A novel flexible exponential-G family: Properties, estimation, and applications in environmental and engineering sciences

1. Introduction

In applied science, selecting the appropriate probability distribution for modeling and analyzing data is essential for more precise conclusions. Real-life datasets, particularly in reliability and failure times fields, often exhibit unimodal, modified unimodal, or bathtub-shaped (U-shaped) FRF patterns. Employing traditional distributions to accommodate these data sets could lead to inaccurate results. Thus, numerous probability distributions have been suggested to suit various types of real-life data. Moreover, enhancing the flexibility of the conventional distributions is an important requirement. The literature on probability distribution techniques contains a range of expansions and enhancements of continuous, discrete, symmetric, and asymmetric distributions. One of the most important of these techniques is to build a new family to generate many flexible probability distributions.

Recently, several methods and approaches have been introduced to construct new statistical distributions, reflecting their importance in modeling real data across diverse fields such as biology, reliability engineering, economics, and environmental sciences. Among the most notable families are Azzalini’s skewed family (Azzalini, 1985), Marshall-Olkin family (Marshall and Olkin, 1997), exponentiated family (Gupta et al., 1998), beta-generated family (Eugene et al., 2002), Kumaraswamy-G (Cordeiro and De Castro, 2011), McDonald-G (Alexander et al., 2012), exponentiated transformed transformer (Alzaghal et al., 2013), exponentiated generalized (Cordeiro et al., 2013), Weibull-G (Bourguignon et al., 2014), new power Topp-Leone family (Bantan et al., (2019), generalized exponential-G (Tahir, et al., 2016), alpha power transformation (Mahdavi and Kundu, 2017), Marshall-Olkin-Weibull-H (Afify et al., 2022), weighted Lindley-G (Alnssyan et al., 2023), sine generalized family (Oramulu et al., 2024), generalized Kavya-Manoharan-G (Mahran et al., 2024), modified Kies flexible generalized family (Ferreira and Cordeiro, 2024), Pi-power logistic-G (Sapkota et al., 2025), shifted Lomax-X (Atchadé et al., 2025), sine modified-Kies generalized family (Mulagala and Nagarjuna, 2025), modified Kies Kavya-Manoharan-G (Hamdi et al., 2025), and Marshall-Olkin alpha log-power transformed-G (Musekwa and Makubate, 2025).

While these families provide valuable extensions, classical distributions still face limitations in capturing diverse hazard rate (HR) shapes and tail behaviors. In particular, many generalized models achieve flexibility only by introducing additional parameters, which often increases complexity and complicates estimation. This limitation motivates the development of new frameworks that achieve both parsimony and adaptability, a gap that the proposed flexible exponential-G (FEx-G) family seeks to address.

Motivated by these challenges, we introduce the FEx-G family, we propose the FEx-G family. This family not only generalizes and extends several well-known distributions but also offers high flexibility in modeling diverse data behaviors. A particularly interesting feature is the presence of a special subfamily capable of combining two different baseline distributions without introducing any additional parameters, thereby achieving a unique balance between simplicity and adaptability. Furthermore, the FEx-G family can accommodate a wide spectrum of HR behaviors, including modified bathtub, decreasing, bathtub, increasing, J-shape, reversed J-shape, and unimodal shapes. These properties, illustrated through specific submodels and graphical analyses in later sections, demonstrate the strong potential of the FEx-G family for both theoretical research and practical applications, particularly in survival analysis and reliability studies.

To strengthen practical applicability, we investigate seven parameter estimation techniques for a notable special case of the proposed FEx-G family, namely the FExILL model. Using extensive simulation studies under various parameter settings and for both small and large sample sizes, we assess and compare the performance of these methods. The findings provide deeper insights into the properties of the FEx-G family while offering applied researchers, engineers, and statisticians clear guidance on selecting effective estimation strategies for real-world applications.

The paper is organized as follows: Section 2 defines the FEx-G family. Section 3 discusses four special sub-models of the FEx-G family. Section 4 provides some mathematical properties of the FExILL model. Section 5 discusses parameter estimation of the FExILL model. Section 6 presents simulation results to examine the behavior of different estimators. Section 7 investigates practical applications using real-world data. Finally, Section 8 concludes the paper with a summary of the main findings and remarks.

2. The New Family

In this section, we develop a new family of distributions called the FEx-G class and derive some of its properties. The cumulative distribution function (CDF) of the FEx-G family is defined by

where $G_1\left(x;{\xi }_1\right)$ and $G_2\left(x;{\xi }_2\right)$ are two baseline CDFs with vectors of unknown parameters ${\xi }_1$and ${\xi }_2$ respectively, $\upsilon >$0 and $\omega >0$ are two extra shape parameters.

The probability density function (PDF) corresponding to Equation (2.1) is given by

where $g_1\left(x;{\xi }_1\right)$ is the PDF of the first baseline distribution with parameter vector ${\xi }_1$ and $h_{g_2}\left(x;{\xi }_2\right)$is the HR function (HRF) corresponding to the second baseline distribution with parameter vector ${\xi }_2$.

From Equations (2.1) and (2.2), the HRF of the FEx-G family reduces to

Equation (2.3) shows that the HRF depends on the values of the parameters $\nu$ and w.

Furthermore, the FEx-G family has two important special cases as follows. Two baseline distributions are identical with the same parameter vector

The PDF and FRF corresponding to Equation (2.4) are given by

and

An important special case of the FEx-G family arises when $\upsilon =\omega =1$. The reduced family is capable of generating new flexible distributions without introducing any additional parameters to the baseline distributions.

The CDF, PDF, and HRF of the reduced family are defined as follows

and

3. Special Sub-Models

This section presents several special sub-models of the FEx-G family by employing different baseline distributions, namely the uniform (U), inverse exponential (IEx), exponential (Ex), Fréchet (F), Weibull (W), inverse Lomax (IL), and Lomax (L). The resulting sub-models include the FEx-uniform uniform (FExUU), FEx-inverse exponential exponential (FExIExEx), FEx-Fréchet Weibull (FExFW), and FEx-inverse Lomax Lomax (FExILL) distributions.

3.1 The FExUU distribution

Let $G_1\left(x\right)=$ $G_2\left(x\right)=\frac{x}{\theta }$ for $0<x<\theta$ be a common baseline U distribution with a scale parameter $\theta >0$. Then, from Equation (2.4), the CDF of the FExUU distribution reduces to

$F\left(x;\nu ,\omega \right)=1-\exp \left(-\frac{{\left(\frac{x}{\theta }\right)}^{\nu }}{{\left(1-\frac{x}{\theta }\right)}^{\omega }}\right),\nu ,\omega >0, 0<x<\theta .$

The PDF and HRF of the FExUU distribution are given by

$f\left(x;\nu ,\omega \right)=\frac{\left(\left(\nu -\omega \right)x-\nu \theta \right){\left(\frac{x}{\theta }\right)}^{\nu }{\left(\frac{\theta -x}{\theta }\right)}^{-\omega }}{x\left(x-\theta \right)}\text{exp}\left(-{\left(\frac{x}{\theta }\right)}^{\nu }{\left(\frac{\theta -x}{\theta }\right)}^{-\omega }\right)$

and

$h\left(x;\nu ,\omega \right)=\frac{\left(\left(\nu -\omega \right)x-\nu \theta \right){\left(\frac{x}{\theta }\right)}^{\nu }{\left(\frac{\theta -x}{\theta }\right)}^{-\omega }}{x\left(x-\theta \right)}.$

This distribution is commonly referred to in the literature as the Kies distribution (Kies, 1958). Illustrative shapes of the PDF and HRF of the FExUU distribution have been presented in Fig. 1

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Fig. 1.

PDF and HRF Plots of the FExUU density for different parametric values.

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.

3.2 The FExIExEx distribution

We define the FExIExEx distribution by taking the IEx and Ex distributions as baseline distributions in the FEx-G family. Consider the CDF of the IEx distribution with parameter $\beta >0$, $G_1\left(x; \beta \right)=\exp \left(- \frac{\beta }{x}\right)$for $x>0$, and the CDF of the Ex distribution with parameters $\alpha >0$, $G_2\left(x;\alpha \right)=1-\exp \left(- \frac{x}{\alpha }\right)$ for $x>0$. Then, the CDF of the FExIExEx distribution can be derived from Equation (2.7) as

$F\left(x;\alpha ,\beta \right)=1-\text{exp}\left(-\text{exp}\left(\frac{x}{\alpha }-\frac{\beta }{x}\right)\right), x>0.$

The PDF and HRF of the FExIExEx distribution are defined by

$f\left(x;\alpha ,\beta \right)=\left(\frac{1}{\alpha }+\frac{\beta }{x^2}\right)\exp \left(\frac{x}{\alpha }-\frac{\beta }{x}\right)\exp \left(-\exp \left(\frac{x}{\alpha }-\frac{\beta }{x}\right)\right),x>0$

and

$h\left(x;\alpha ,\beta \right)=\left(\frac{1}{\alpha }+\frac{\beta }{x^2}\right)\exp \left(\frac{x}{\alpha }-\frac{\beta }{x}\right).$

This distribution is referred to in the literature as the flexible Weibull distribution (Bebbington et al, 2007). Representative shapes of its density and HR functions have been illustrated in Fig. 2

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Fig. 2.

PDF and HRF plots of the FExIExEx density for different parametric values.

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.

3.3 The FExFW distribution

The CDFs of the F and W distributions are, respectively, $G_1\left(x; \beta ,\nu \right)=\exp \left(-\frac{\beta }{x^{\nu }}\right),x > 0,\beta ,\nu >0$ and $G_2\left(x; \alpha ,\omega \right)=1-\exp \left(-\frac{x^{\omega }}{\alpha }\right),$ $x > 0,\alpha ,\omega >0$JKSUS820_039 - Copy.eps]. The CDF of the FExFW distribution can then be derived from Equation (2.7) as follows:

$F\left(x;\alpha ,\beta ,\nu ,\omega \right)=1-\exp \left(-\exp \left(\frac{x^{\omega }}{\alpha }-\frac{\beta }{x^{\nu }}\right)\right), \alpha ,\beta ,\nu ,\omega >0,x>0.$

The PDF and HRF of the FExFW distribution are derived as

$\begin{array}{c}f\left(x;\alpha ,\beta ,\nu ,\omega \right)\\ =\left(\frac{\omega x^{\omega -1}}{\alpha }+\frac{\beta \nu }{x^{\nu +1}}\right)\exp \left(\frac{x^{\omega }}{\alpha }-\frac{\beta }{x^{\nu }}\right)\exp \left(-\exp \left(\frac{x^{\omega }}{\alpha }-\frac{\beta }{x^{\nu }}\right)\right)\end{array}$

and

$h\left(x;\alpha ,\beta ,\nu ,\omega \right)=\left(\frac{\omega x^{\omega -1}}{\alpha }+\frac{\beta \nu }{x^{\nu +1}}\right)\exp \left(\frac{x^{\omega }}{\alpha }-\frac{\beta }{x^{\nu }}\right).$

This distribution is known in the literature as the new flexible extended Weibull distribution, and it is introduced by Qinghu Liao et. al. (2020). Some shapes of the density and failure rate functions of the FExIExEx distribution have been displayed in Fig. 3.

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Fig. 3.

PDF and HRF Plots of the FExFW density for different parametric values.

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This distribution is known in the literature as the new flexible extended Weibull distribution (Liao et al., 2020). Some possible shapes of its PDF and HRF have been displayed in Fig. 3.

3.4 The FExILL distribution

In this section, we introduce the FExILL distribution as a special case of the FEx-G family, with support [0, $\infty$), and derive some of its properties. The CDF of the FExILL distribution is obtained by substituting the CDFs of the IL and L baseline distributions into the general CDF form of the FEx-G family.

The CDFs of the IL and L distributions are given by

$\begin{array}{c}{G}_1\left(x; \beta ,\nu \right)={\left(1+\frac{\beta }{x}\right)}^{-\nu }, x > 0,\beta ,\nu >0and\\ G_2\left(x; \alpha ,\omega \right)=1-{\left(1+\frac{x}{\alpha }\right)}^{-\omega },x>0,\alpha ,\omega >0.\end{array}$

Then, the CDF of the FExILL distribution follows from Equation (2.7) as

$F\left(x;\alpha ,\beta ,\upsilon ,\omega \right)=1-\exp \left(-{\left(1+\frac{x}{\alpha }\right)}^{\omega }{\left(1+\frac{\beta }{x}\right)}^{-\nu }\right),x>0,\alpha ,\beta ,\upsilon ,\omega >0.$

The cumulative HRF, HRF and PDF of the FExILL distribution are expressed as follows

$H\left(x;\alpha ,\beta ,\upsilon ,\omega \right)={\left(1+\frac{x}{\alpha }\right)}^{\omega }{\left(1+\frac{\beta }{x}\right)}^{-\nu },$

$h\left(x;\alpha ,\beta ,\upsilon ,\omega \right)={\left(1+\frac{x}{\alpha }\right)}^{\omega -1}{\left(1+\frac{\beta }{x}\right)}^{-\nu }\left(\frac{\omega }{\alpha }+\frac{\nu \beta \left(1+\frac{x}{\alpha }\right)}{x^2\left(1+\frac{\beta }{x}\right)}\right)$

and

$\begin{array}{c}f\left(x;\alpha ,\beta ,\upsilon ,\omega \right)={\left(\frac{x+\alpha }{\alpha }\right)}^{\omega }{\left(\frac{x+\beta }{x}\right)}^{-\nu }\left(\frac{\alpha \beta \nu +\beta \left(\nu +\omega \right)+\omega x^2}{x\left(x+\alpha \right)\left(x+\beta \right)}\right)\\ \exp \left(-{\left(\frac{\alpha +x}{\alpha }\right)}^{\omega }{\left(\frac{x+\beta }{x}\right)}^{-\nu }\right). \end{array}$

Let us consider the special case where the IL and L distributions share a common scale parameter, i.e., $\alpha =\beta =\theta$. Then:

(3.1)

$F\left(x;\theta ,\upsilon ,\omega \right)=1-\exp \left(-{\left(1+\frac{x}{\theta }\right)}^{\omega }{\left(1+\frac{\theta }{x}\right)}^{-\nu }\right),x>0,\theta ,\upsilon ,\omega >0$

and

$\begin{array}{c}f\left(x;\theta ,\upsilon ,\omega \right)\\ ={\left(\frac{x+\theta }{\theta }\right)}^{\omega }{\left(\frac{x+\theta }{x}\right)}^{-\nu }\left(\frac{\nu \theta +\omega x}{x\left(x+\theta \right)}\right)\exp \left(-{\left(\frac{x+\theta }{\theta }\right)}^{\omega }{\left(\frac{x+\theta }{x}\right)}^{-\nu }\right). \end{array}$

The HRF of the FExILL distribution has the form

$h\left(x;\theta ,\upsilon ,\omega \right)={\left(\frac{x+\theta }{\theta }\right)}^{\omega }{\left(\frac{x+\theta }{x}\right)}^{-\nu }\left(\frac{\nu \theta +\omega x}{x\left(x+\theta \right)}\right).$

Representative shapes of the PDF and HRF of the FExILL distribution have been displayed in Fig. 4.

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Fig. 4.

PDF and HRF Plots of the FExILL density for different parametric values.

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4. Properties of the FExILL Model

The shape of the HRF of the FExILL model can be described analytically.

$\begin{array}{c}\ln h\left(x;\theta ,\upsilon ,\omega \right)\\ =\omega \ln \left(\frac{x+\theta }{\theta }\right)-\nu \ln \left(\frac{x+\theta }{x}\right)+\ln \left(\nu \theta +\omega x\right)-\ln \left(x\left(x+\theta \right)\right).\end{array}$

$\frac{d}{dx}\left(\omega \ln \left(\frac{x+\theta }{\theta }\right)-\nu \ln \left(\frac{x+\theta }{x}\right)+\ln \left(\nu \theta +\omega x\right)-\ln \left(x\left(x+\theta \right)\right)\right).$

The critical points of the FRF are the solutions of the equation:

$\frac{\omega }{x+\theta }+\frac{\omega }{\nu \theta +\omega x}-\frac{\nu \left(\frac{1}{x}-\frac{x+\theta }{x^2}\right)}{x+\theta }-\frac{2x+\theta }{x\left(x+\theta \right)}=0.$

The two change points $x^{*}$, the turning points of the HRF, are

$x^{*}=\frac{\theta \left(\nu \left(1-\omega \right)+\sqrt{-\nu \left(\omega -1\right)\left(-\omega +\nu \right)}\right)}{\omega \left(\omega -1\right)}.$

Clearly, the shape of the HRF of the FExILL model, denoted by, $h\left(x\right)$, is influenced by the parameters $\upsilon$ and $\omega ,$ through the terms ${\left(\frac{x+\theta }{\theta }\right)}^{\omega }$ and ${\left(\frac{x+\theta }{x}\right)}^{\nu }.$

Case (i): $0<\upsilon <1$and $0<\omega <1$:

$h\left(x\right)\to \infty$ as $x\to 0+$ and $h\left(x\right)\to 0$ as $x\to -\infty$.

$h\left(x\right)$ is decreasing in $x$, implying it has a decreasing shape.

Case (ii): $0<\upsilon <1$and $\omega >1$:

$h\left(x\right)\to \infty$ as $x\to \phi +$, and $h\left(\text{x}\right)\to \infty$ as $x\to -\infty$.

$h\left(x\right)$ initially decreases and then increases, implying it has a bathtub shape.

Case (iii): $\upsilon >1$and $0<\omega <1$:

$h\left(x\right)\to 0$ as $x\to 0+$, and $h\left(x\right)\to 0$ as $x\to \infty -$.

$h\left(x\right)$ initially increases and then decreases, implying it has a unimodal shape.

Case (iv): $\upsilon >1$and $\omega >1$:

$h\left(x\right)\to 0$ as $x\to 0+$ and $h\left(x\right)\to \infty$ as $x\to \infty -$.

$h\left(x\right)$ is increasing in $x$, implying it has an increasing shape.

Thus, the FExILL distribution can accommodate various HR shapes, including increasing, decreasing, bathtub, and unimodal, demonstrating its flexibility for modeling survival and reliability data.

4.3 Skewness and kurtosis

In this section, we derive the skewness ( ${\gamma }_1$) and kurtosis ( ${\gamma }_2$) of the FExILL distribution. Khan et al. (2021) proposed a measure for the skewness based on the CDF. This measure uses the CDF evaluated at the mean as a key component of the skewness measure aligns naturally with the tendency of the mean to shift in the direction of skewness. According to Khan et al. (2021), the skewness measure is defined as:

${\gamma }_1=2 F\left(\mu \right)-1,$

where $F\left(\mu \right)$ is the value of the CDF at the mean $\mu$. Using the CDF (3.1) and the mean of the FExILL distribution, we can write

${\gamma }_1=1-2 \exp \left(-{\left(1+\frac{\mu }{\theta }\right)}^{\omega }{\left(1+\frac{\theta }{\mu }\right)}^{-\nu }\right),$

where $\mu$ is the mean of the FExILL distribution, which can be obtained from the first non-central moment ( $r=1$) as follows (for $\omega >\nu$):

$\mu ={{\mu }^{\prime }}_1=\sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!}\theta \left(\begin{array}{c}\nu \frac{\Gamma \left(\left(n+1\right)\nu +1\right)\Gamma \left(\left(n+1\right)\left(\omega -\nu \right)\right)}{\Gamma \left(\left(n+1\right)\omega +1\right)}\\ +\omega \frac{\Gamma \left(\left(n+1\right)\nu +2\right)\Gamma \left(\left(n+1\right)\left(\omega -\nu \right)\right)}{\Gamma \left(\left(n+1\right)\omega +2\right)}\end{array}\right).$

For the kurtosis measure, we adopt the following definition: ${\gamma }_2={\mu }_4/{\mu }_2^2.$

Fig. 5 illustrates the behavior of skewness ( ${\gamma }_1$) and kurtosis ( ${\gamma }_2$) for the FExILL distribution across different values of $\nu$ and $\omega$ with $\theta$ fixed at $1$. Both measures ${\gamma }_1$ and ${\gamma }_2$ increase as the parameters $\nu \text{and}\omega$ increase. The skewness values remain within the range of $-1$ to $1$ as expected (see Khan et al., 2021), indicating moderate asymmetry in the distribution.

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Fig. 5.

Plots of the skewness and kurtosis of the FExILL distribution.

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5. Estimation methods

In this section, we present seven methods for estimating the parameters of the FExILL distribution. These methods include maximum likelihood (ML), least squares (LS), weighted least squares (WLS), Cramér-von Mises (CVM), maximum product of spacings (MPS), Anderson-Darling (AD), and right-tail Anderson-Darling (RTAD) estimators.

Let $x_1,\dots , x_n$ be a random sample from the FExILL distribution with parameters $\theta ,\nu ,$ and $\omega$. Denote the ordered statistics as $X_{1:n}<X_{2:n}<\dots <X_{n:n}$.

The log-likelihood function of the FExILL model can be expressed as follows

$l=\omega \sum _{i=1}^n \log \left(k_i\right)- \nu \sum _{i=1}^n\log \left(m_i\right)+\sum _{i=1}^n\log \left(\frac{\nu \theta +\omega x_i}{x_i\left(x_i+\theta \right)}\right)-\sum _{i=1}^n{k}_i^{\omega } m_i^{-\nu },$

where $k_i=\left(1+\frac{x_i}{\theta }\right)$ and $m_i=\left(1+\frac{\theta }{x_i}\right)$.

The MLEs for $\theta ,\nu ,$ and $\omega$ can be obtained by maximizing the previous equation with respect to these parameters or by solving the provided nonlinear equations:

$\begin{array}{c}\frac{\partial l}{\partial \theta }=- \sum _{i=1}^n\frac{\omega x_i}{k_i {\theta }^2}-\sum _{i=1}^n\frac{\nu }{m_i x_i}+\sum _{i=1}^n\left(\frac{\nu }{\left(\nu \theta +\omega x_i\right)}-\frac{1}{\left(x_i+\theta \right)}\right)\\ +\sum _{i=1}^n\frac{\nu k_i^{\omega } m_i^{-\nu -1}}{x_i}+\frac{\omega x_i m_i^{-\nu } k_i^{\omega -1}}{{\theta }^2}=0,\end{array}$

$\frac{\partial l}{\partial \nu }=-\sum _{i=1}^n\log \left(m_i\right)+\sum _{i=1}^n\left(\frac{\theta }{\nu \theta +\omega x_i}\right)+\sum _{i=1}^n{k}_i^{\omega } m_i^{-\nu }\log \left(m_i\right)=0$

and

$\frac{\partial l}{\partial \omega }=\sum _{i=1}^n \log \left(k_i\right)+\sum _{i=1}^n\left(\frac{ x_i}{\nu \theta +\omega x_i}\right)-\sum _{i=1}^n{m}_i^{-\nu }{k}_i^{\omega }\log \left(k_i\right)=0.$

The LS and WLS methods are employed to estimate the parameters of the beta distribution (Swain et al., 1988). The LS estimators (LSEs) and WLS estimators (WLSEs) for the FExILL parameters can be obtained by minimizing the following:

$V\left(\theta ,\nu ,\omega \right)=\sum _{i=1}^n{\upsilon }_i{\left(1-\exp \left(-k_{i:n}^{\omega } m_{i:n}^{-\nu }\right)-\frac{i}{n+1}\right)}^2,$

where ${\upsilon }_i=1$ for the LS method, ${\upsilon }_i={\left(n+1\right)}^2\left(n+2\right)/\left(i\left(n-i+1\right)\right)$ for the WLS approach, and $k_{i:n}=\left(1+\frac{x_{i:n}}{\theta }\right)$ and $m_{i:n}=\left(1+\frac{\theta }{x_{i:n}}\right)$.

Additionally, the LSEs and WLSEs can be derived by solving the nonlinear equations (for $s=1,2,3$):

$\sum _{i=1}^n{\upsilon }_i\left(1-\exp \left(-k_{i:n}^{\omega } m_{i:n}^{-\nu }\right)-\frac{i}{n+1}\right){\Delta }_s\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)=0,$

where

and

The CVM estimators (CVMEs) (Cramér, 1928 and Von Mises, 1928) can be derived from the difference between the estimated CDF and the empirical CDF. The CVMEs for the FExILL parameters are found by minimizing the following function:

$C\left(\theta ,\nu ,\omega \right)=\frac{1}{12n}+\sum _{i=1}^n{\left(1-\exp \left(-k_{i:n}^{\omega } m_{i:n}^{-\nu }\right)-\frac{2i-1}{2n}\right)}^2.$

Further, the CVMEs follow by solving the nonlinear equations,

$\sum _{i=1}^n\left(1-\exp \left(-k_{i:n}^{\omega } m_{i:n}^{-\nu }\right)-\frac{2i-1}{2n}\right){\Delta }_s\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)=0,$

where ${\Delta }_s\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)=0$are defined in (1)-(3) for $s=1,2,3$.

The MPS method is used for parameter estimation in continuous univariate models as an alternative to the ML method (Cheng and Amin, 1979, 1983). The niform spacings of a random sample of size $n$ from the FExILL distribution can be characterized by:

$D_i=F \left(x_{i:n}\Delta \theta ,\nu ,\omega \right)-F \left(x_{i-1:n}\Delta \theta ,\nu ,\omega \right),$

where $D_i$ denotes the uniform spacings, where $F \left(x_{0:n}\Delta \theta ,\nu ,\omega \right)=0, F \left(x_{n+1:n}\Delta \theta ,\nu ,\omega \right)=1$ and $\sum _{i=1}^{n+1}{D}_i\left(\theta ,\nu ,\omega \right)=1$. The MPS estimators (MPSEs) of the FExILL parameters can be obtained by maximizing

$G\left(\theta ,\nu ,\omega \right)=\frac{1}{n+1}\sum _{i=1}^{n+1}\log D_i\left(\theta ,\nu ,\omega \right).$

Additionally, the MPSEs of the FExILL parameters can also be obtained by solving:

$\begin{array}{c}\frac{1}{n+1}\sum _{i=1}^{n+1}\frac{1}{D_i\left(\theta ,\nu ,\omega \right)}\left({\Delta }_s\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)-{\Delta }_s\left(x_{i-1:n}\Delta \theta ,\nu ,\omega \right)\right)=0,\\ s=1, 2,3.\end{array}$

The AD estimators (ADEs) are another form of minimum distance estimator. The ADEs for the FExILL parameters are obtained by minimizing:

$\begin{array}{c}A\left(\theta ,\nu ,\omega \right)\\ =-n-\frac{1}{n}\sum _{i=1}^n\left(2i-1\right) \left(\log F\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)+\log \overline{F}\left(x_{n+1-i:n}\Delta ,\nu ,\omega \right)\right).\end{array}$

The ADEs can also be determined by solving the corresponding nonlinear equations:

$\sum _{i=1}^n\left(2i-1\right)\left(\frac{{\Delta }_s\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)}{F\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)}-\frac{{\Delta }_j\left(x_{n+1-i:n}\Delta ,\nu ,\omega \right)}{S\left(x_{n+1-i:n}\Delta \theta ,\nu ,\omega \right)}\right)=0.$

The RTAD estimators (RTADEs) for the FExILL parameters $\theta ,\nu ,$ and $\omega$ are obtained by minimizing the following function with respect to these parameters:

$\begin{array}{c}R\left(\theta ,\nu ,\omega \right)\\ =\frac{n}{2}-2\sum _{i=1}^{n}F\left(x_{i:n}\Delta \theta ,\nu ,\omega \right)-\frac{1}{n}\sum _{i=1}^n\left(2i-1\right) \log \overline{F}\left(x_{n+1-i:n}\Delta \theta ,\nu ,\omega \right).\end{array}$