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A new WeibullX family of distributions: properties, characterizations and applications
Journal of Statistical Distributions and Applications volume 5, Article number: 5 (2018)
Abstract
We propose a new family of univariate distributions generated from the Weibull random variable, called a new WeibullX family of distributions. Two special submodels of the proposed family are presented and the shapes of density and hazard functions are investigated. General expressions for some statistical properties are discussed. For the new family, three useful characterizations based on truncated moments are presented. Three different methods to estimate the model parameters are discussed. Monti Carlo simulation study is conducted to evaluate the performances of these estimators. Finally, the importance of the new family is illustrated empirically via two real life applications.
1. Introduction
In the field of reliability theory, modeling of lifetime data is very crucial. A number of statistical distributions such as Weibull, Pareto, Gompertz, linear failure rate, Rayleigh, Exonential etc., are available for modeling lifetime data. However, in many practical areas, these classical distributions do not provide adequate fit in modeling data, and there is a clear need for the extended version of these classical distributions. In this regard, serious attempts have been made to propose new families of continuous probability distributions that extend the existing wellknown distributions by adding additional parameter(s) to the model of the baseline random variable. The wellknown family of distributions are: the betaG by Eugene et al. (2002), Jones (2004), GammaG (type1) due to Zografos and Balakrishnan (2009), McG proposed by Alexander et al. (2012), LogGammaG Type2 of Amini et al. (2012), GammaG (type2) studied by Risti’c and Balakrishnan (2012), GammaG (type3) of Torabi and Montazeri (2012), WeibullX family of distributions of Alzaatreh et al. (2013), exponentiated generalized class of Cordeiro et al. (2013), LogisticG introduced by Torabi and Montazeri (2014), GammaX family of Alzaatreh et al. (2014), odd generalized exponentialG of Tahir et al. (2015a, b), type I halflogistic family of Cordeiro et al. (2016), Kumaraswamy Weibullgenerated family of Hassan and Elgarhy (2016), new WeibullG family of Tahir et al. (2016), generalized transmutedG of Nofal et al. (2017) and a new generalized family of distributions of Ahmad (2018). Let v(t) be the probability density function (pdf) of a random variable, say T, where T ∈ [m, n] for − ∞ ≤ m < n < ∞ and let W[F(x; ξ)] be a function of cumulative distribution function (cdf) of a random variable, say X, depending on the vector parameterξ satisfying the conditions given below:

i.
W[F(x; ξ)] is differentiable and monotonically increasing, and

ii.
W[F(x; ξ)] → m as x → − ∞ and W[F(x; ξ)] → n as x → ∞.
Recently, Alzaatreh et al. (2013), defined the cdf of the TX family of distributions by
where W[F(x; ξ)] satisfies the conditions stated above. The pdf corresponding to (1) is
Using the TX idea, several new classes of distributions have been introduced in the literature. Table 1 provides some W[F(x; ξ)] functions for some members of the TX family.
The main goal of this article is to introduce a new family of continuous distributions, called the new WeibullX (“NWX” for short) family. We discuss three special submodels of this family, capable of modeling with monotonic and nonmonotonic hazard rates. For each special submodel of NWX family, a real life application is presented.
The rest of this paper is structured as follows: In Section “The New WeibullX family”, we define the NWX family of distributions. Section “Special subModels” offers some special submodels of this family. Useful expansions for the pdf and cdf of NWX are provided in section “Useful expansions of the NWX family”. Statistical properties of the NWX family are investigated in section “Basic mathematical properties”. Section “Estimation” provides estimation of the model parameters using maximum likelihood method. Simulation results are reported in section “Simulation Study”. Section “Characterizations” contains some useful characterizations of the proposed class. Section “Applications” provides analysis to real data sets. Finally, section “Concluding Remarks”, concludes the article.
2. The new WeibullX family
If X is a Weibull random variable with shape parameter α > 0 and scale parameterβ > 0, then its cdf is given by
Setting β = 1 in (2), we obtain the cdf of the one parameter Weibull random variable given by
The density function corresponding to (3) is
If v(t; α) follows (4) and setting \( W\left[F\left(x;\xi \right)\right]=\frac{\left[\log \left\{1F\left(x;\xi \right)\right\}\right]}{1F\left(x;\xi \right)} \) in (1), we define the cdf of the NWX family by
where, F(x; ξ) is the cdf of the baseline distribution which depends on the vector parameter ξ. We can also write (5) as follows
where, H(x; ξ)is the cumulative hazard rate function (chrf) of the baseline random variable. The density function corresponding to (6) is
Henceforward, a random variable X with density (7) is denoted by X ∼ NW − X(x; α, ξ). Moreover, we may omit the dependence on the vector ξ of the parameters and simply write G(x) = G(x; ξ). The survival function (sf), hazard rate function (hrf) and chrf of X are given by (8)–(10), respectively,
The basic motivations for using the NWX family in practice are:

A prominent method of introducing additional parameter(s) to generate an extended version of the baseline model.

To improve the characteristics of the traditional distributions.

To make the kurtosis more flexible compared to the baseline model.

To generate distributions with symmetric, rightskewed, leftskewed and reversedJ shaped.

To define special models with all types of hrf.

To define special models having closed form for cdf, sf as well as hrf.

To provide consistently better fits than other generated distributions having the same or higher number of parameters.
3. Special submodels
Most of the extended versions of the distributions are introduced for one of the following properties: an alternative model to the existing distribution that has previously been utilized successfully, a statistical model whose empirical distribution is a good fit to the data, and offers greater flexibility. A statistical model having closed forms of cdf, sf as well as hrf and reducing the estimation difficulties. Here, we introduce two special submodels of NWX family that can possess at least one of these properties.
3.1. The NWFrechet (NWF) distribution
The density and cdf of the Frechet random variable are \( f\left(x;\xi \right)={\theta \gamma}^{\theta }{x}^{\left(\theta +1\right)}{e}^{{\left(\gamma /x\right)}^{\theta }},\kern1.75em x>0,\xi >0, \) and \( F\left(x;\xi \right)={e}^{{\left(\gamma /x\right)}^{\theta }}, \) where ξ = (γ, θ). Then the cdf and pdf of the NWF model are given by
and
Plots of the NWF density and hrf for selected parameter values are presented in Fig. 1.
3.2. The NWWeibull (NWW) distribution
Now considering the cdf of the twoparameter Weibull model with shape parameter θ > 0 and scale parameterγ > 0, given by \( F\left(x;\xi \right)=1{e}^{\gamma {x}^{\theta }},\kern0.5em x\ge 0,\gamma, \theta >0 \), and pdf \( f\left(x;\xi \right)=\gamma \theta {x}^{\theta 1}{e}^{\gamma {x}^{\theta }}, \) where ξ = (θ, γ). Then, the cdf and pdf of the NWW model are
and
respectively.
Figure 2 displays plots of the NWW pdf and hrf for selected parameter values.
Useful expansions of the NWX family
In this section, we provide useful expansions for the pdf and cdf of the NWX family. Using the series
from (7), we obtain
Hence, the pdf (16) can be written as
where \( {\eta}_{i,j}=\alpha \frac{{\left(1\right)}^{i+j}}{i!}\left(\begin{array}{c}\alpha \left(i+1\right)\\ {}j\end{array}\right),{\tau}_{i,j}=f\left(x;\xi \right){\left\{H\left(x;\xi \right)\right\}}^{\alpha \left(i+1\right)}{\left\{F\left(x;\xi \right)\right\}}^j \) and \( {\tau}_{i,j}^{/}=f\left(x;\xi \right){\left\{H\left(x;\xi \right)\right\}}^{\alpha \left(i+1\right)1}{\left\{F\left(x;\xi \right)\right\}}^j. \)
Furthermore, we provide an expansion for {G(x; α, ξ)}^{s}, where s is an integer. Using the binomial theorem, we have
Using the series (15) in (18), we are arrive at
Finally,
5. Basic mathematical properties
In this section, we provide some basic properties of the NWX family.
5.1. Moments
Let X follows the NWX family with density (17), then its r^{th} moment is
where.
\( {\Lambda}_{r,i,j}=\underset{\infty }{\overset{\infty }{\int }}{x}^r{\tau}_{i,j} dx \) and \( {\Lambda}_{r,i,j}^{/}=\underset{\infty }{\overset{\infty }{\int }}{x}^r{\tau}_{i,j}^{/} dx \).
Furthermore, the moment generating function (mgf) of X is given by
5.2. Probability weighted moments
The probability weighted moment (pwm) of NWX random variable is
Using (17) and (19) in (22), we have
Where \( {\kappa}_{i,j,k,l,m}={\eta}_{i,j}\sum \limits_{k,l,m=0}^{\infty }{\left(1\right)}^{l+m}\left(\begin{array}{c}s\\ {}k\end{array}\right)\left(\begin{array}{c}\alpha l\\ {}m\end{array}\right){k}^l,{\rho}_{i,j,m}=f\left(x;\xi \right){\left\{H\left(x;\xi \right)\right\}}^{\alpha \left(i+l+1\right)}{\left\{F\left(x;\xi \right)\right\}}^{m+j} \) and \( {\rho}_{i,j,m}^{/}=f\left(x;\xi \right){\left\{H\left(x;\xi \right)\right\}}^{\alpha \left(i+l+1\right)1}{\left\{F\left(x;\xi \right)\right\}}^{m+j}. \)
Then,
where, \( {\psi}_{i,j,m,s}=\underset{\infty }{\overset{\infty }{\int }}{x}^s{\rho}_{i,j,m} dx \) and \( {\psi}_{i,j,m,s}^{/}=\underset{\infty }{\overset{\infty }{\int }}{x}^s{\rho}_{i,j,m}^{/} dx \).
5.3. Quantile function
The quantile function of X with cdf (5) is given by
where u is the uniform random number in (0,1). The expression (25) does not have a closed form, therefore, computer software can be used to obtain a closed form solution of the quantile function.
5.4. Order statistics
Let X_{1}, X_{2}, ⋯, X_{k} be independent and identically distributed (i.i.d) random variables from NWX distribution with parameters α and ξ. Let X_{1 : k}, X_{2 : k}, ⋯, X_{k : k} be the corresponding order statistics. Then, the density of X_{r : k}for (r = 1, 2,. .., k) is given by
where, B(., .) represents the beta function. Using (17) and (19) in (26) and replacing s with v + r1, we have
where, \( {\Upsilon}_{i,j,k,l,m,v}={\left(1\right)}^v\left(\begin{array}{l}kr\\ {}\kern0.75em v\end{array}\right){\kappa}_{i,j,k,l,m}. \)
6. Estimation
Here, we describe three methods of estimation of the unknown parameters of the NWX family. The methods are: ordinary least square (OLS) estimation, percentile based estimation and maximum likelihood estimation. The performance of these estimation methods are studied through Monte Carlo simulation.
6.1. Ordinary Least Square estimation
Let x_{1}, x_{2}, …, x_{k} be the observed ordered values from the sample X_{1}, X_{2}, …, X_{k} from NWX family. Then, the expectation of the empirical cdf is defined as
The OLS estimates \( \widehat{\alpha} \) and \( \widehat{\xi} \) of α and ξ can be determined by maximizing
Consequently, ordinary least square estimators (OLSEs) \( \widehat{\alpha} \) and \( \widehat{\xi} \) of α and ξ can be obtained as the solution of the following equations
The OLS estimates of (α, ξ) are the simultaneous solutions of \( \frac{\partial Z\left(\alpha, \xi \right)}{\partial \alpha }=0 \) and \( \frac{\partial Z\left(\alpha, \xi \right)}{\partial \xi }=0 \).
6.2. Percentile estimation
In this subsection, we estimate the unknown parameters by using the percentile estimation method. Let \( {p}_i=\frac{1}{k+1} \)be an estimate of G(x_{(i)}; α, ξ), then the percentile estimators of (α, ξ) can be obtained by maximizing the function
Henceforth, the percentile estimates \( \widehat{\alpha} \) and \( \widehat{\xi} \) of α and ξ are obtained as the simultaneous solution of the expressions given in (33) and (34), below
Setting \( \frac{\partial \varphi }{\partial \alpha } \) and \( \frac{\partial \varphi }{\partial \xi } \) equal to zero and solving numerically these expressions simultaneously yield the percentile estimators (PEs) of (α, ξ).
6.3. Maximum likelihood estimation
In this subsection, we determine the maximum likelihood estimates of the parameters of the NWX family. Let x_{1}, x_{2}, ⋯, x_{k} be the observed values from the NWX distribution with parameters α and ξ. The total log likelihood function corresponding to (7) is given by
The partial derivatives of (35) are
Setting \( \frac{\partial }{\partial \alpha}\log L\left({x}_i;\alpha, \xi \right) \) and \( \frac{\partial }{\partial \xi}\log L\left({x}_i;\alpha, \xi \right) \) equal to zero and solving numerically these expressions simultaneously yield the maximum likelihood estimators (MLEs) of (α, ξ).
7. Simulation study
In this section, we carry out simulation study for NWW model. The process is described below:

i.
Random samples of sizes n = 30, 100 are generated from NWW model and parameters have been estimated via ordinary least square, percentile and maximum likelihood methods.

ii.
1000 repetitions are made to calculate the bias and mean square error (MSE) of these estimators.

iii.
Formulas used for calculating bias and MSE are given by \( Bias\left(\widehat{\alpha}\right)=\frac{1}{1000}\sum \limits_{i=1}^{1000}\left({\widehat{\alpha}}_i\alpha \right) \) and
\( MSE\left(\widehat{\alpha}\right)=\frac{1}{1000}\sum \limits_{i=1}^{1000}{\left({\widehat{\alpha}}_i\alpha \right)}^2 \), respectively.

iv.
Step (iii) is also repeated for the other parameters (θ, γ).
8. Characterizations
This section deals with the characterizations of the NWX distribution in different directions: (i) based on the ratio of two truncated moments; (ii) in terms of the hazard function and (iii) based on the conditional expectation of certain function of the random variable. Note that (i) can be employed also when the cdf does not have a closed form. We would also like to mention that due to the nature of NWX distribution, our characterizations may be the only possible ones. We present our characterizations (i)(iii) in three subsections.
8.1. Characterizations based on two truncated moments
This subsection is concerned with the characterizations of NWX distribution based on the ratio of two truncated moments. Our first characterization employs a theorem due to Glänzel (1987); see Theorem 1 of Appendix. The result, however, holds also when the interval H is not closed, since the condition of the Theorem is on the interior of H.
Proposition 8.1. Let X:Ω → ℝ be a continuous random variable and let q_{1}(x) ≡ 1 and \( {q}_2(x)=\exp \left[{\left\{\frac{\log \left(1F\left(x;\xi \right)\right)}{1F\left(x;\xi \right)}\right\}}^{\alpha}\right] \) for x ∈ ℝ. The random variable X has pdf (7) if and only if the function η(x) defined in Theorem 1 is of the form
Proof. Suppose the random variable X has pdf (7), then
and
Further,
Conversely, if η(x) is of the above form, then
Now, according to Theorem 1, X has density (7).
Corollary 8.1. Let X:Ω → ℝ be a continuous random variable and letq_{1}(x)be as in Proposition 8.1. The random variable X has pdf (7) if and only if there exist functions q_{2}(x)andη(x) defined in Theorem 1 satisfying the following differential equation
Corollary 8.2. The general solution of the differential equation in Corollary 8.1 is
where D is a constant. We like to point out that one set of functions satisfying the above differential equation is given in Proposition 8.1 with D = 1. Clearly, there are other triplets (q_{1}(x), q_{2}(x), η(x)) which satisfy conditions of Theorem 1.
8.2. Characterization in terms of hazard function
The hazard function h_{G}(x) of a twice differentiable distribution function, G(x), satisfies the following first order differential equation
It should be mentioned that for many univariate continuous distributions, the above equation is the only differential equation available in terms of the hazard function. In this subsection we present a nontrivial characterization of NWW distribution in terms of the hazard function.
Proposition 8.2. Let X: Ω → ℝ be a continuous random variable. The random variable X has pdf (7) if and only if its hazard function h_{G}(x) satisfies the following differential equation
Proof. If X has density (7), then clearly the above differential equation holds. Now, if the differential equation holds, then
from which we arrive at the hazard function of (9).
8.3. Characterization based on the conditional expectation of certain function of the random variable
In this subsection we employ a single function ψ of X and characterize the distribution of X in terms of the truncated moment of ψ(x). The following proposition has already appeared in Hamedani’s previous work (2013), so we will just state it here which can be used to characterize the NWX distribution.
Proposition 8.3. Let X: Ω → (e, f) be a continuous random variable with cdf F. Let ψ(x) be a differentiable function on (e, f) with \( {\lim}_{x\to {e}^{+}}\psi (x) \). Then for δ ≠ 1,
if and only if
Remark 8.1. For \( \left(e,f\right)=\mathrm{\mathbb{R}},\kern0.5em \psi (x)=\exp \left[{\left\{\frac{\log \left(1F\left(x;\xi \right)\right)}{1F\left(x;\xi \right)}\right\}}^{\alpha}\right] \) and \( \delta =\frac{1}{2} \), Proposition 8.3 provides a characterization of NWX distribution.
9. Applications
In the following section, we provide two applications of the NWX family using real data for illustrative purposes. These applications show the flexibility and usefulness of the new generator. For these data sets, we compare the fits of the proposed distribution to other wellknown distributions. In order to compare the models, we consider the following analytical measures: Akaike information criterion (AIC), Consistent Akaike Information Criterion (CAIC), Bayesian information criterion (BIC), HannanQuinn information criterion (HQIC), KolmogorovSmirnov (KS) test statistic, Cramervon Mises (CM) statistic and AndersonDarling (AD) test statistic.
Data 1 The first data set obtained from Lee and Wang (2003) representing the remission times (in months) of a random sample of 128 bladder cancer patients. We applied the NWF distribution to this data set in competition with Frechet (F) and exponentiated Frechet (EF) distributions. The cdf of the EF distribution proposed by Nadarajah Kotz (2003) is given by
The maximum likelihood estimates of the model parameters and analytical results are provided in Tables 4 and 5; Figs. 3 and 4, respectively.
Data 2 The second data set representing strength of the Alumina (Al2O3) material taken from the website: http://www.ceramics.nist.gov/srd/summary/ftmain.htm. This data set can also be found in Nadarajah and Kotz (2008). We applied another submodel of the proposed family NWW distribution to this data set in competition with five other wellknown competing distributions. The distribution functions of the competing models are

Exponentiated Weibull (EW) of Mudholkar and Srivastava (1993)
$$ G(x)={\left(1{e}^{\gamma {x}^{\theta }}\right)}^{\alpha },\kern8.00em x,\alpha, \theta, \gamma >0. $$ 
Modified Weibull (MW) of Lai et al. (2003)
$$ G(x)=1{e}^{\gamma {x}^{\theta }{e}^{\alpha x}},\kern9.00em x,\alpha, \theta, \gamma >0. $$ 
Kumaraswamy Weibull (KuW) of Cordeiro et al. (2010)
$$ G(x)=1{\left(1{\left(1{e}^{\gamma {x}^{\theta }}\right)}^a\right)}^b,\kern2.75em \ \kern2.00em x,a,b,\theta, \gamma >0. $$ 
Alpha power transformed Weibull (APTW) of Dey et al. (2017)
$$ G(x)=\frac{\alpha^{\left(1{e}^{\gamma {x}^{\theta }}\right)}1}{\alpha 1},\kern8.75em x,\alpha, \theta, \gamma >0,\kern0.5em \alpha \ne 1. $$ 
Flexible Weibull extended (FWE) of Ahmad and Hussain (2017)
$$ G(x)=1\exp \left({e}^{\alpha {x}^2\frac{\gamma }{x^{\theta }}}\right),\kern6.75em x,\alpha, \theta, \gamma >0. $$
The maximum likelihood estimates of the model parameters and analytical results are provided in Tables 6 and 7; Figs. 5 and 6, respectively.
10. Concluding remarks
We have introduced a new family of distributions, called a new WeibullX family. Two special submodels of this family are discussed. The densities of two special models can be leftskewed, rightskewed, symmetrical, reverse Jshaped and can have increasing, decreasing, unimodal or most importantly bathtub shaped failure rates. The parameters are estimated using three different methods, namely, ordinary least square, percentile estimation and maximum likelihood. A simulation study is presented to evaluate the performances of the model parameters estimators. Some mathematical properties of the new family are also derived. Two real applications of the proposed family are provided, and reveal better fits to data than other wellknown distributions.
Abbreviations
 AD:

AndersonDarling
 AIC:

Akaike information criterion
 APTW:

Alpha power transformed Weibull
 BIC:

Bayesian information criterion
 CAIC:

Consistent Akaike Information Criterion
 cdf:

Cumulative distribution function
 chrf:

Cumulative hazard rate function
 CM:

Cramervon Mises
 EF:

Exponentiated Frechet
 EW:

Exponentiated Weibull
 F:

Frechet
 FWE:

Flexible Weibull extended
 HQIC:

HannanQuinn information criterion
 hrf:

Hazard rate function
 i.i.d:

Independent and identically distributed
 KS:

KolmogorovSmirnov
 KuW:

Kumaraswamy Weibull
 mgf:

Moment generating function
 MLEs:

Maximum likelihood estimators
 MSE:

Mean square error
 MW:

Modified Weibull
 NWF:

NWFrechet
 NWW:

NWWeibull
 NWX :

New WeibullX
 OLS:

Ordinary least square
 OLSEs:

Ordinary least square estimators
 pdf:

Probability density function
 PEs:

Percentile estimators
 pwm:

Probability weighted moment
 sf:

Survival function
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Acknowledgements
The authors are grateful to the EditorinChief, the Associate Editor and anonymous referees for many of their valuable comments and suggestions which lead to this improved version of the manuscript.
Funding
Prof. G.G. Hamedani (coauthor of the manuscript) is an Associate Editor of JSDA, a 100% discount on Article Processing Charge (APC) for accepted article.
Availability of data and materials
The idea of TX family of distributions proposed by Alzaatreh et al. (2013) has been used.
Data 1 The first data set taken from Lee and Wang (2003) are as follows: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69
Data 2 The second data set taken from the website: http://www.ceramics.nist.gov/srd/summary/ftmain.htm are as follows: 5.5, 5, 4.9, 6.4, 5.1, 5.2, 5.2, 5, 4.7, 4, 4.5, 4.2, 4.1, 4.56, 5.01, 4.7, 3.13, 3.12, 2.68, 2.77, 2.7, 2.36, 4.38, 5.73, 4.35, 6.81, 1.91, 2.66, 2.61, 1.68, 2.04, 2.08, 2.13, 3.8, 3.73, 3.71, 3.28, 3.9, 4, 3.8, 4.1, 3.9, 4.05, 4, 3.95, 4, 4.5, 4.5, 4.2, 4.55, 4.65, 4.1, 4.25, 4.3, 4.5, 4.7, 5.15, 4.3, 4.5, 4.9, 5, 5.35, 5.15, 5.25, 5.8, 5.85, 5.9, 5.75, 6.25, 6.05, 5.9, 3.6, 4.1, 4.5, 5.3, 4.85, 5.3, 5.45, 5.1, 5.3, 5.2, 5.3, 5.25, 4.75, 4.5, 4.2, 4, 4.15, 4.25, 4.3, 3.75, 3.95, 3.51, 4.13, 5.4,5, 2.1, 4.6, 3.2, 2.5, 4.1, 3.5, 3.2, 3.3, 4.6, 4.3, 4.3, 4.5, 5.5, 4.6, 4.9, 4.3, 3, 3.4, 3.7, 4.4, 4.9, 4.9,5
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The authors, ZA, ME and GGH with the consultation of each other carried out this research and drafted the manuscript together. All authors read and approved the final version of the manuscript.
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Appendix
Appendix
Theorem 1. Let (Ω, Ƒ, P) be a given probability space and let H = [a; b] be an interval for some a < b (a = − ∞ ; b = ∞ might as well be allowed). Let X: Ω → H be a continuous random variable with the distribution function F and let q_{1}(x) and q_{2}(x) be two real functions defined on H such that
is defined with some real function η(x). Assume that q_{1}(x), q_{2}(x) ∈ C^{1}(H), η(x) ∈ C^{2}(x) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξq_{1}(x) = q_{2}(x) has no real solution in the interior of H. Then F is uniquely determined by the functions q_{1}(x), q_{2}(x) and η(x) particularly
where the function s(u) is a solution of the differential equation \( {s}^{/}(u)=\frac{\eta^{/}(u){q}_1(u)}{\eta (u){q}_1(u){q}_2(u)} \) and C is the normalization constant, such that \( \underset{H}{\int } dF=1. \).
We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, Glänzel (1990)), in particular, let us assume that there is a sequence {X_{n}} of random variables with distribution functions {F_{n}} such that the functions q_{1n}, q_{2n} and η_{n}(n ∈ ℕ) satisfy the conditions of Theorem 1 and let q_{1n} → q_{1}, q_{2n} → q_{2} for some continuously differentiable real functions q_{1}(x) and q_{2}(x). Let, finally, X be a random variable with distribution F(x). Under the condition that q_{1n}(x) and q_{2n}(x) are uniformly integrable and the family{F_{n}} is relatively compact, the sequence X_{n} converges to X in distribution if and only if η_{n} converges to η, where
This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions q_{1}(x), q_{2}(x) and η(x) respectively. It guarantees, for instance, the ‘convergence’ of characterization of the Wald distribution to that of the LevySmirnov distribution if α → ∞. A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions q_{1}(x), q_{2}(x) and, specially,η(x) should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose η(x) as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.
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Ahmad, Z., Elgarhy, M. & Hamedani, G.G. A new WeibullX family of distributions: properties, characterizations and applications. J Stat Distrib App 5, 5 (2018). https://doi.org/10.1186/s4048801800876
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Keywords
 Weibull distribution
 TX family
 Moment
 Characterizations
 Order statistics
 Estimation