The Ristić–Balakrishnan–Topp–Leone–Gompertz-G Family of Distributions with Applications

Abstract

In this paper, we introduce the newly generated Ristić–Balakrishnan–Topp–Leone–Gompertz-G family of distributions. Statistical and mathematical properties of this new family including moments, moment generating function, incomplete moments, conditional moments, probability weighted moments, distribution of the order statistics, stochastic ordering, and Rényi entropy are derived. The unknown parameters of the family are inferred using the maximum likelihood estimation technique. A Monte Carlo simulation study is performed to investigate the convergence of the maximum likelihood estimation. Three real-life data sets are used to demonstrate the flexibility and capacity of the new family of distributions.

1 Introduction

Probability distributions are utilized extensively in statistical analysis especially when modeling and predicting real-world phenomena. There is a growing demand for more flexible distributions due to the variety and complexity of big data encountered nowadays. Recent work on the extensions of the existing distributions includes the Gompertz–Topp Leone-G family of distributions [33], the Marshall–Olkin-Odd power generalized Weibull-G family of distributions [10], Topp–Leone odd Fréchet generated family of distributions [4], a new extended Rayleigh distribution [6], the odd Weibull inverse Topp–Leone distribution [5], Type I half logistic Burr X-G family of distributions [2], and the Marshall–Olkin–Weibull-H family of distributions [1].

The gamma generator proposed by [35] has been one of the most explored approaches for creating new distributions. The cumulative distribution function (cdf) of the gamma generator is provided by

$$\begin{aligned} F_{RB}(x;\delta )=1-\dfrac{1}{\varGamma (\delta )}\int _0^{-\log \left( G(x)\right) } t^{\delta -1}\textrm{e}^{-t}dt, \quad \delta >0 \end{aligned}$$

(1)

with a probability density function (pdf)

$$\begin{aligned} f_{RB}(x;\delta )=\dfrac{1}{\varGamma (\delta )}\left[ -\log \left( G(x)\right) \right] ^{\delta -1}g(x), \quad x\in {\mathbb {R}}. \end{aligned}$$

(2)

Some existing distributions developed via the gamma transformation method include the gamma odd power generalized Weibull-G family of distributions by [14], a new gamma generalized Lindley-log-logistic distribution by [24], the gamma modified Weibull distribution by [11], the gamma-exponentiated Weibull distribution by [31], the gamma odd Burr III-G family of distributions by [30], the gamma Weibull-G family of distributions by [28], the Zografos–Balakrishnan-G family of distributions by [27], Zografos–Balakrishnan Burr XII distribution by [8], the Zografos–Balakrishnan Lindley distribution by [21], the Zografos–Balakrishnan odd log-logistic generalized half-normal distribution by [26], the gamma-generalized inverse Weibull distribution by [29], the Ristić and Balakrishnan Lindley–Poisson distribution by [13], and Ristić–Balakrishnan extended exponential distribution by [16].

In this paper, we develop a novel family of distributions using the gamma generator method in conjunction with the Topp–Leone-G (TL-G) family of distributions [7] and the Gompertz-G (Gom-G) family of distributions [3]. The cdf and pdf of TL-G are given by

$$\begin{aligned} F_{TL-G}(x; b, \psi )=\left[ 1- {\overline{G}}^2(x;\psi )\right] ^b \end{aligned}$$

(3)

and

$$\begin{aligned} f_{TL-G}(x; b, \psi )=2 b\left[ 1- {\overline{G}}^2(x;\psi )\right] ^{b-1}{\overline{G}}(x;\psi ) g(x;\psi ), \end{aligned}$$

respectively, for $b>0$ and parameter vector $\psi$. Note that ${\overline{G}}(x;\psi )=1-G(x;\psi )$ is the baseline survival function (also called reliability function), and $g(x;\psi )=dG(x;\psi )/dx$ where $G(x;\psi )$ is the cdf of any continuous distribution. The cdf and pdf of the Gompertz-G (Gom-G) family of distributions are

$$\begin{aligned} F(x; \theta ,\gamma , \psi )= 1-\exp \Big \{\frac{\gamma }{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}, \end{aligned}$$

(4)

and

$$\begin{aligned} f(x; \theta , \gamma , \psi )=\gamma \left[ 1-{G}(x;\psi )\right] ^{-\theta -1}\exp \Big \{\frac{\gamma }{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}{g}(x;\psi ), \end{aligned}$$

where $\theta , \gamma >0$, and $\psi$ is the baseline parameter vector.

The objectives of developing this new family of distributions are as follows: to create new statistical distributions with more shape properties than previous distributions; to extend the possible shapes of the density and hazard rate functions of the baseline distribution; to skew any symmetrical distribution; and to modulate the weight of the tails of any parental distribution.

The remaining sections are organized as follows. In Sect. 2, we introduce the new RB–TL–Gom-G distribution family, its subfamilies, the hazard rate function, and the quantile function. Some special cases of the new family of distributions are described in Sect. 3. The expansion of density functions is demonstrated in Sect. 4. Section 5 derives the mathematical and statistical properties of the RB–TL–Gom-G family of distributions. In Sect. 6, the model parameters of the RB–TL–Gom-G family of distributions are estimated using the maximum likelihood approach, and in Sect. 7, convergence properties are validated using a simulation study. The flexibility and applicability of the RB–TL–Gom-G family of distributions are exemplified by three real-world data sets in Sect. 8. Section 9 concludes with some final remarks.

In this study, all simulations, parameter estimates, and model comparisons are conducted in R (an open-source software environment maintained by R Core Team and the R Foundation for Statistical Computing (https://www.r-project.org/).

2 The New Family of Distributions

Based on the TL-G and Gom-G families of distributions, we define a new family of distributions namely the Ristić–Balakrishnan–Topp–Leone–Gompertz-G (RB–TL–Gom-G) distribution.¹ The cdf of the RB–TL–Gom-G family of distributions is denoted by

$$\begin{aligned}&F_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi ) =1-\dfrac{1}{\varGamma (\delta )}\int _0^{ -\log \left( \left[ 1-\exp \Big \{\frac{2}{\theta }\left( 1-[1-G(x;\psi )]^{-\theta }\right) \Big \}\right] ^b \right) } t^{\delta -1}\textrm{e}^{-t}dt \nonumber \\&\quad =1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}, \end{aligned}$$

(5)

where $\gamma (\delta , y)= \int _{0}^y t^{\delta -1} e^{-t} dt$ is the lower incomplete gamma function, and $\varGamma (\delta )$ is the gamma function. Note that $g(x;\psi )=dG(x;\psi )/dx$ is the pdf of any continuous baseline distribution, and $G(x;\psi )$ is the corresponding cdf. The pdf of the RB–TL–Gom-G family of distributions is

$$\begin{aligned} f_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi )& = \frac{2b}{\varGamma (\delta )}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\nonumber \\ & \quad \times \left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ), \end{aligned}$$

(6)

for $\delta , b, \theta >0$ and baseline parameter vector $\psi .$

2.1 Sub-families

Several sub-families of the RB–TL–Gom-G family of distributions are presented in this sub-section.

• When $\delta =1,$ we obtain the Topp–Leone–Gompertz-G (TL-Gom-G) family of distributions with the cdf

  $$\begin{aligned} F(x; b, \theta , \psi )= \left[ 1-\exp \Big \{\frac{2}{\theta }\left( 1-[1-G(x;\psi )]^{-\theta }\right) \Big \}\right] ^b, \end{aligned}$$

  for $b, \theta >0$ and parameter vector $\psi .$

• Setting $b=1$, we find a new family of distributions namely Ristić–Balakrishnan–Gompertz-G (RB-Gom-G) with the cdf

  $$\begin{aligned} F(x; \delta , \theta , \psi )&=1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] \right) }{\varGamma (\delta )}, \end{aligned}$$

  for $\delta , \theta >0$ and parameter vector $\psi .$

• When $\theta =1$, a new family of distributions is generated with the cdf

  $$\begin{aligned} F(x; \delta , b, \psi )&=1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{{2}\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-1}\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}, \end{aligned}$$

  (7)

  for $\delta , b>0$ and parameter vector $\psi .$

• When $\delta =b=1,$ it becomes the Gompertz-G (Gom-G) family of distributions with the cdf

  $$\begin{aligned} F(x; \theta , \psi )= \left[ 1-\exp \Big \{\frac{2}{\theta }\left( 1-[1-G(x;\psi )]^{-\theta }\right) \Big \}\right] , \end{aligned}$$

  for $\theta >0$ and parameter vector $\psi .$

• When $\delta =\theta =1,$ another new family of distributions is obtained with the cdf

  $$\begin{aligned} F(x; b, \psi )= \left[ 1-\exp \Big \{{2}\left( 1-[1-G(x;\psi )]^{-1}\right) \Big \}\right] ^b, \end{aligned}$$

  for $b>0$ and parameter vector $\psi .$

• Let $b=\theta =1$, then we obtain one more new family of distributions with the cdf

  $$\begin{aligned} F(x; \delta , \psi )&=1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{{2}\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-1}\right) \Big \}\right] \right) }{\varGamma (\delta )}, \end{aligned}$$

  for $\delta >0$ and parameter vector $\psi .$

• As $\delta =b=\theta =1,$ we obtain a special case of the Gom-G family of distributions with the cdf

  $$\begin{aligned} F(x; \psi )= \left[ 1-\exp \Big \{{2}\left( 1-[1-G(x;\psi )]^{-1}\right) \Big \}\right] , \end{aligned}$$

  for parameter vector $\psi .$

2.2 Hazard Rate and Quantile Functions

For a continuous random variable X with cdf F(x), survival function ${\bar{F}}(x)=1-F(x)$, and pdf f(x), its hazard rate function (hrf), reverse hazard function (rhf), and mean residual life function are defined as $h_F(x) = f(x)/{\bar{F}}(x)$, $\tau _{F}(x)=f(x)/F(x)$, and $\displaystyle \delta _F(x)=\int _x^\infty \dfrac{{\bar{F}}(u)}{{\bar{F}}(x)} du$ respectively. It has been shown by [36] that the behaviors of $h_F(x)$, $\tau _{F}(x)$, and $\delta _F(x)$ are equivalent. Here we will only present the hazard rate function, but the reverse hazard and residual life functions can be obtained in a similar way. Given the cdf (Eq. (5)) and pdf (Eq. (6)), the hrf of the RB–TL–Gom-G family of distributions is

$$\begin{aligned} h_{_{RB-TL-Gom-G}}(x)& = 2b\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\nonumber \\ & \quad \times \dfrac{\left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi )}{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) }. \nonumber \end{aligned}$$

(8)

The quantile function of the RB–TL–Gom-G family of distributions can be obtained by solving for the inverse cumulative distribution function as

$$\begin{aligned} & F_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi )\\ & \quad =1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}=p, \end{aligned}$$

for $0\le p \le 1$, which is equivalent to

$$\begin{aligned} -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] =\dfrac{\gamma ^{-1}[\varGamma (\delta )(1-p),\delta ]}{b}, \end{aligned}$$

and

$$\begin{aligned} \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \} =1-{\displaystyle \textrm{e}^{-\gamma ^{-1}(\varGamma (\delta )(1-p),\delta )/b}}. \end{aligned}$$

Thus, it is sufficient to solve

$$\begin{aligned} {G}(x;\psi ) =1-\left[ 1-\frac{\theta }{2}\log \left( 1-{\displaystyle \textrm{e}^{-\gamma ^{-1}(\varGamma (\delta )(1-p),\delta )/b}}\right) \right] ^{-1/\theta }:=q. \end{aligned}$$

(9)

Therefore, the quantile function of the RB–TL–Gom-G family of distributions reduces to the quantile $x_q$ of the baseline distribution with cdf ${G}(x;\psi )$ which is given by

$$\begin{aligned} x_q=G^{-1}(q), \end{aligned}$$

(10)

where q is defined in Eq. (9).

3 Some Special Cases

Special cases of the RB–TL–Gom-G family of distributions are presented in this section when the baseline cdf $G(x; \psi )$ is specified as the Burr XII, Weibull, or Uniform distributions.

3.1 Ristić–Balakrishnan–Topp–Leone–Gompertz–Burr XII (RB–TL–Gom–BXII) Distribution

Suppose the cdf and pdf of the baseline distribution are given by $G(x;c, \lambda )=1-(1+x^{c})^{-\lambda }$ and $g(x;c, \lambda )=\lambda cx^{c-1}(1+x^{c})^{-\lambda -1}$ for $c, \lambda >0$ and $x>0.$ The cdf and pdf of the new RB–TL–Gom–BXII distribution are

$$\begin{aligned}&F_{_{RB-TL-Gom-BXII}}(x; \delta , b, \theta , c, \lambda )\nonumber \\&\quad =1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}, \end{aligned}$$

(11)

and

$$\begin{aligned}{} f_{_{RB-TL-Gom-BXII}}(x; \delta , b, \theta , c, \lambda )& =\frac{2b}{\varGamma (\delta )}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \} & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\nonumber \\ & \quad \times \left[ (1+x^{c})^{-\lambda }\right] ^{-\theta -1} \lambda cx^{c-1}(1+x^{c})^{-\lambda -1}, \end{aligned}$$

for $\delta , b, \theta , c, \lambda >0.$ The hrf of the RB–TL–Gom–BXII distribution is given by

$$\begin{aligned}&h_{_{RB-TL-Gom-BXII}}(x; \delta , b, \theta , c, \lambda )\\&\quad ={2b}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\\&\qquad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \\&\qquad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\\&\qquad \times \left[ (1+x^{c})^{-\lambda }\right] ^{-\theta -1} \lambda cx^{c-1}(1+x^{c})^{-\lambda -1}\\&\qquad \times \left( \gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ (1+x^{c})^{-\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) \right) ^{-1}, \end{aligned}$$

for $\delta , b, \theta , c, \lambda >0.$ If $\lambda =1,$ we obtain the RB–TL–Gom-Log-Logistic (RB–TL–Gom-LLoG) distribution. Also, if $c=1,$ the RB–TL–Gom–BXII distribution reduces to the RB–TL–Gom–Lomax (RB–TL–Gom–Lx) distribution.

Fig. 1

Density and hazard plots for RB–TL–Gom–BXII distribution

Figure 1 shows several typical configurations of the pdf and hrf of the RB–TL–Gom–BXII distribution. The pdf can take various shapes including almost symmetric, reverse-J, left-skewed, and right-skewed. Furthermore, plots of the hrf for the RB–TL–Gom–BXII distribution exhibit monotonically increasing, monotonically decreasing, bathtub, and upside-down bathtub shapes.

3.2 Ristić–Balakrishnan–Topp–Leone-Gompertz–Weibull (RB–TL–Gom–W) Distribution

If we consider the Weibull distribution with cdf and pdf given by $G(x;\lambda )=1-\exp (- x^{\lambda })$ and $g(x;\lambda )=\lambda x^{\lambda -1}\exp (- x^{\lambda })$ respectively, for $\lambda >0$ and $x>0$, as the baseline distribution, then the RB–TL–Gom–W distribution has cdf and pdf given by

$$\begin{aligned} F_{_{RB-TL-Gom-W}}(x; \delta , b, \theta , \lambda )&=1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}, \end{aligned}$$

and

$$\begin{aligned} f_{_{RB-TL-Gom-W}}(x; \delta , b, \theta , \lambda )& = \frac{2b}{\varGamma (\delta )}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\nonumber \\ & \quad \times \left[ \exp (- x^{\lambda })\right] ^{-\theta -1} \lambda x^{\lambda -1}\exp (- x^{\lambda }), \end{aligned}$$

for $\delta , b, \theta , \lambda >0.$ The corresponding hrf is given by

$$\begin{aligned} h_{_{RB-TL-Gom-W}}(x; \delta , b, \theta , \lambda )& = {2b}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\nonumber \\ & \quad \times \left[ \exp (- x^{\lambda })\right] ^{-\theta -1} \lambda x^{\lambda -1}\exp (- x^{\lambda })\nonumber \\ & \quad \times \left( \gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ \exp (- x^{\lambda })\right] ^{-\theta }\right) \Big \}\right] ^b \right) \right) ^{-1}, \end{aligned}$$

for $\delta , b, \theta , \lambda >0.$

Fig. 2

Density and hazard plots for RB–TL–Gom–W distribution

Figure 2 illustrates the adaptability of the RB–TL–Gom-W pdf and hrf distributions. The pdf produces a variety of forms, such as unimodal, reverse-J, left-skewed, and right-skewed. In addition, hrf plots for the RB–TL–Gom–W distribution exhibit growing, decreasing, bathtub, and inverted bathtub forms.

3.3 Ristić–Balakrishnan–Topp–Leone–Gompertz-Uniform (RB–TL–Gom-U) Distribution

If we consider the uniform distribution as the baseline distribution with cdf and pdf $G(x; \lambda )=\frac{x}{\lambda }$ and $g(x; \lambda )=\frac{1}{\lambda }$, for $\lambda >0$ and $0<\lambda <x$, then the RB–TL–Gom-U distribution has the cdf and pdf given by

$$\begin{aligned} F_{_{RB-TL-Gom-U}}(x; \delta , b, \theta , \lambda )&=1- \frac{\gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) }{\varGamma (\delta )}, \end{aligned}$$

and a pdf of

$$\begin{aligned} f_{_{RB-TL-Gom-U}}(x; \delta , b, \theta , \lambda )& = \frac{2b}{\varGamma (\delta )}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\left[ 1-\frac{x}{\lambda }\right] ^{-\theta -1} \frac{1}{\lambda }, \end{aligned}$$

for $\delta , b, \theta , \lambda >0.$ The hrf for the RB–TL–Gom-U distribution is

$$\begin{aligned} h_{_{RB-TL-Gom-U}}(x; \delta , b, \theta , \lambda )& = \frac{2b}{\lambda }\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1}\nonumber \\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \nonumber \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\left[ 1-\frac{x}{\lambda }\right] ^{-\theta -1} \nonumber \\ & \quad \times \left( \gamma \left( \delta ,-\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-\frac{x}{\lambda }\right] ^{-\theta }\right) \Big \}\right] ^b \right) \right) ^{-1}, \end{aligned}$$

for $\delta , b, \theta , \lambda >0.$

Figure 3 presents the pdf and hrf of the RB–TL–Gom-U distribution. The pdf can take numerous shapes such as almost symmetric, J, reverse-J, left-skewed, and right-skewed. Furthermore, the hrf of the RB–TL–Gom-U distribution captures a variety of possibilities including decreasing, bathtub, and bathtub followed upside-down bathtub shapes.

Fig. 3

Density and hazard plots for RB–TL–Gom-U distribution

4 Expansion of the Density Function

In this section, we will present a series expansion of the RB–TL–Gom-G density function. First, let $\displaystyle y=\exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}$, and note that $\displaystyle -\log (1-y)=\sum \nolimits _{i=0}^\infty \dfrac{y^{i+1}}{i+1}$. Then

$$\begin{aligned}&\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1} \nonumber \\&\quad =b^{\delta -1}y^{\delta -1}\left[ \sum _{m=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) y^m\left( \sum _{s=0}^\infty \dfrac{y^s}{s+2} \right) ^m\right] . \end{aligned}$$

(12)

Moreover, let $a_s=(s+2)^{-1}$, then $\displaystyle \left( \sum \nolimits _{s=0}^\infty \dfrac{y^s}{s+2} \right) ^m=\left( \sum \nolimits _{s=0}^\infty a_sy^s \right) ^m=\sum \nolimits _{s=0}^\infty b_{s,m}y^s,$ where $\displaystyle b_{s,m}=(sa_0)^{-1}\sum \nolimits _{l=1}^s[m(l+1)-s]a_lb_{s-l,m}$, and $b_{0,m}=a_0^m$ [17, 32]. Then Eq. (12) can be simplified as

$$\begin{aligned}&\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta -1} \nonumber \\&\quad =b^{\delta -1}y^{\delta -1}\left[ \sum _{m=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) y^m \sum _{s=0}^\infty b_{s,m}y^s\right] \nonumber \\&\quad = b^{\delta -1}\left[ \sum _{m,s=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) b_{s,m} y^{m+s+\delta -1} \right] . \end{aligned}$$

(13)

Now, the pdf in Eq. (6) can be written as

$$\begin{aligned} f_{_{RB-TL-Gom-G}}(x;\delta , b, \theta , \psi )&=\frac{2b^\delta }{\varGamma (\delta )}\left[ \sum _{m,s=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) b_{s,m} y^{m+s+\delta -1}\right] \nonumber \\&\qquad \times (1- y)^{b-1}y\left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ) \nonumber \\&\quad =\frac{2b^\delta }{\varGamma (\delta )}\left[ \sum _{m,s=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) b_{s,m} y^{m+s+\delta }\right] \nonumber \\&\qquad \times \sum _{n=0}^\infty \left( {\begin{array}{c}b-1\\ n\end{array}}\right) (-1)^n y^n \left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ) \nonumber \\&\quad =\frac{2b^\delta }{\varGamma (\delta )}\left[ \sum _{m,s,n=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) \left( {\begin{array}{c}b-1\\ n\end{array}}\right) (-1)^n b_{s,m} y^{m+s+\delta +n}\right] \nonumber \\&\qquad \times \left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ). \end{aligned}$$

(14)

Note that $\displaystyle y=\exp \left( \frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \right)$ and

$$\begin{aligned} y^{m+s+\delta +n}&=\left[ \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{m+s+\delta +n} \nonumber \\&=\sum _{p=0}^\infty \dfrac{2^p(m+s+\delta +n)^p}{\theta ^p p!} \sum _{q=0}^\infty \left( {\begin{array}{c}p\\ q\end{array}}\right) (-1)^q \left[ 1-{G}(x;\psi )\right] ^{-\theta q} \nonumber \\&=\sum _{p,q=0}^\infty \left( {\begin{array}{c}p\\ q\end{array}}\right) \dfrac{2^p(m+s+\delta +n)^p}{\theta ^p p!} (-1)^q \left[ 1-{G}(x;\psi )\right] ^{-\theta q}. \end{aligned}$$

(15)

Substitute Eq. (15) into Eq. (14), and the pdf of RB–TL–Gom-G distribution can be written as

$$\begin{aligned}&f_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi )\nonumber \\&\quad = \frac{2b^\delta }{\varGamma (\delta )} \sum _{m,s,n,p,q=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) \left( {\begin{array}{c}b-1\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \nonumber \\&\qquad \times \dfrac{2^p(m+s+\delta +n)^p}{\theta ^p p!} (-1)^{n+q} b_{s,m} \left[ 1-{G}(x;\psi )\right] ^{-\theta (q+1)-1} {g}(x;\psi ) \nonumber \\&\quad = \frac{2b^\delta }{\varGamma (\delta )} \sum _{m,s,n,p,q,u=0}^\infty \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) \left( {\begin{array}{c}b-1\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \left( {\begin{array}{c}-\theta (q+1)-1\\ u\end{array}}\right) \nonumber \\&\qquad \times \dfrac{2^p(m+s+\delta +n)^p}{\theta ^p p!} (-1)^{n+q+u} b_{s,m} G^u(x;\psi ) {g}(x;\psi ). \end{aligned}$$

(16)

To simplify the notation, we let

$$\begin{aligned} \displaystyle w_{u+1}=&\sum _{m,s,n,p,q=0}^\infty \frac{2b^\delta }{\varGamma (\delta )} \left( {\begin{array}{c}\delta -1\\ m\end{array}}\right) \left( {\begin{array}{c}b-1\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \left( {\begin{array}{c}-\theta (q+1)-1\\ u\end{array}}\right) \nonumber \\&\times \dfrac{2^p(m+s+\delta +n)^p}{(u+1)\theta ^p p!} (-1)^{n+q+u} b_{s,m} \end{aligned}$$

(17)

be the weights. Then the RB–TL–Gom-G density function can be written as

$$\begin{aligned} f_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi )&= \sum _{u=0} ^\infty w_{u+1}(u+1)G^u(x;\psi ) {g}(x;\psi ) \end{aligned}$$

(18)

$$\begin{aligned}&=\sum _{u=0} ^\infty w_{u+1}{g_{_{u+1}}}(x;\psi ), \end{aligned}$$

(19)

where ${g_{_{u+1}}}(x;\psi )$ is the pdf of the exponentiated-G (Exp-G) distribution [20] with power parameter $u+1$.

Therefore, we can obtain the statistical properties of the RB–TL–Gom-G family of distributions from the well-established properties of the Exp-G distributions.

5 Mathematical and Statistical Properties

The moments, moment generating function, incomplete and conditional moments, Reńyi entropy, order statistics, stochastic orderings, and probability weighted moments of the RB–TL–Gom-G family of distributions are presented in this section. Throughout this section X denotes a random variable with a density function $f_{_{RB-TL-Gom-G}}(x)$ as in Eq. (6), and $Y_{u+1}$ is a random variable with the exponentiated-G distribution with power parameter $u+1$.

5.1 Moments and Generating Functions

The $r^{th}$ moment of the RB–TL–Gom-G family of distributions can be derived as

$$\begin{aligned} E(X^r)&=\int _{-\infty }^\infty x^rf_{_{RB-TL-Gom-G}}(x) dx =\sum _{u=0}^\infty w_{u+1} E(Y^r_{u+1}), \end{aligned}$$

(20)

where $E(Y^r_{u+1})$ is the $r^{th}$ moment of the Exp-G distribution with power parameter $u+1$, and $w_{u+1}$ is given in Eq. (17). Furthermore, the moment generating function (mgf) for $s<1$ is

$$\begin{aligned} M_X(s)=\sum _{u=0}^\infty w_{u+1} M_{u+1}(s), \end{aligned}$$

(21)

where $M_{u+1}(s)$ is the mgf of $Y_{u+1}$, and $w_{u+1}$ is defined in Eq. (17).

5.2 Incomplete and Conditional Moments

Incomplete and conditional moments are widely used in lifetime models and measures of inequality such as Bonferroni and Lorenz curves. The rth incomplete moment of X can be obtained as

$$\begin{aligned} \phi _r(z)=\int _{-\infty }^z x^rf_{_{RB-TL-Gom-G}}(x) dx =\sum _{u=0}^\infty w_{u+1} \int _{-\infty }^z x^rg_{u+1}(x;\psi )dx, \end{aligned}$$

(22)

where $g_{u+1}(x;\psi )$ is the pdf of Exp-G distribution with power parameter $u+1$. By setting $r=1$ in Eq. (22), we obtain the first incomplete moment of the RB–TL–Gom-G family of distributions. The rth conditional moments of the RB–TL–Gom-G family of distributions is given by

$$\begin{aligned} E(X^r|X\ge a)&=\frac{1}{{\bar{F}}_{_{RB-TL-Gom-G}}(a; \delta ,b, \theta , \psi )}\int _{t}^\infty x^rf_{_{RB-TL-Gom-G}}(x; \delta , b, \theta , \psi ) dx \nonumber \\&=\frac{1}{{\bar{F}}_{_{RB-TL-Gom-G}}(a; \delta ,b, \theta , \psi )}\sum _{u=0}^\infty w_{u+1} E(Y^r_{u+1}I_{Y^r_{u+1}\ge t}), \end{aligned}$$

(23)

where ${\bar{F}}_{_{RB-TL-Gom-G}}(a; \delta ,b, \theta , \psi )=1-F_{_{RB-TL-Gom-G}}(a; \delta ,b, \theta , \psi )$ and

$$\begin{aligned} E(Y^r_{u+1}I_{Y^r_{u+1}\ge t})&=\int _{t}^\infty y^r{g_{_{u+1}}}(x;\psi ) dy \nonumber \\&=(u+1)\int _{G(z;\psi )}^1[Q_G(z;\psi )]^rz^{u+1}dz, \end{aligned}$$

(24)

for parameter vector $\psi$. The mean residual life function is given by $E(X-a|X>a)=E(X|X>a)-a=V_F(a)-a$, where $V_F(a)$ is referred to as the vitality function of the distribution function F. The mean deviations, Bonferroni, and Lorenz curves can be readily obtained from the conditional and incomplete moments.

5.3 Reńyi Entropy

In information theory, the Reńyi entropy [34] is a measurement that generalizes various related entropies including Hartley entropy, Shannon entropy, collision entropy, and min-entropy. Reńyi entropy plays an important role as an index of diversity in fields such as ecology and statistics. The Reńyi entropy of the RB–TL–Gom-G family of distributions is given by

$$\begin{aligned} I_R(v)=\frac{1}{1-v}\log \left( \int _0^\infty f^v_{_{RB-TL-Gom-G}}(x) dx\right) , v>0 \text { and } v\ne 1, \end{aligned}$$

(25)

where

$$\begin{aligned} f^v_{_{RB-TL-Gom-G}}(x)&= \left( \frac{2b}{\varGamma (\delta )}\right) ^v \left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{v(\delta -1)} \nonumber \\&\quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{v(b-1)} \nonumber \\&\quad \times \exp \Big \{\frac{2v}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\nonumber \\&\quad \times \left[ 1-{G}(x;\psi )\right] ^{-v\theta -v} {g}^v(x;\psi ). \end{aligned}$$

(26)

Let $\displaystyle y=\exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}$. Consider a similar expansion for the pdf, as in Eq. (13), we can obtain that

$$\begin{aligned}&\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{v(\delta -1)} \nonumber \\&\quad = b^{v\delta -v}\left[ \sum _{m,s=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) b_{s,m} y^{m+s+v\delta -v} \right] . \end{aligned}$$

(27)

Therefore, Eq. (26) can be written as

$$\begin{aligned} f^v_{_{RB-TL-Gom-G}}(x)&= \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )} \left[ \sum _{m,s=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) b_{s,m} y^{m+s+v\delta -v} \right] \nonumber \\&\quad \times ( 1- y)^{v(b-1)}y^v\left[ 1-{G}(x;\psi )\right] ^{-v\theta -v} {g}^v(x;\psi ) \nonumber \\&= \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )} \left[ \sum _{m,s,n=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) (-1)^n b_{s,m} y^{m+s+v\delta +n} \right] \nonumber \\&\quad \times \left[ 1-{G}(x;\psi )\right] ^{-v\theta -v} {g}^v(x;\psi ). \end{aligned}$$

(28)

Note that with $\displaystyle y=\exp \left( \frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \right)$, then

$$\begin{aligned} y^{m+s+v\delta +n}&=\exp \left( \frac{2(m+s+v\delta +n)}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \right) \nonumber \\&=\sum _{p=0}^\infty \frac{2^p(m+s+v\delta +n)^p}{\theta ^p}\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) ^p \nonumber \\&=\sum _{p,q=0}^\infty \left( {\begin{array}{c}p\\ q\end{array}}\right) \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!} (-1)^q \left[ 1-{G}(x;\psi )\right] ^{-\theta q}. \end{aligned}$$

(29)

It follows that

$$\begin{aligned} f^v_{_{RB-TL-Gom-G}}(x)&= \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )}\sum _{m,s,n,p,q=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!}\\&\quad \times (-1)^{n+q} b_{s,m} \left[ 1-{G}(x;\psi )\right] ^{-\theta (v+q)-v} {g}^v(x;\psi ) \\&= \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )}\sum _{m,s,n,p,q,u=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \left( {\begin{array}{c}-\theta (v+q)-v\\ u\end{array}}\right) \\&\quad \times \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!} (-1)^{n+q+u} b_{s,m} G^u(x;\psi ) {g}^v(x;\psi ). \end{aligned}$$

Thus, the Reńyi entropy of the RB–TL–Gom-G family of distributions can be written as

$$\begin{aligned} I_R(v)&=\frac{1}{1-v}\log \left( \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )}\sum _{m,s,n,p,q,u=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \left( {\begin{array}{c}-\theta (v+q)-v\\ u\end{array}}\right) \right. \nonumber \\&\quad \times \left. \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!} (-1)^{n+q+u} b_{s,m} \int _0^\infty G^u(x;\psi ) {g}^v(x;\psi ) dx\right) \nonumber \\&=\frac{1}{1-v}\log \left( \frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )}\sum _{m,s,n,p,q=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!} \right. \nonumber \\&\quad \times \sum _{u=0}^\infty \left( {\begin{array}{c}-\theta (v+q)-v\\ u\end{array}}\right) (-1)^{n+q+u} b_{s,m} \dfrac{1}{(1+u/v)^v} \nonumber \\&\quad \times \left. \int _0^\infty \left[ \left( 1+u/v\right) G^{u/v}(x;\psi ) {g}(x;\psi )\right] ^v dx \right) \nonumber \\&=(1-v)^{-1}\log \left( \sum _{u=0}^\infty C_u \textrm{e}^{(1-v)I_{REG}}\right) , \end{aligned}$$

(30)

where $\displaystyle I_{REG}=(1-v)^{-1} \log \left[ \int _0^{\infty }\left( \left( 1+u/v\right) G^{u/v}(x;\psi ) {g}(x;\psi )\right) ^v dx \right]$ is the Reńyi entropy of exponentiated-G distribution with power parameter $\displaystyle 1+u/v$, and the weights are

$$\begin{aligned} c_u&=\frac{2^vb^{v\delta }}{\displaystyle \varGamma ^v(\delta )}\sum _{m,s,n,p,q=0}^\infty \left( {\begin{array}{c}v\delta -v\\ m\end{array}}\right) \left( {\begin{array}{c}vb-v\\ n\end{array}}\right) \left( {\begin{array}{c}p\\ q\end{array}}\right) \dfrac{2^p(m+s+v\delta +n)^p}{\theta ^p p!} \nonumber \\&\quad \times \left( {\begin{array}{c}-\theta (v+q)-v\\ u\end{array}}\right) (-1)^{n+q+u} b_{s,m} \dfrac{1}{(1+u/v)^v}. \end{aligned}$$

(31)

As a result, we can obtain the Reńyi entropy of the RB–TL–Gom-G family of distributions from that of the exponentiated-G distribution by Eq. (30).

5.4 Order Statistics

Order statistics have a wide range of applications such as in actuarial science, modeling auctions, optimizing production processes, and estimating parameters of distributions. Let $X_1, X_2, \ldots , X_n$ be independent and identically distributed RB–TL–Gom-G random variables, f(x) and F(x) be the pdf and cdf of the RB–TL–Gom-G family of distributions, and denote ${\bar{F}}(x)=1-F(x)$. The pdf of the ith order statistic, $f_{_{i:n}}(x)$, can be expressed as

$$\begin{aligned} f_{_{i:n}}(x)&=\frac{n!f(x)}{(i-1)!(n-i)!}\left[ F(x)\right] ^{i-1}\left[ 1-F(x)\right] ^{n-i} \nonumber \\&=\frac{n!f(x)}{(i-1)!(n-i)!}\sum _{m=0}^{n-i}\left( {\begin{array}{c}n-i\\ m\end{array}}\right) (-1)^m\left[ F(x)\right] ^{i-1+m} \nonumber \\&=\frac{n!}{(i-1)!(n-i)!}\sum _{m=0}^{n-i}\left( {\begin{array}{c}n-i\\ m\end{array}}\right) (-1)^m \left[ \sum _{u=0} ^\infty w_{u+1}(u+1)G^u(x;\psi ) {g}(x;\psi ) \right] \nonumber \\&\quad \times \left[ \sum _{u=0} ^\infty w_{u+1} G^{u+1}(x;\psi ) \right] ^{i+m-1}. \end{aligned}$$

(32)

Here we can write

$$\begin{aligned} \left[ \sum _{u=0} ^\infty w_{u+1} G^{u+1}(x;\psi )\right] ^{i+m-1}=\sum _{u=0} ^\infty d_{u,i+m-1} G^{u+1}(x;\psi ), \end{aligned}$$

(33)

where $\displaystyle d_{0,i+m-1}=(w_1)^{i+m-1}$,

$$\begin{aligned} \displaystyle d_{u,i+m-1}=[(u+1)w_1]^{-1}\sum _{z=1}^u[z(i+m)-u-1]w_{z+1} d_{u-z,i+m-1}, \end{aligned}$$

for $u=1,2,\ldots$, and $w_{_{u+1}}$ is given in Eq. (17). Then

$$\begin{aligned} f_{_{k:n}}(x)&= \sum _{m=0}^{n-i} \sum _{u,r=0} ^\infty s_{m,u,r} g_{_{u+r+1}}(x;\psi ), \end{aligned}$$

(34)

where

$$\begin{aligned} s_{m,u,r}=\frac{n!w_{u+1} d_{u,i+m-1}(u+1)}{(i-1)!(n-i)!(u+r+1)} (-1)^m \left( {\begin{array}{c}n-i\\ m\end{array}}\right) , \end{aligned}$$

(35)

and $g_{_{u+r+1}}(x;\psi )$ is the exponentiated-G distribution with parameter $u+r+1$. Therefore, we can derive properties of the distribution of the order statistics from the RB–TL–Gom-G family of distributions from those of the exponentiated-G distribution.

5.5 Stochastic Orderings

In this subsection, we present the commonly used stochastic orders for the RB–TL–Gom-G family of distributions including stochastic order, hazard rate order and the likelihood ratio order [37].

Let $F_X(t)$ and $F_Y(t)$ be the cdfs of two random variables X and Y, and define ${\overline{F}}_X(t)=1-F_X(t)$ and ${\overline{F}}_Y(t)=1-F_Y(t)$ as the corresponding reliability or survival functions. The random variable X is stochastically smaller than Y if, for all t, $\displaystyle {\overline{F}}_X(t)\le {\overline{F}}_Y(t)$ (or $\displaystyle {F}_X(t)\ge {F}_Y(t)$). It is represented by $\displaystyle X<_{st}Y$ or ${\displaystyle X\preceq Y}$. Moreover, if ${\displaystyle {\overline{F}}_X(t)< {\overline{F}}_Y(t)}$ for some t, then X is stochastically strictly less than Y and denoted as ${\displaystyle X\prec Y}$. In the hazard rate order given by $\displaystyle X\preceq _{hr}Y$ X, $h_X(t)\ge h_Y(t)$ for all t. Similarly, X is said to be smaller than Y in the likelihood ratio order denoted by $\displaystyle X\preceq _{lr}Y$ if $\displaystyle \frac{f_X(t)}{f_Y(t)}$ is decreasing in t. It has been shown that $\displaystyle \displaystyle X\preceq _{lr}Y \implies X\preceq _{hr}Y \implies X\preceq Y$ [37].

Now consider two independent random variables $X_1$ and $X_2$ following RB–TL–Gom-G family of distributions with $X_1\sim F_1(x; \delta _1,b, \theta , \psi )$ and $X_2\sim F_2(x; \delta _2,b,\theta , \psi )$ and their pdfs given by

$$\begin{aligned} f_{1}(x)& = \frac{2b}{\varGamma (\delta _1)}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta _1-1}\\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\\ & \quad \times \left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ), \end{aligned}$$

and

$$\begin{aligned} f_{2}(x)& = \frac{2b}{\varGamma (\delta _2)}\left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta _2-1}\\ & \quad \times \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^{b-1} \\ & \quad \times \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\\ & \quad \times \left[ 1-{G}(x;\psi )\right] ^{-\theta -1} {g}(x;\psi ) \end{aligned}$$

respectively. Then

$$\begin{aligned} \dfrac{f_1(x)}{f_2(x)}=\frac{\varGamma (\delta _2)}{\varGamma (\delta _1)} \left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) ^{\delta _1-\delta _2}. \end{aligned}$$

(36)

Differentiating Eq. (36) with respect to x yields

$$\begin{aligned} \frac{d}{dx}\left( \dfrac{f_1(x)}{f_2(x)}\right)&=\frac{\varGamma (\delta _2)}{\varGamma (\delta _1)} \left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}\right] \right) ^{\delta _1-\delta _2-1} \nonumber \\&\quad \times (\delta _1-\delta _2)b^{\delta _1-\delta _2} \frac{\exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}}{ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x;\psi )\right] ^{-\theta }\right) \Big \}}\nonumber \\&\quad \times 2\left[ 1-{G}(x;\psi )\right] ^{\theta -1}g(x;\psi ), \end{aligned}$$

(37)

which is negative if $\delta _1<\delta _2$. Therefore, the likelihood ratio order $\displaystyle X\preceq _{lr}Y$ exists, and we can conclude that the random variables $X_1$ and $X_2$ are stochastically ordered.

5.6 Probability Weighted Moments

Distributions may be characterized by the probability weighted moments (PWMs) [18], defined as

$$\begin{aligned} M_{i,j,k}=E\left( X^iF^j(X)\left[ 1-F(X)\right] ^{k}\right) =\int _{-\infty }^\infty x^iF^j(x)\left[ 1-F(x)\right] ^{k} f(x)\,dx, \end{aligned}$$

where the random variable X is distributed as $\displaystyle F \equiv F(x)=P(X\le x)$, i, j and k are real numbers. When j and k are nonnegative integers, then the probability weighted moment of order (i, j, k) is proportional to the ith moment about the origin of the $(j + 1)$th order statistic for a sample of size $n=k + j + 1$. If X is a random variable that follows the RB–TL–Gom-G family of distributions, then

$$\begin{aligned}&F^j(x)\left[ 1-F(x)\right] ^{k} f(x)= \sum _{l=0}^k \left( {\begin{array}{c}k\\ l\end{array}}\right) (-1)^l\left[ F(x)\right] ^{j+l}f(x) \nonumber \\&\quad = \sum _{l=0}^k \left( {\begin{array}{c}k\\ l\end{array}}\right) (-1)^l\left[ \sum _{u=0} ^\infty w_{u+1} G^{u+1}(x;\psi ) \right] ^{j+l} \left[ \sum _{u=0} ^\infty w_{u+1}(u+1)G^u(x;\psi ) {g}(x;\psi ) \right] , \end{aligned}$$

(38)

and apply a similar expansion in Eq. (33),

$$\begin{aligned}&F^j(x)\left[ 1-F(x)\right] ^{k} \nonumber \\&\quad = \sum _{l=0}^k \left( {\begin{array}{c}k\\ l\end{array}}\right) (-1)^l \left[ \sum _{u=0} ^\infty d_{u,j+l} G^{u+1}(x;\psi )\right] \left[ \sum _{u=0} ^\infty w_{u+1}(u+1)G^u(x;\psi ) {g}(x;\psi ) \right] \nonumber \\&\quad = \sum _{l=0}^{k} \sum _{u,r=0} ^\infty s_{l,u,r} g_{_{u+r+1}}(x;\psi ), \end{aligned}$$

(39)

where

$$\begin{aligned} s_{m,u,r}= \frac{w_{u+1}(u+1)d_{u,j+l}}{u+r+1}\left( {\begin{array}{c}k\\ l\end{array}}\right) (-1)^l, \end{aligned}$$

(40)

$\displaystyle d_{0,j+l}=(w_1)^{j+l}$,

$$\begin{aligned} \displaystyle d_{u,j+l}=[(u+1)w_1]^{-1}\sum _{z=1}^u[z(j+l+1)-u-1]w_{z+1} d_{u-z,j+l}, \end{aligned}$$

for $u=1,2,\ldots$, and $w_{_{u+1}}$ is given in Eq. (17). As a result, the PWMs of the RB–TL–Gom-G family of distributions can be written as

$$\begin{aligned} M_{i,j,k}&=\sum _{l=0}^{k} \sum _{u,r=0} ^\infty s_{l,u,r}\int _{-\infty }^\infty x^i g_{_{u+r+1}}(x;\psi )\, dx, \end{aligned}$$

(41)

i.e., the PWMs of the RB–TL–Gom-G family of distributions can be obtained from the moments of the exponentiated-G distribution.

6 Maximum Likelihood Estimation

In this section, we estimate the unknown parameters of the RB–TL–Gom-G family of distributions using the maximum likelihood estimation (MLE) method. Assuming that the independent random sample $\left( X_{1}, X_{2}, \ldots ,X_{n} \right)$ is observed from the RB–TL–Gom-G family of distributions with the vector of model parameters ${\mathbf {\Delta }}=( \delta , b, \theta , \psi )^{T}$. Then, the log-likelihood function $\ell _{n}=\ell _{n}(\mathbf{{\Delta }})$ for the parameters from the observed values has the form

$$\begin{aligned} \ell _{n}(\mathbf{{\Delta }})& = n\ln (2b)+(b-1)\sum _{i=1}^{n} \log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x_{i};\psi )\right] ^{-\theta }\right) \Big \}\right] \nonumber \\ & \quad +({\delta -1})\sum _{i=1}^{n} \log \left( -\log \left[ 1- \exp \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x_{i};\psi )\right] ^{-\theta }\right) \Big \}\right] ^b \right) \nonumber \\ & \quad +\sum _{i=1}^{n} \Big \{\frac{2}{\theta }\left( 1-\left[ 1-{G}(x_{i};\psi )\right] ^{-\theta }\right) \Big \}+({-\theta -1})\sum _{i=1}^{n} \log \left[ 1-{G}(x_{i};\psi )\right] \nonumber \\ & \quad -n\ln (\varGamma (\delta ))+\sum _{i=1}^{n} \log ( {g}(x_{i};\psi )). \end{aligned}$$

(42)

To obtain the maximum likelihood estimates (MLEs), we maximize the log-likelihood function $\ell _{n}(\mathbf{{\Delta }})$ numerically. This can be obtained by setting the nonlinear system of equations $(\frac{\partial \ell _{n}}{\partial \delta },\frac{\partial \ell _{n}}{\partial b }, \frac{\partial \ell _{n}}{\partial \theta }, \frac{\partial \ell _{n}}{\partial \psi _{k}})^{T}={{\textbf{0}}}$ and solving them simultaneously. However, since these equations are not in closed form, the MLEs can be found by maximizing $\ell _{n}(\mathbf{{\Delta }})$ numerically with respect to the parameters using a numerical method such as Newton–Raphson procedure. The partial derivatives of the log-likelihood function with respect to each component of the parameter vector are presented in the Supplementary Information.

7 Monte Carlo Simulations

This section is devoted to assessing the asymptotic convergence property of the estimated parameters of the RB–TL–Gom-LLoG distribution. A Monte Carlo simulation study was performed based on the following: N = 1000 samples of size n = 25, 50, 100, 200, 400, 800, 1600 generated from the RB–TL–Gom-LLoG distribution with different parameter values. The assessment was done based on the average bias (ABIAS) and root mean squared errors (RMSEs).

The ABIAS and RMSE for the estimated parameter, ${\hat{\theta }},$ are given by:

$$\begin{aligned} ABIAS({{\hat{\theta }}})=\frac{\sum _{i=1}^{N}{\hat{\theta _i}}}{N}-\theta \quad \text {and} \quad RMSE({{\hat{\theta }}})=\sqrt{\frac{\sum _{i=1}^{N}({\hat{\theta _i}}-\theta )^2}{N}}, \end{aligned}$$

respectively.

The results of the simulation are summarized in Tables 1 and 2. These Tables report the average estimates (mean), average bias (ABIAS), and root mean squared errors (RMSEs). These results indicate the convergence of the mean estimations to the true parameters as the sample size (n) increases. Moreover, the RMSEs and ABIAS converge to zero as n increases further validating that the estimates are robust and consistent.

Table 1 Monte Carlo simulation results

Table 2 Monte Carlo simulation results

8 Applications

In this section we illustrate the flexibility and functionality of the RB–TL–Gom-G family of distributions by fitting its special case, the RB–TL–Gom-LLoG distribution, to several real-life data sets. The computations in the applications are based on the definitions in Eqs. (5) and (6). Parameters are estimated using the maximum likelihood estimation method, with the box-constrained optimization using PORT routines [15]. Moreover, we apply the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm, a quasi-Newton method, to find the best-fit parameter sets and their corresponding goodness-of-fit. The performance of the new distribution is compared to that of other recent models including: Topp-Leone odd Burr III Log-logistic (TL-OBIII-LLoG) by [25], generalized Weibull log-logistic (GWLLoG) distribution by [12], a new gamma exponentiated Lindley-log-logistic (GELLoG) distribution by [24], heavy-tailed log-logistic distribution, named alpha power log-logistic distribution by [38], and alpha power Topp–Leone Weibull (APTLW) distribution by [9]. The pdf’s of the models of comparison are given in the Supplementary Information.

To assess the goodness-of-fit of all the fitted distributions, the well-known goodness-of-fit statistics such as −2log-likelihood statistic ($-2\ln (L)$), Akaike Information Criterion ($AIC=2p-2\ln (L)$), Consistent Akaike Information Criterion ($CAIC=AIC+2\frac{p(p+1)}{n-p-1}$), Bayesian Information Criterion ($BIC=p\ln (n)-2\ln (L)$), (n is the number of observations, and p is the number of estimated parameters), Cramér-von Mises ($W^{*}$) statistic, Anderson-Darling statistic ($A^{*}$), Kolmogorov-Smirnov (K-S) statistic and its $p-$value were performed. It is known that, except for the $p-$value of the K-S statistic, the smaller the values of all the goodness-of-fit statistics, the better the model for fitting the data set.

8.1 Survival Times of Guinea Pigs Data

This data set consists of the survival times of guinea pigs injected with tubercle bacilli and was analyzed by [23] and [19]. The observations are as follows:

12, 15, 22, 24, 24, 32, 32, 33, 34, 38, 38, 43, 44, 48, 52, 53, 54, 54, 55, 56, 57, 58, 58, 59, 60, 60, 60, 60, 61, 62, 63, 65, 65, 67, 68, 70, 70, 72, 73, 75, 76, 76, 81, 83, 84, 85, 87, 91, 95, 96, 98, 99, 109, 110, 121, 127, 129, 131, 143, 146, 146, 175, 175, 211, 233, 258, 258, 263, 297, 341, 341, 376.

Table 3 MLEs and goodness-of-fit statistics for guinea pigs data

Estimates of the parameters of the fitted distributions (standard error in parentheses), AIC, BIC, W$^*$, and A$^*$ are given in Table 3. Plots of the fitted densities along with the histogram and the observed probability vs predicted probability are presented in Fig. 4. We can conclude that the RB-TL-Gom-LLoG distribution is significantly better than other distributions since the values of all goodness-of-fit statistics are the smallest, and the p-value of the K-S statistic is the largest.

Fig. 4

Fitted density superposed on the histogram (left) and observed probability vs expected probability plots (right) for the Guinea Pigs Data

Fig. 5

Fitted Kaplan–Meier survival curve, theoretical and empirical cumulative distribution functions, the total time on test statistics, and the hazard rate function for the Guinea Pigs Data

Graphical representations such as Kaplan–Meier (K–M) survival curve, theoretical and empirical cumulative distribution functions (ECDF), and total time on test (TTT) scaled are plotted in Fig. 5. It is clear that the fitted empirical and theoretical plots are close to each other, hence we conclude that our model provides a very good fit for the data. Moreover, the TTT scaled plot clearly demonstrates that the model fits a non-monotonic hazard rate structure.

8.2 Active Repair Times Data

The data set was reported by [22], and it represents active repair times for airborne communication transceivers. The observations are

0.50, 0.60, 0.60, 0.70, 0.70, 0.70, 0.80, 0.80, 1.00, 1.00, 1.00, 1.00, 1.10, 1.30, 1.50, 1.50, 1.50, 1.50, 2.00, 2.00, 2.20, 2.50, 2.70, 3.00, 3.00, 3.30, 4.00, 4.00, 4.50, 4.70, 5.00, 5.40, 5.40, 7.00, 7.50, 8.80, 9.00, 10.20, 22.00, 24.50.

Table 4 MLEs and goodness-of-fit statistics for active repair times data

Table 4 reports the maximum likelihood estimates (MLEs) of the parameters of all fitted models, as well as the standard errors (in parenthesis) and various goodness-of-fit statistics for the active repair times data. We can see that the RB–TL–Gom-LLoG distribution is the best model for describing the data set since it has the smallest values among all goodness-of-fit statistics and the largest p-value of the K–S statistic. Plots of the fitted densities and the histogram, observed probability against predicted probability are given in Fig. 6.

Fig. 6

Fitted density vs histogram (left) and observed probability vs expected probability plots (right) for the Active Repair Times Data

Fig. 7

Fitted Kaplan–Meier survival curve, theoretical and empirical cumulative distribution functions, the total time on test statistics, and the hazard rate function for the Active Repair Times Data

Moreover, the fitted Kaplan–Meier survival curves, theoretical and ECDF plots, and TTT scaled plot are presented in Fig. 7. From the Kaplan-Meier and ECDF plots, it is clear that the RB–TL–Gom-LLoG distribution provides a perfect description of the active repair times data. The TTT scaled plot demonstrates that the data follow a non-monotonic hazard rate shape.

8.3 Eruptions Data

The last data set was reported by Professor Jim Irish, which can be accessed at http://www.statsci.org/data/oz/kiama.html. Regarding the Kiama Blowhole eruptions, the following data is provided:

83, 51, 87, 60, 28, 95, 8, 27, 15, 10, 18, 16, 29, 54, 91, 8, 17, 55, 10, 35, 47, 77, 36, 17, 21, 36, 18, 40, 10, 7, 34, 27, 28, 56, 8, 25, 68, 146, 89, 18, 73, 69, 9, 37, 10, 82, 29, 8, 60, 61, 61, 18, 169, 25, 8, 26, 11, 83, 11, 42, 17, 14, 9, 12.

Table 5 MLEs and goodness-of-fit statistics for eruptions data

The parameter estimates (standard errors in parentheses), the goodness-of-fit statistics: AIC, BIC, CAIC, $W^{*}$, $A^{*}$, K-S statistic, and its p-value are given in Table 5. The numbers in Table 5 illustrate that the RB–TL–Gom-LLoG distribution has the smallest values for the goodness-of-fit statistics and the largest p-value of the K–S statistic. Thus, the RB–TL–Gom-LLoG distribution can fit the eruptions data better than the rest of the distributions. Plots of the fitted densities and the histogram, observed probability vs predicted probability are given in Fig. 8.

Fig. 8

Fitted density vs histogram (left) and observed probability vs expected probability plots (right) for the Eruptions Data

Fig. 9

Fitted Kaplan–Meier survival curve, theoretical and empirical cumulative distribution functions, the total time on test statistics, and the hazard rate function for the Eruptions Data

Figure 9 shows the observed and the fitted Kaplan-Meier survival curves, ECDF, and TTT scaled plot. We can see that the new proposed distribution follows the Kaplan-Meier survival and empirical cdf curves very closely. The TTT scaled plot demonstrates a non-monotonic hrf for the eruptions data.

9 Concluding Remarks

The Ristić–Balakrishnan–Topp–Leone–Gompertz-G (RB–TL–Gom-G) family of distributions has been proposed and studied. Major statistical properties of this novel distribution family are derived. Using the maximum likelihood estimation approach, the parameters of the RB–TL–Gom-G distribution family have been estimated. The accuracy and convergence of the maximum likelihood estimates are assessed using Monte Carlo simulations. By fitting three real-world data sets, the flexibility and properties of the RB–TL–Gom-G family of distributions are presented. In conclusion, the RB–TL–Gom-G family of distributions can leverage the performance of existing distributions to generate a range of densities and hazard rate functions with a variety of shapes.