_{3:3},& \text{with probability}{p}_3,\end{array}$$

where $\stackrel{d}{=}$ denotes convergence in distribution and $p_1+p_2+p_3=1$. Thus, the cdf of $Y$is given by

$$\begin{array}{rl}{F}_Y\left(x\right)& =p_{1}P\left(X_{1:3}\le x\right)+p_{2}P\left(X_{2:3}\le x\right)+p_{3}P\left(X_{3:3}\le x\right)\\ & =p_1\left[1-{\left(1-G\left(x\right)\right)}^3\right]+6p_2{\int }_0^{x}G\left(t\right)\left(1-G\left(t\right)\right)g\left(t\right)\mathrm{d}t+p_3{G}^3\left(x\right)\\ & =3p_{1}G\left(x\right)+\left(3p_2-3p_1\right)G^2\left(x\right)+\left(1-3p_2\right)G^3\left(x\right)\end{array}$$ (5)

If it is substituted $3p_1={\lambda }_1,3p_2={\lambda }_2$ in Equation (5), the cdf and pdf of cubic rank transmuted distribution are constructed by

$$F_Y\left(x\right)={\lambda }_{1}G\left(x\right)+({\lambda }_2-{\lambda }_1)G^2\left(x\right)+(1-{\lambda }_2)G^3\left(x\right),$$ (6)

and

$$f_Y\left(x\right)=g\left(x\right)[{\lambda }_1+2({\lambda }_2-{\lambda }_1)G\left(x\right)+3(1-{\lambda }_2)G^2\left(x\right)],$$ (7)

respectively, where ${\lambda }_1\in [0,1],{\lambda }_2\in [-1,1]$. Granzotto et al. [13] have suggested cubic rank transmuted Weibull and cubic rank transmuted log-logistic distributions via CRTM. Then, Saraçoğlu and Tanış [23] suggested a new special case of the family of cubic rank transmuted distribution based on Kumaraswamy distribution [21]. Bhatti et al. [8] introduced cubic rank modified Burr III Pareto distribution. Aslam et al. [4] introduced a new family of distributions called as cubic transmuted-G family of distributions. This new family of distributions is a generalization of cubic rank transmuted distributions proposed by Granzotto et al. [13]. Hameldarbandi and Yılmaz [16] discussed some distributional properties of family of cubic rank transmuted distributions in detail. On the other hand, it is clearly seen that there are a few studies on cubic rank transmuted distributions in the literature. In this regard, we motivated to extend the special cases of the family of cubic rank transmuted distributions.

The main purpose of this paper is to introduce a new lifetime distribution called cubic rank transmuted generalized Gompertz distribution (CRTGG) by using CRTM and provide a new special case of cubic rank transmuted family of distributions. The paper is organized as follows: In Section 2, we introduce CRTGG distribution and examine its some distributional properties. In Section 3, maximum-likelihood (ML), least-square (LS), maximum product spacing (MPS) estimators for CRTGG distribution are obtained. In Section 4, an MC simulation study is presented based on performances of these estimators. Finally, three real data applications are performed to compare the fits of CRTGG distribution with other fitted models.

2. Cubic rank transmuted generalized Gompertz distribution

A new five parameters CRTGG distribution is constructed by using generalized Gompertz distribution (GG) defined in Equations (1) and (2) as baseline distribution in Equations (6) and (7).

Let X be a random variable from CRTGG distribution with $a,b,\alpha ,{\lambda }_1,{\lambda }_2$ parameters. It is denoted by CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$. The cdf, pdf, reliability function (rf) and hazard function (hf) of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution are given in Equations (8)–(11), respectively.

$$\begin{array}{rl}F\left(x\right)& ={\lambda }_1{[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha }\\ & +({\lambda }_2-{\lambda }_1){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{2\alpha }\\ & +(1-{\lambda }_2){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{3\alpha },\end{array}$$ (8)

$$\begin{array}{rl}f\left(x\right)& =\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha -1}\\ & \times [{\lambda }_1+2({\lambda }_2-{\lambda }_1)[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]\alpha \\ & +3(1-{\lambda }_2){(1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)))}^{2\alpha }],\end{array}$$ (9)

$$\begin{array}{rl}R\left(x\right)& =1-F\left(x\right),\end{array}$$ (10)

$$\begin{array}{rl}h\left(x\right)& =\frac{f\left(x\right)}{1-F\left(x\right)},\end{array}$$ (11)

where $a,b,\alpha >0$, ${\lambda }_1\in [0,1],{\lambda }_2\in [-1,1]$ and $x\in {\mathrm{R}}^{+}$. Substituting ${\lambda }_1=1$ and ${\lambda }_2=1$ in Equation (8), CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ reduces to GG $(a,b,\alpha )$ distribution, and substituting $\alpha =1$, ${\lambda }_1=1$ and ${\lambda }_2=1$ in Equation (8), CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ reduces to Gompertz $(a,b)$ distribution. Figures 1 and 2 illustrate possible shapes of the pdfs and hfs of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution for selected parameter values, respectively.

Figure 1.

Density shapes of CRTGG distribution for selected parameter values.

Figure 2.

Hazard function shapes of CRTGG distribution for selected parameter values.

Figure 2 shows that the CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution has an increasing, decreasing and bathtub hf.

2.1. Quantile function and median of CRTGG distribution

The $p\mathrm{th}$ quantile $Q(p)$ of the CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution is positive real solution of Equation (12):

$$Q\left(p\right)=\frac{\log \{1-\frac{b}{a}\log (1-h^{\frac{1}{\alpha }})\}}{b}$$ (12)

where

$$\begin{array}{rl}h& ={(\varpi +\sqrt{{\varpi }^2+{\psi }^3})}^{\frac{1}{3}}+{(\varpi -\sqrt{{\varpi }^2+{\psi }^3})}^{\frac{1}{3}}-\frac{({\lambda }_2-{\lambda }_1)}{3-3{\lambda }_2},\\ \varpi & =\frac{9(1-{\lambda }_2)({\lambda }_2-{\lambda }_1){\lambda }_1+27{(1-{\lambda }_2)}^{2}p-2{({\lambda }_2-{\lambda }_1)}^3}{54{(1-{\lambda }_2)}^3},\\ \psi & =\frac{3(1-{\lambda }_2){\lambda }_1-{({\lambda }_2-{\lambda }_1)}^2}{9{(1-{\lambda }_2)}^2}\end{array}$$

and, p $\in [0,1].$ Thus, the median can be obtained by substituting $p=\frac{1}{2}$ in Equation (12).

2.2. Moments of CRTGG distribution

Theorem 2.1

$r\mathrm{th}$ moment of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution is given by,

$$\begin{array}{rl}E(X^r)& ={\lambda }_1{\omega }_{j,k,r}(1,\alpha ,a,b)+2({\lambda }_2-{\lambda }_1){\omega }_{j,k,r}(2,\alpha ,a,b)\\ & +3(1-{\lambda }_2){\omega }_{j,k,r}(3,\alpha ,a,b)\end{array}$$ (13)

where

$$\begin{array}{rl}& {\omega }_{j,k,r}(m,\alpha ,a,b)\\ & =\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{m\alpha a\mathrm{\Gamma }\left(m\alpha \right)\mathrm{\Gamma }(r+1){(-1)}^{j+k+r+1}\exp (a(j+1)/b)b^{-(k+r+1)}}{\mathrm{\Gamma }\left(m\alpha -j\right)\mathrm{\Gamma }\left(j+1\right)\mathrm{\Gamma }(k+1){\left[a(j+1)\right]}^{-k}{(k+1)}^{r+1}},\end{array}$$ (14)

$m=1,2,3,r=1,2,\dots$ and $\mathrm{\Gamma }(.)$ is gamma function.

Proof.

$r\mathrm{th}$ moment of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution are written as follows:

$$\begin{array}{rl}E\left(X^r\right)& ={\int }_0^{\mathrm{\infty }}{x}^{r}g\left(x\right)\left[{\lambda }_1+2\left({\lambda }_2-{\lambda }_1\right)G\left(x\right)+3\left(1-{\lambda }_2\right)G^2\left(x\right)\right]\mathrm{d}x\\ & ={\lambda }_1{I}_1+2({\lambda }_2-{\lambda }_1)I_2+3(1-{\lambda }_2)I_3\end{array}$$ (15)

where $G(x;a,b,\alpha )$ and $g(x;a,b,\alpha )$ are defined in Equations (1) and (2).

$$\begin{array}{rl}I{}_1& ={\int }_0^{\mathrm{\infty }}{x}^{r}g\left(x\right)\mathrm{d}x\\ & ={\int }_0^{\mathrm{\infty }}{x}^r\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha -1}\mathrm{d}x\end{array}$$ (16)

The binomial series expansion of $[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){]}^{\alpha -1}$in right side of Equation (16) can be written as follows.

$${[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{\alpha -1}=\sum _{j=0}^{\mathrm{\infty }}\frac{{(-1)}^j\mathrm{\Gamma }\left(\alpha \right)\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)j)}{\mathrm{\Gamma }\left(\alpha -j\right)\mathrm{\Gamma }(j+1)}$$ (17)

 [14]. If Equation (17) is substituted into Equation (16), $I_1$ is obtained by

$$I_1=\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{\alpha a\mathrm{\Gamma }\left(\alpha \right)\mathrm{\Gamma }(r+1){(-1)}^{j+k+r+1}\exp (a(j+1)/b)b^{-(k+r+1)}}{\mathrm{\Gamma }\left(\alpha -j\right)\mathrm{\Gamma }(j+1)\mathrm{\Gamma }(k+1){\left[a(j+1)\right]}^{-k}{(k+1)}^{r+1}}$$ (18)

 [12]. where $r=1,2,\dots$ Similarly, $I_2$ and $I_3$ are obtained as follows:

$$\begin{array}{rl}{I}_2& ={\int }_0^{\mathrm{\infty }}{x}^{r}g\left(x\right)G\left(x\right)\mathrm{d}x\\ & ={\int }_0^{\mathrm{\infty }}{x}^r\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{2\alpha -1}\mathrm{d}x\\ & =\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{2\alpha a\mathrm{\Gamma }\left(2\alpha \right)\mathrm{\Gamma }(r+1){(-1)}^{j+k+r+1}\exp (a(j+1)/b)b^{-(k+r+1)}}{\mathrm{\Gamma }\left(2\alpha -j\right)\mathrm{\Gamma }(j+1)\mathrm{\Gamma }(k+1){\left[a(j+1)\right]}^{-k}{(k+1)}^{r+1}}\end{array}$$ (19)

and

$$\begin{array}{rl}{I}_3& ={{\int }_0^{\mathrm{\infty }}{x}^{r}g\left(x\right)G}^2\left(x\right)\mathrm{d}x\\ & ={\int }_0^{\mathrm{\infty }}{x}^r\alpha a{\mathrm{e}}^{bx}\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1)){[1-\exp (-\frac{a}{b}({\mathrm{e}}^{bx}-1))]}^{3\alpha -1}\mathrm{d}x\\ & =\sum _{j=0}^{\mathrm{\infty }}\sum _{k=0}^{\mathrm{\infty }}\frac{3\alpha a\mathrm{\Gamma }\left(3\alpha \right)\mathrm{\Gamma }(r+1){(-1)}^{j+k+r+1}\exp (a(j+1)/b)b^{-(k+r+1)}}{\mathrm{\Gamma }\left(3\alpha -j\right)\mathrm{\Gamma }(j+1)\mathrm{\Gamma }(k+1){\left[a(j+1)\right]}^{-k}{(k+1)}^{r+1}},\end{array}$$ (20)

respectively. If Equations (18), (19) and (20) are substituted into Equation (15), the proof is completed.

2.3. Coefficients of Skewness and Kurtosis for CRTGG distribution

Coefficient of skewness (CS) and coefficient of kurtosis (CK) of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution are given in Equations (21) and (22).

$$\mathrm{CS}=\frac{E\left(X^3\right)-3E\left(X\right)E\left(X^2\right)+2E{\left(X\right)}^3}{Var{\left(X\right)}^{\frac{3}{2}}}$$ (21)

and

$$\mathrm{CK}=\frac{E\left(X^4\right)-4E\left(X\right)E\left(X^3\right)+6E{\left(X\right)}^{2}E\left(X^2\right)-3E{\left(X\right)}^4}{\mathrm{Var}{\left(X\right)}^2},$$ (22)

respectively, where first four moments can be obtained by taking r = 1, 2, 3, 4 in Equation (13) and $\mathrm{Var}(X)=E(X^2)-[E(X){]}^2$

From Figures 4 and 5, it is clearly seen that as the value of a parameter increases, expected value and variance decreases and approaches to zero while CS increases. An increase in value of α parameter lead to a significant increase in expected value and variance while a decrease in CS and CK. Moreover, as the value of α increases, the CS and CK approximate to zero (Figure 3).

Figure 4.

Expected value, Variance, CS, CK for $a=3,b=2,\alpha \in [0,2],{\lambda }_1=0.5,{\lambda }_2\in \{-0.9,0,0.9\}$.

Figure 5.

Fitted cdfs for milk product data.

Figure 3.

Expected value, Variance, CS, CK for $a\in [0,2],b\in \{0.5,1.5,3\},\alpha =0.5,{\lambda }_1=0.5,{\lambda }_2=0.8$.

2.4. Moment generating function of CRTGG distribution

The moment generating function of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution, $M_x(t)$, is

$$M_x\left(t\right)={{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{tx}f\left(x\right)\mathrm{d}x=\sum _{r=0}^{\mathrm{\infty }}\frac{t^r}{r!}E(X}^r),$$ (23)

where $E(X^r)$ is given in Equation (13).

2.5. Order statistics

Let $X_1,X_2,\dots ,X_n$ is be random sample taken from $CRTGG(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution and $X_{1:n},X_{2:n},\dots ,X_{n:n}$ denote the order statistics of this random sample. The pdf of $X_{j:n},$ $j=1,\dots ,n$ is given as follows:

$$f_{X_{j:n}}\left(x\right)=\frac{n!}{\left(j-1\right)!\left(n-j\right)!}f\left(x\right){\left[F\left(x\right)\right]}^{j-1}{\left[1-F\left(x\right)\right]}^{n-j}$$ (24)

where $F(x)$ and $f(x)$ are defined in Equations (3) and (4).

Theorem 2.2

Let T ${}_{n:n}$ be the largest order statistics of a random sample $T_1,T_2,\dots ,T_n$ from the $CRTGG(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution.

$$\underset{n\to \mathrm{\infty }}{lim}P(\frac{T_{n:n}-a_n}{b_n}\le t)=\exp \{-\exp (-t)\},$$ (25)

where,

$$\begin{array}{rl}{a}_n& =F^{-1}(1-\frac{1}{n})\\ b_n& =F^{-1}(1-{\left(ne\right)}^{-1})-F^{-1}(1-\frac{1}{n})\end{array}$$

Proof.

The proof can be easily shown from Theorem 8.3.1, Theorem 8.3.3 and Theorem 8.3.4 in Arnold et al. [2].

3. Point estimation for CRTGG distribution

In this section, we consider three methods to estimate of parameters of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution. Therefore, ML estimators (MLEs), LS estimators (LSEs) and MPS estimators (MPSEs) are obtained for point estimation of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution.

3.1. Maximum-likelihood estimation

Let $X_1,X_2,\dots ,X_n$ be independent random variables from CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution with $a,b,\alpha ,{\lambda }_1,{\lambda }_2$ parameters. The log-likelihood function is given by

$$\begin{array}{rl}\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})& =n\log \alpha +n\log a+b\sum _{i=1}^n{x}_i-{\displaystyle \frac{a}{b}}\sum _{i=1}^n({\mathrm{e}}^{b{x}_i}-1)\\ & +(\alpha -1)\sum _{i=1}^n\log \left[w(x_i,a,b)\right]\\ & +\sum _{i=1}^n\log [{\lambda }_1+k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)],\end{array}$$ (26)

where $\mathbf{x}=x_1,x_2,\dots ,x_n$, $w(x_i,a,b)=1-\exp (-\frac{a}{b}({\mathrm{e}}^{b{x}_i}-1))$,

$k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)=2({\lambda }_2-{\lambda }_1)w(x_i,a,b{)}^{\alpha }+3(1-{\lambda }_2)w(x_i,a,b{)}^{2\alpha }.$ Then, the log-likelihood equations are given as follows:

$$\begin{array}{rl}{\displaystyle \frac{\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})}{\mathrm{\partial }a}}& ={\displaystyle \frac{n}{a}}-{\displaystyle \frac{1}{b}}\sum _{i=1}^n({\mathrm{e}}^{b{x}_i}-1)\\ & +(\alpha -1)\sum _{i=1}^n{\displaystyle \frac{({\mathrm{e}}^{b{x}_i}-1)(1-w(x_i,a,b))}{bw(x_i,a,b)}}\\ & +\sum _{i=1}^n{\displaystyle \frac{\alpha ({\mathrm{e}}^{b{x}_i}-1)z(x_i,a,b,\alpha )h(x_i,a,b,{\lambda }_1,{\lambda }_2)}{b({\lambda }_1+k(x_i,a,b,{\lambda }_1,{\lambda }_2))}}=0,\end{array}$$ (27)

$$\begin{array}{rl}{\displaystyle \frac{\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})}{\mathrm{\partial }b}}& =\sum _{i=1}^n{x}_i+{\displaystyle \frac{a}{b^2}}\sum _{i=1}^n({\mathrm{e}}^{b{x}_i}-1)-{\displaystyle \frac{a}{b}}\sum _{i=1}^n{x}_i{\mathrm{e}}^{b{x}_i}\\ & -(\alpha -1)\sum _{i=1}^n{\displaystyle \frac{s(x_i,a,b)(1-w(x_i,a,b))}{w(x_i,a,b)}}\\ & -\sum _{i=1}^n{\displaystyle \frac{\alpha s(x_i,a,b)z(x_i,a,b,\alpha )h(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{{\lambda }_1+k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}}=0,\end{array}$$ (28)

$$\begin{array}{rl}\frac{\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})}{\mathrm{\partial }\alpha }& =\frac{n}{\alpha }+\sum _{i=1}^n\frac{\alpha \log \left(w(x_i,a,b)\right)w{(x_i,a,b)}^{\alpha }h(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{{\lambda }_1+k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}=0,\end{array}$$ (29)

$$\begin{array}{rl}\frac{\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})}{\mathrm{\partial }{\lambda }_1}& =\sum _{i=1}^n\frac{1-2w{(x_i,a,b)}^{\alpha }}{{\lambda }_1+k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}=0\end{array}$$ (30)

and

$$\frac{\ell (a,b,\alpha ,{\lambda }_1,{\lambda }_2|\mathbf{x})}{\mathrm{\partial }{\lambda }_2}=\sum _{i=1}^n\frac{2w{(x_i,a,b)}^{\alpha }-3w{(x_i,a,b)}^{2\alpha }}{{\lambda }_1+k(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)}=0$$ (31)

where $h(x_i,a,b,\alpha ,{\lambda }_1,{\lambda }_2)=2({\lambda }_2-{\lambda }_1)+6(1-{\lambda }_2)w(x_i,a,b{)}^{\alpha },s(x_i,a,b)=\frac{a}{b^2}({\mathrm{e}}^{x_i}-1)-\frac{a}{b}{x}_i{\mathrm{e}}^{b{x}_i}$and $z(x_i,a,b,\alpha )=w(x_i,a,b{)}^{\alpha -1}-w(x_i,a,b{)}^{\alpha }$. The MLEs  is obtained maximizing the log-likelihood function in Equation (26). We get the MLEs ${\hat{a}}_{\mathrm{MLE}},$ ${\hat{b}}_{\mathrm{MLE}},$ ${\hat{\alpha }}_{\mathrm{MLE}},$ ${\hat{\lambda }}_{1_{\mathrm{MLE}}}$ and ${\hat{\lambda }}_{2_{\mathrm{MLE}}}$ of a,b, $\alpha ,{\lambda }_1$ and ${\lambda }_2$ with simultaneously the solutions of log-likelihood equations in (27)–(31). The log-likelihood equations can be solved using numerical method such as Newton Raphson.

3.2. Least-square estimation

Let $X_1,X_2,\dots ,X_n$ be a random sample from CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution and $X_{1:n}\le X_{2:n}\le \cdots \le X_{n:n}$ be order statistics of this sample. The LSEs , ${\hat{a}}_{\mathrm{LSE}},{\hat{b}}_{\mathrm{LSE}},{\hat{\alpha }}_{\mathrm{LSE}},{\hat{\lambda }}_{1_{\mathrm{LSE}}},{\hat{\lambda }}_{2_{\mathrm{LSE}}}$, of $a,b,\alpha ,{\lambda }_1$ and ${\lambda }_2$ can be obtained by minimizing

$$Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)=\sum _{i=1}^n{[F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1}]}^2$$ (32)

with respect to $a,b,\alpha ,{\lambda }_1$ and ${\lambda }_2$. Therefore, ${\hat{a}}_{\mathrm{LSE}},{\hat{b}}_{\mathrm{LSE}},{\hat{\alpha }}_{\mathrm{LSE}},{\hat{\lambda }}_{1_{\mathrm{LSE}}}$and ${\hat{\lambda }}_{2_{\mathrm{LSE}}}$ can be obtained by simultaneously solving of the following system of equations:

$$\begin{array}{rl}& \frac{\mathrm{\partial }Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{\mathrm{\partial }a}=\sum _{i=1}^n{F}_a^{\prime }(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)(F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1})=0\\ & \frac{\mathrm{\partial }Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{\mathrm{\partial }b}=\sum _{i=1}^n{F}_b^{\prime }(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)(F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1})=0,\\ & \frac{\mathrm{\partial }Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{\mathrm{\partial }\alpha }=\sum _{i=1}^n{F}_{\alpha }^{\prime }(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)(F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1})=0,\\ & \frac{\mathrm{\partial }Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{\mathrm{\partial }{\lambda }_1}=\sum _{i=1}^n{F}_{{\lambda }_1}^{\prime }(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)(F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1})=0\end{array}$$

and

$$\frac{\mathrm{\partial }Z(a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{\mathrm{\partial }{\lambda }_2}=\sum _{i=1}^n{F}_{{\lambda }_2}^{\prime }(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)(F(x_{i:n},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-\frac{i}{n+1})=0$$

3.3. Maximum product spacing method

MPS method was proposed by Cheng and Amin [10] as an alternative to the ML method. The MPS method provides consistent estimates under more general conditions than the MLEs. MPSEs are effective and asymptotically normal [10]. MPS method is based on the idea that the difference between consecutive observations should be identically distributed. The geometric mean of the spacings is given by

$$GM=\sqrt[n+1]{\prod _{i=1}^{n+1}{D}_i}$$ (33)

where the spacing $D_i$ is defined by

$$D_i={\int }_{x_{(i-1)}}^{x_{\left(i\right)}}f(a,b,\alpha ,{\lambda }_1,{\lambda }_2)\mathrm{d}x;i=1,2,\dots ,n+1,$$ (34)

respectively. The MPSEs, ${\hat{a}}_{\mathrm{MPSE}},{\hat{b}}_{\mathrm{MPSE}},{\hat{\alpha }}_{\mathrm{MPSE}},{\hat{\lambda }}_{1_{\mathrm{MPSE}}},{\hat{\lambda }}_{2_{\mathrm{MPSE}}}$, are obtained by maximizing the geometric mean (GM) of the spacings. Logarithm of Equation (33) is given by

$$\mathrm{LogGM}=\frac{1}{n+1}\sum _{i=1}^{n+1}\log [F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)],$$ (35)

where $F(x_{(0)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)=0$ and $F(x_{(n+1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)=1$. The MPSEs ${\hat{a}}_{\mathrm{MPSE}},{\hat{b}}_{\mathrm{MPSE}},{\hat{\alpha }}_{\mathrm{MPSE}},{\hat{\lambda }}_{1_{\mathrm{MPSE}}}$ and ${\hat{\lambda }}_{2_{\mathrm{MPSE}}}$ can be obtained by the solution of the following system of equations;

$$\begin{array}{rl}& \frac{\mathrm{\partial }\mathrm{LogGM}}{\mathrm{\partial }a}=\frac{1}{n+1}\sum _{i=1}^{n+1}\left[\frac{F_a^{\prime }(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F_a^{\prime }(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}\right]=0,\\ & \frac{\mathrm{\partial }\mathrm{LogGM}}{\mathrm{\partial }b}=\frac{1}{n+1}\sum _{i=1}^{n+1}\left[\frac{F_b^{\prime }(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F_b^{\prime }(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}\right]=0,\\ & \frac{\mathrm{\partial }\mathrm{LogGM}}{\mathrm{\partial }\alpha }=\frac{1}{n+1}\sum _{i=1}^{n+1}\left[\frac{F_{\alpha }^{\prime }(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F_{\alpha }^{\prime }(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}\right]=0,\\ & \frac{\mathrm{\partial }\mathrm{LogGM}}{\mathrm{\partial }{\lambda }_1}=\frac{1}{n+1}\sum _{i=1}^{n+1}\left[\frac{F_{{\lambda }_1}^{\prime }(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F_{{\lambda }_1}^{\prime }(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}\right]=0\end{array}$$

and

$$\frac{\mathrm{\partial }\mathrm{LogGM}}{\mathrm{\partial }{\lambda }_2}=\frac{1}{n+1}\sum _{i=1}^{n+1}\left[\frac{F_{{\lambda }_2}^{\prime }(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F_{{\lambda }_2}^{\prime }(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}{F(x_{\left(i\right)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)-F(x_{(i-1)},a,b,\alpha ,{\lambda }_1,{\lambda }_2)}\right]=0$$

4. Simulation study

In this section, we consider an MC simulation study to compare the performances of MLEs, MPSEs and LSEs of $a,b,\alpha ,{\lambda }_1$ and ${\lambda }_2$ in terms of MSEs and biases. In the simulation study, the biases and MSEs of the MLEs, LSEs and MPSEs are emprically estimated by 5000 trials. The sample sizes are fixed as 50,100,200,500 and four different parameter settings are considered as follows: $Case1=(0.5,0.2,0.7,0.3,0.8),$ $Case2=(0.4,0.7,0.8,0.1,0.9)$ and $Case3=(1.2,1.5,0.5,0.6,0.95)$ with parameter values are given in Table 1.

Table 1.

Biases and MSEs of MLEs,LSEs and MPSEs for CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution parameters.

			$\hat{a}$		$\hat{b}$		$\hat{\alpha }$		${\hat{\lambda }}_1$		${\hat{\lambda }}_2$
Case	n		bias	MSE	bias	MSE	bias	MSE	bias	MSE	bias	MSE
1	50	MLE	0.0321	0.0409	–0.0599	0.0163	0.5430	0.5757	0.2232	0.0973	–1.0787	1.2893
		LSE	–0.0209	0.0722	0.0033	0.0365	0.4082	0.7052	0.2334	0.1137	–1.1105	1.5127
		MPSE	0.0460	0.0462	–0.0978	0.0229	0.3181	0.3198	0.2056	0.0748	–1.3984	2.0247
	100	MLE	0.0162	0.0265	–0.0647	0.0114	0.4844	0.4345	0.1759	0.0789	–1.0115	1.1596
		LSE	0.0366	0.0573	–0.0588	0.0216	0.5521	0.7489	0.2272	0.1113	–1.0719	1.3943
		MPSE	0.0239	0.0294	–0.0869	0.0153	0.3334	0.2933	0.1490	0.0610	–1.2245	1.5884
	200	MLE	–0.0231	0.0163	–0.0536	0.0073	0.4304	0.3235	0.1374	0.0666	–0.8210	0.7981
		LSE	0.0282	0.0410	–0.0693	0.0158	0.5713	0.6680	0.2064	0.1058	–0.9282	1.0866
		MPSE	–0.0189	0.0176	–0.0671	0.0093	0.3132	0.2381	0.0961	0.0505	–0.9864	1.0598
	500	MLE	–0.0743	0.0131	–0.0343	0.0035	0.3652	0.2170	0.0773	0.0438	–0.5401	0.3767
		LSE	–0.0235	0.0214	–0.0547	0.0090	0.5102	0.4499	0.1548	0.0811	–0.6538	0.5872
		MPSE	–0.0712	0.0133	–0.0420	0.0042	0.2779	0.1693	0.0322	0.0356	–0.6733	0.5177
2	50	MLE	0.2602	0.1657	–0.1987	0.0915	0.9641	2.1528	0.3335	0.1489	–1.2354	1.6457
		LSE	0.1172	0.2007	0.0874	0.3030	0.6768	2.7224	0.3256	0.1657	–1.1893	1.6814
		MPSE	0.2511	0.1671	–0.2397	0.1138	0.5417	1.0984	0.3181	0.1220	–1.5686	2.5090
	100	MLE	0.2303	0.1190	–0.2072	0.0757	0.7794	1.2497	0.2761	0.1114	–1.1564	1.4731
		LSE	0.2418	0.2176	–0.1362	0.1421	0.9616	2.9003	0.3289	0.1662	–1.1746	1.6324
		MPSE	0.2201	0.1182	–0.2273	0.0878	0.5088	0.7725	0.2549	0.0900	–1.3985	2.0260
	200	MLE	0.1608	0.0670	–0.1750	0.0531	0.6140	0.7672	0.2181	0.0796	–0.9805	1.0936
		LSE	0.2638	0.1794	–0.2070	0.1022	0.9988	2.2656	0.3173	0.1564	–1.0751	1.3828
		MPSE	0.1516	0.0666	–0.1843	0.0588	0.4213	0.5275	0.1904	0.0616	–1.1800	1.4679
	500	MLE	0.0659	0.0229	–0.1173	0.0270	0.4259	0.3812	0.1413	0.0446	–0.6959	0.5915
		LSE	0.1667	0.0851	–0.1727	0.0610	0.7610	1.2040	0.2387	0.1053	–0.7911	0.7987
		MPSE	0.0609	0.0233	–0.1217	0.0290	0.2910	0.2803	0.1103	0.0317	–0.8793	0.8440
3	50	MLE	0.1635	0.3438	–0.4916	0.4715	0.7560	0.8050	0.0093	0.0540	–1.2796	1.1756
		LSE	–0.0832	0.5321	–0.1309	0.6306	0.5518	0.5901	0.0143	0.0523	–1.1592	1.6162
		MPSE	0.1883	0.3763	–0.6191	0.6164	0.5227	0.4642	–0.0467	0.0399	–1.5131	2.3755
	100	MLE	0.0728	0.2046	–0.4879	0.3706	0.6872	0.6275	–0.0102	0.0577	–1.1887	1.5504
		LSE	–0.0102	0.3963	–0.3346	0.4467	0.6266	0.6192	0.0016	0.0539	–1.0977	1.4632
		MPSE	0.0842	0.2224	–0.5603	0.4524	0.5315	0.4334	–0.0719	0.0536	–1.3355	1.8915
	200	MLE	–0.0588	0.1255	–0.4254	0.2656	0.6541	0.5309	–0.0053	0.0547	–1.0061	1.1453
		LSE	–0.0746	0.2646	–0.3590	0.3299	0.6324	0.5519	0.0060	0.0552	–0.9767	1.1792
		MPSE	–0.0580	0.1375	–0.4638	0.3089	0.5354	0.4049	–0.0789	0.0604	–1.1006	1.3241
	500	MLE	–0.2457	0.1172	–0.3019	0.1380	0.6064	0.4196	–0.0041	0.0440	–0.7150	0.5996
		LSE	–0.2241	0.1895	–0.2840	0.2012	0.6077	0.4605	0.0039	0.0526	–0.7409	0.7075
		MPSE	–0.2406	0.1188	–0.3271	0.1568	0.5291	0.3489	–0.0751	0.0551	–0.7699	0.6708

According to Table 1, as sample size increases, it is seen that the MSEs and biases of MLEs, LSEs and MPSEs of the unknown parameters of the CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution decrease and approaches each other. As a result of simulation study, we recommend ML method to estimate a, b and ${\lambda }_2$ parameters while MPS method to estimate α and ${\lambda }_1$ parameters according to MSE criterion.

5. Real-data analysis

In this section, three real-data applications are presented to illustrate the applicability of the CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ model in real life and to compare the fits of other fitted distributions. For this reason, we consider $-2\times$ log-likelihood value, Akaike's Information Criterion (AIC), Bayesian Information Criterion (BIC), Anderson–Darling statistics (A*), Cramer-von-Mises statistics (W*), Kolmogorov–Smirnov statistics (K-S) and its p-value for the selection of models.

5.1. Milk product data

Firstly we analyze the data about the total milk production proportion in the first birth of 107 cows living in the Carnaúba farm in Brazil. The data exist in studies of Cordeiro and Brito [11], Brito [9].The first data set is fitted to CRTGG, TGG, GG, transmuted Weibull (TW) [3], Weibull (W), Exponentiated Exponential (EE, [15]) and transmuted exponentiated exponential (TEE) [19] distributions. The results of real-data application are presented for milk product data in Tables 2 and 3. MLEs and standard errors of unknown parameters of fitted distributions are given in Table 2 and selection criteria statistics of fitted distributions are given in Table 3. Also, Figures 1–4 show fitted cdfs and pdfs for milk product data set, respectively.

Table 2.

The MLEs and standard errors for milk product data.

	$\hat{a}$	$\hat{b}$	$\hat{\alpha }$	$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_2}$	SE( $\hat{a}$)	SE( $\hat{b}$)	SE( $\hat{\alpha }$)	SE( $\widehat{{\lambda }_1}$)	SE( $\widehat{{\lambda }_2}$)
CRTGG	2.2770	2.2448	1.5739	0.6770	–0.8487	0.9838	0.7995	0.4987	0.2305	0.5827
TGG	0.5567	4.3758	0.9789	–0.4864	–	0.3236	0.9575	0.3473	0.5070	–
GG	0.3427	5.0464	1.0365	–	–	0.2075	0.9756	0.2619	–	–
Weibull	2.6012	0.5236	–	–	–	0.2098	0.0202	–	–	–
TW	2.3230	0.4717	–0.4801	–	–	0.2388	0.0290	0.2084	–	–
TEE	3.1199	4.7499	–0.6874	–	–	0.5967	0.4033	0.1415	–	–
EE	3.7139	4.2007	–	–	–	0.5658	0.3728	–	–	–

Table 3.

Selection criteria statistics for milk product data.

Distributions	–2logL	AIC	BIC	K-S	A*	W*	p-value
CRTGG	–61.3198	–51.3198	–37.9556	0.0441	0.1324	0.0192	0.9851
TGG	–58.5449	–50.5449	–39.8536	0.0458	0.2036	0.0282	0.9780
GG	–58.2715	–52.2715	–44.253	0.0504	0.2376	0.0345	0.9483
Weibull	–42.695	–38.695	–33.3494	0.0832	1.4840	0.1894	0.4486
TW	–45.8377	–39.8377	–31.8192	0.0676	1.1366	0.1201	0.7121
TEE	–20.1764	–14.1764	–6.15791	0.1231	3.4382	0.5414	0.0776
EE	–10.0775	–6.0775	–0.7318	0.1476	4.6866	0.8125	0.0188

From Table 3, it is observed that the best-fitted model is CRTGG model according to $-2\times$ log-likelihood value, A*, W*, K-S and its p-value while GG is the best-fitted model according to AIC and BIC (Figure 6).

Figure 6.

Fitted pdfs for milk product data.

5.2. Carbon fibres data

The second real data set consists of 100 observations based on breaking stress of carbon fibres (in Gba) given by Nichols and Padgett [20]. These data are fitted CRTGG, TGG, GG, Gompertz, Weibull, TW, TEE and EE distributions. The MLEs and standard errors of unknown parameters of fitted distributions and selection criteria statistics of fitted distributions for carbon fibres data set are given, respectively, in Tables 4 and 5. Further, the graphs of fitted cdfs and pdfs for carbon fibres data are presented in Figures 7 and 8, respectively.

Table 4.

The MLEs and standard errors for carbon fibres dataset.

	$\hat{a}$	$\hat{b}$	$\hat{\alpha }$	$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_2}$	SE( $\hat{a}$)	SE( $\hat{b}$)	SE( $\hat{\alpha }$)	SE( $\widehat{{\lambda }_1}$)	SE( $\widehat{{\lambda }_2}$)
CRTGG	0.7383	0.2159	5.0388	0.8268	0.1046	0.2924	0.1219	2.5399	0.5165	0.8863
TGG	0.5200	0.2852	3.3506	–0.4304	–	0.2180	0.1339	1.4039	0.7646	–
GG	0.4094	0.3433	3.3574	–	–	0.1464	0.1173	1.0653	–	–
Gompertz	0.0769	0.7911	–	–	–	0.0174	0.0776	–	–	–
Weibull	2.7929	2.9437	–	–	–	0.2141	0.1111	–	–	–
TW	2.9935	3.4126	0.6789	–	–	0.2413	0.3377	0.3798	–	–
TEE	6.1874	1.1020	–0.6837	–	–	1.9283	0.0947	0.2793	–	–
EE	7.7882	1.0132	–	–	–	1.4962	0.0875	–	–	–

Table 5.

Selection criteria statistics for carbon fibres data set.

Distributions	–2logL	AIC	BIC	K-S	A*	W*	p-value
CRTGG	282.0365	292.0365	305.0623	0.0537	0.3331	0.0478	0.9348
TGG	282.5691	290.5691	300.9898	0.0634	0.3988	0.0672	0.8161
GG	282.7799	288.7799	296.5954	0.0637	0.4197	0.0708	0.8116
Gompertz	298.25	302.25	307.4604	0.0962	1.7537	0.2280	0.3129
Weibull	283.0586	287.0586	292.2689	0.0604	0.4176	0.0633	0.8578
TW	282.2698	288.2698	296.0853	0.0642	0.3813	0.0650	0.8038
TEE	288.8303	294.8303	302.6458	0.0966	0.9273	0.1748	0.3078
EE	292.3646	296.3646	301.5749	0.1077	1.2246	0.2291	0.1961

Figure 7.

Fitted cdfs for carbon fibres data.

Figure 8.

Fitted pdfs for carbon fibres data.

For carbon fibres data, we observe that CRTGG distribution is best-fitted model in terms of $-2\times$ log-likelihood value, A*, W*, K-S and its p-value. On the other hand, one of the competitor distributions TW is the best-fitted distribution according to AIC and BIC.

5.3. Earthquake data

The third real data consists of 41 observations are magnitudes of earthquakes in the Aegean Sea on 21 November 2020. The data are obtained by (http://www.koeri.boun.edu.tr/). The earthquake data set is fitted CRTGG, GG, Gompertz, Weibull, TW and EE distributions. The data are as follows: 2.2, 3.3, 2.2, 1.7, 1.5, 2.5, 2.1, 1.4, 2.5, 1.8, 1.7, 2.1, 2, 3.1, 2.2, 2.1, 2, 2, 2.1, 2.2, 2.2, 2.1, 1.2, 1.4, 2.5, 2.1, 1.7, 1.6, 1.5, 1.5, 1.7, 1.2, 2.1, 1.5, 1.8, 1.6, 1.7, 1.7, 0.9, 1.8, 1.7.

The MLEs and standard errors for unknown parameters and selection criteria statistics of these fitted distributions for earthquake data set are given in Tables 6 and 7, respectively. Figures 9 and 10 provide fitted cdfs and pdfs for earthquake data, respectively.

Table 6.

The MLEs and standard errors for earthquake data.

	$\hat{a}$	$\hat{b}$	$\hat{\alpha }$	$\widehat{{\lambda }_1}$	$\widehat{{\lambda }_2}$	SE( $\hat{a}$)	SE( $\hat{b}$)	SE( $\hat{\alpha }$)	SE( $\widehat{{\lambda }_1}$)	SE( $\widehat{{\lambda }_2}$)
CRTGG	2.5183	0.0735	34.1913	0.2823	–0.3989	1.6075	0.3239	57.7962	0.3340	0.7925
GG	1.1193	0.4220	16.3672	–	–	0.6598	0.3029	13.9670	–	–
Gompertz	0.0426	1.7460	–	–	–	0.0182	0.2082	–	–	–
Weibull	4.1736	2.0888	–	–	–	0.4623	0.0829	–	–	–
TW	4.6063	2.3100	0.7412	–	–	0.5159	0.1087	0.2316	–	–
EE	55.5557	2.3899	–	–	–	25.4031	0.2834	–	–	–

Table 7.

Selection criteria statistics for earthquake data.

Distributions	–2logL	AIC	BIC	K-S	A*	W*	p-value
CRTGG	49.95028	59.95028	68.51814	0.121041	0.506931	0.093244	0.585236
GG	51.35952	57.35952	62.50024	0.126012	0.566597	0.093701	0.53301
Gompertz	67.67717	71.67717	75.10432	0.206779	2.073927	0.31944	0.060024
Weibull	56.59084	60.59084	64.01798	0.166954	1.036345	0.156801	0.203204
TW	54.12297	60.12297	65.26369	0.14451	0.790718	0.117913	0.358746
EE	53.43354	57.43354	60.86068	0.129815	0.719847	0.119653	0.494282

Figure 9.

Fitted cdfs for earthquake data.

Figure 10.

Fitted pdfs for earthquake data.

Table 7 illustrates that the best-fitted model is CRTGG model according to $-2\times$ log-likelihood value, A*, W*, K-S and its p-value while GG is the best-fitted model according to AIC and BIC for earthquake data.

6. Conclusion

In this paper, we propose a new lifetime distribution called CRTGG distribution with parameters $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ using CRTM suggested by Granzotto et al. [13]. This new distribution can be used in modelling the data having to increasing, decreasing and bathtub hazard rates. We discuss some statistical properties such as, moments, variance, skewness and kurtosis coefficients, moment generating function, order statistics, quantile function and median of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution are obtained. Also, MLEs, LSEs and MPSEs of the parameters are obtained. These estimators via an MC simulation study have been compared according to MSE criterion. We recommend ML method to estimate $a,b,{\lambda }_1$ parameters while MPS method to estimate α and ${\lambda }_2$ parameters according to MSE criterion. Finally, three real data applications are presented to compare the fits of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution and some well-known models. According to real-data analysis results, CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ is the best-fitted model for three data sets according to $-2\times$ log-likelihood value, A*, W*, K-S and its p-value. This situation has indicated the usefulness of the CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ model in real life. One of the advantages of CRTGG $(a,b,\alpha ,{\lambda }_1,{\lambda }_2)$ distribution is that it can be used to model various real data sets having to increasing, decreasing and bathtube shape hazard rates.

Disclosure statement

No potential conflict of interest was reported by the author(s).