# New Rayleigh Flexible Weibull Extension (RFWE) Distribution with Applications to Real and Simulated Data

Authors: Muneeb Javed; Syed Muhammad Asim; Alamgir Khalil; Said Farooq Shah; Zahra Almaspoor
Journal: Modelling and Simulation in Engineering
 (2022)

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## Abstract

The Rayleigh flexible Weibull extension (RFWE) distribution, a new three-parameter model introduced in this paper, is a generalization of the flexible Weibull extension. This model produces best fit for failure time of electronic device obtained from power-linkage voltage spikes during electronic storms. We derive the statistical properties of the RFWE distribution. The parameters of this new distribution are estimated using the maximum likelihood method, which also yielded asymptotic confidence bounds. This model is examined using both real and simulated data. Under various priors, an additional Bayesian estimate is also carried out. The Bayes estimates and other posterior results are calculated using simulations.


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## Body

## 1. Introduction

To model the lifetime components, Weibull distribution is very useful in fields like physics and engineering. Farooq et al. [1] focused on investigations to derive a new probability model for data sets with extreme values in engineering. Ijaz et al. [2] developed a new modification with three parameters of the Lomax distribution. Kumaraswamy Weibull distribution is studied by Corderio et al. [3]. Moreover, El-Morshedy et al. [4] proposed a three-parameter model by exponentiating the inverse flexible Weibull extension distribution. They called it exponentiated inverse flexible Weibull extension (EIFW) distribution. Manisha and Tiensuwan [5] introduced a beta transmuted Weibull distribution, which contains several distributions as special cases, and properties of the distribution are also discussed. Mustafa et al. [6] introduced a four-parameter model called the Weibull generalized flexible Weibull extension (WGFWE) distribution which exhibits a bathtub-shaped hazard rate. Bebbington et al. [7] discussed applications of the flexible Weibull distribution that includes life testing experiments and applied statistics. Nadarajah and Kotz [8] and Murthy et al. [9] discussed the extensions of the Weibull distribution.In this article, a new generalization of the flexible Weibull extension (FWE) distribution called Rayleigh flexible Weibull extension (RFWE) distribution is constructed by using a method developed by Alzaatreh et al. [10] to generate a family of distributions. This class of distributions is defined as
(1)Gz=∫0−ln1−Fzgxdx.Alzaatreh et al. [10] derived the Weibull-Pareto distribution by taking gx to be the probability density function (pdf) of the Weibull distribution and Fz to be the cumulative density function (cdf) of the Pareto distribution, and Alzaatreh et al. [11] derived the gamma-normal distribution by taking gx to be the pdf of the gamma distribution and Fz to be the cdf of the normal distribution.Now, we consider pdfgx of Rayleigh distribution as a parent distribution given as
(2)gx=2θxe−θx2.Substituting Equation (2) in Equation (1), we get
(3)Gz=1−exp−λ−ln1−Fz2,and pdf
(4)gz=2λfz1−Fz−ln1−Fzexp−λ−ln1−Fz2.In the above expression, the FWE distribution is used asfz as follows:
(5)fz=β+γz2expβz−γzexp−expβz−γz,z>0,β,γ>0,and cdf
(6)Fz=1−exp−expβz−γz,z>0,β,γ>0.


## 2. Rayleigh Flexible Weibull Extension Distribution (RFWE)

IfZ follows the flexible Weibull extension distribution with the pdf given in Equation (5). Then, from Equation (4), the pdf of RFWE distribution is defined as
(7)gz=2λβ+γz2exp2βz−γzexp−λexpβz−γz2,z>0,β,γ,λ>0.From (3), we obtain the cdf of RFWE distribution as
(8)Gz=1−exp−λexpβz−γz2,z>0,β,γ,λ>0.


### 2.1. Survival and Hazard Functions

From Equation (8), we can define the survival function (sf) of RFWE distribution as follows:
(9)Sz=1−Gz=exp−λexpβz−γz2,z>0,β,γ,λ>0.From Equations (7)–(9), we can define the hazard function (hf) of RFWE distribution as follows:
(10)hz=gzSz=2λβ+γz2exp2βz−γz,z>0,β,γ,λ>0.This hazard function (hf) is plotted along with the pdf (7), cdf (8), and survival function (9) in Figure 1 for different values of the parameters to depict the nature of the proposed distribution.Figure 1
Visual plots for the pdf, cdf, sf, and hf of the proposed model.


## 2.1. Survival and Hazard Functions

From Equation (8), we can define the survival function (sf) of RFWE distribution as follows:
(9)Sz=1−Gz=exp−λexpβz−γz2,z>0,β,γ,λ>0.From Equations (7)–(9), we can define the hazard function (hf) of RFWE distribution as follows:
(10)hz=gzSz=2λβ+γz2exp2βz−γz,z>0,β,γ,λ>0.This hazard function (hf) is plotted along with the pdf (7), cdf (8), and survival function (9) in Figure 1 for different values of the parameters to depict the nature of the proposed distribution.Figure 1
Visual plots for the pdf, cdf, sf, and hf of the proposed model.


## 3. Statistical Properties of RFWE Distribution

In this section, we study the quantile function,rth moment, moment generating function, characteristic function, probability generating function, and factorial moment generating function.


### 3.1. Quantile Function

For a positive continuous random variable, LetZ follows the RFWE distribution, then, the quantile function of Z is derived as
(11)PZ≤zq=q,0<q<1,1−exp−λexpβzq−γzq2=q.By solving the above equation, we getzq as follows:
(12)zq=12βlnln1/1−qλ±lnln1/1−qλ2+4βγ.Since quantilezq is positive, then we get zq as follows:
(13)zq=12βlnln1/1−qλ+lnln1/1−qλ2+4βγ.The median of RFWE can be obtained from Equation (13) by takingq=1/2. That is,
(14)Median=12βlnln1/1−1/2λ+lnln1/1−1/2λ2+4βγ.Now, to generate data from the cdf of RFWE, first generate the random numbers from standard uniform distribution then transform the random numbers into inverse cdf function. The inverse cdf function of RFWE can be written as(15)GRFWEz;β,γ,λ=u,where U is a standard uniform random variable; from (8), we get
(16)1−exp−λexpβz−γz2=u.In Equation (13), if we replace q by u, we get the inverse function
(17)zu=12βlnln1/1−uλ+lnln1/1−uλ2+4βγ.


### 3.2. The Moments

Moments are very important for the statistical analysis to study the average, variation, skewness, and kurtosis.Therth moment of RFWE distribution is introduced by the following theorem.Theorem 1.
Therth moment of a random variable Z∼RFWEΘ, where Θ=β,γ,λ is given by
(18)μr′=∑i=0∞∑j=0∞−1i+jλi+1γjΓr−j−1i!j!βr−j2γ−2ji+1r−2j+1r−jr−j−1+4γβi+12.Proof.
Therth moment of the random variable Z with gz is defined as
(19)μr′=∫0∞zrgzdz.
By putting Equation (7) into Equation (19), we get
(20)μr′=2λ∫0∞zrβ+γz2exp2βz−γzexp−λexpβz−γz2dz,using the Taylor series
(21)exp−λx=∑i=0∞−1ixλii!.
Takex=expβz−γ/z2 in the above equation:
(22)exp−λexpβz−γz2=∑i=0∞−1iλiexpβz−γ/z2ii!,
Equation (20) becomes
(23)μr′=2∑i=0∞−1iλi+1i!∫0∞zrβ+γz2exp2βz−γzexpβz−γz2idz.
By solving the above equation, we obtain(24)μr′=2∑i=0∞−1iλi+1i!∫0∞zrβ+γz2exp2i+1βzexp−2i+1γzdz.
Using the series expansion ofexp−2i+1γ/z, we obtain
(25)μr′=2∑i=0∞∑j=0∞−1i+j2ji+1jγjλi+1i!j!∫0∞zr−jβ+γz2exp2i+1βzdz,by using the gamma function in the form
(26)Γα=βα∫0∞exptβtα−1dt,α,β>0.
Finally, we get therth moment of RFWE in the form
(27)μr′=∑i=0∞∑j=0∞−1i+jλi+1γjΓr−j−1i!j!βr−j2γ−2ji+1r−2j+1r−jr−j−1+4γβi+12.


### 3.3. Moment Generating Function

In this section, we derive the moment generating function of RFWE distribution.Theorem 2.
The moment generating functionMZt of a random variable Z, that is, Z∼RFWEΘ, where Θ=β,γ,λ is given by
(28)Mzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.Proof.
The moment generating functionMZt is defined as
(29)MZt=∫0∞exptzgzdz.
Using series expansion ofetz, we obtain
(30)MZt=∑m=0∞tmm!∫0∞zmgzdz=∑m=0∞tmm!μm′.
Substituting from Equation (27) into Equation (30), we get
(31)Mzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.
Hence, the proof.


### 3.4. Characteristic Function

The characteristic function of RFWE distribution can be given as follows:(32)ϕzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjitmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjitmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


### 3.5. Probability Generating Function

The probability generating function of RFWE distribution can be given as follows:(33)Gα=Eαz,Gα=∫0∞expzlnαgzdz,Gα=2λ∫0∞expzlnαβ+γz2exp2βz−γzexp−λexpβz−γz2dz,Gα=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnmαΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnmαΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


### 3.6. Factorial Moment Generating Function

The factorial moment generating function of RFWE distribution can be given as follows:(34)H0δ=E1+δz,H0δ=∫0∞expzln1+δgz;β,γ,λdz,H0δ=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnm1+δΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnm1+δΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


## 3.1. Quantile Function

For a positive continuous random variable, LetZ follows the RFWE distribution, then, the quantile function of Z is derived as
(11)PZ≤zq=q,0<q<1,1−exp−λexpβzq−γzq2=q.By solving the above equation, we getzq as follows:
(12)zq=12βlnln1/1−qλ±lnln1/1−qλ2+4βγ.Since quantilezq is positive, then we get zq as follows:
(13)zq=12βlnln1/1−qλ+lnln1/1−qλ2+4βγ.The median of RFWE can be obtained from Equation (13) by takingq=1/2. That is,
(14)Median=12βlnln1/1−1/2λ+lnln1/1−1/2λ2+4βγ.Now, to generate data from the cdf of RFWE, first generate the random numbers from standard uniform distribution then transform the random numbers into inverse cdf function. The inverse cdf function of RFWE can be written as(15)GRFWEz;β,γ,λ=u,where U is a standard uniform random variable; from (8), we get
(16)1−exp−λexpβz−γz2=u.In Equation (13), if we replace q by u, we get the inverse function
(17)zu=12βlnln1/1−uλ+lnln1/1−uλ2+4βγ.


## 3.2. The Moments

Moments are very important for the statistical analysis to study the average, variation, skewness, and kurtosis.Therth moment of RFWE distribution is introduced by the following theorem.Theorem 1.
Therth moment of a random variable Z∼RFWEΘ, where Θ=β,γ,λ is given by
(18)μr′=∑i=0∞∑j=0∞−1i+jλi+1γjΓr−j−1i!j!βr−j2γ−2ji+1r−2j+1r−jr−j−1+4γβi+12.Proof.
Therth moment of the random variable Z with gz is defined as
(19)μr′=∫0∞zrgzdz.
By putting Equation (7) into Equation (19), we get
(20)μr′=2λ∫0∞zrβ+γz2exp2βz−γzexp−λexpβz−γz2dz,using the Taylor series
(21)exp−λx=∑i=0∞−1ixλii!.
Takex=expβz−γ/z2 in the above equation:
(22)exp−λexpβz−γz2=∑i=0∞−1iλiexpβz−γ/z2ii!,
Equation (20) becomes
(23)μr′=2∑i=0∞−1iλi+1i!∫0∞zrβ+γz2exp2βz−γzexpβz−γz2idz.
By solving the above equation, we obtain(24)μr′=2∑i=0∞−1iλi+1i!∫0∞zrβ+γz2exp2i+1βzexp−2i+1γzdz.
Using the series expansion ofexp−2i+1γ/z, we obtain
(25)μr′=2∑i=0∞∑j=0∞−1i+j2ji+1jγjλi+1i!j!∫0∞zr−jβ+γz2exp2i+1βzdz,by using the gamma function in the form
(26)Γα=βα∫0∞exptβtα−1dt,α,β>0.
Finally, we get therth moment of RFWE in the form
(27)μr′=∑i=0∞∑j=0∞−1i+jλi+1γjΓr−j−1i!j!βr−j2γ−2ji+1r−2j+1r−jr−j−1+4γβi+12.


## 3.3. Moment Generating Function

In this section, we derive the moment generating function of RFWE distribution.Theorem 2.
The moment generating functionMZt of a random variable Z, that is, Z∼RFWEΘ, where Θ=β,γ,λ is given by
(28)Mzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.Proof.
The moment generating functionMZt is defined as
(29)MZt=∫0∞exptzgzdz.
Using series expansion ofetz, we obtain
(30)MZt=∑m=0∞tmm!∫0∞zmgzdz=∑m=0∞tmm!μm′.
Substituting from Equation (27) into Equation (30), we get
(31)Mzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjtmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.
Hence, the proof.


## 3.4. Characteristic Function

The characteristic function of RFWE distribution can be given as follows:(32)ϕzt=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjitmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjitmΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


## 3.5. Probability Generating Function

The probability generating function of RFWE distribution can be given as follows:(33)Gα=Eαz,Gα=∫0∞expzlnαgzdz,Gα=2λ∫0∞expzlnαβ+γz2exp2βz−γzexp−λexpβz−γz2dz,Gα=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnmαΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnmαΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


## 3.6. Factorial Moment Generating Function

The factorial moment generating function of RFWE distribution can be given as follows:(34)H0δ=E1+δz,H0δ=∫0∞expzln1+δgz;β,γ,λdz,H0δ=∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnm1+δΓm−j−1i!j!m!βm−ji+1m−2j+12m−2jm−jm−j−1+∑i=0∞∑j=0∞∑m=0∞−1i+jλi+1γjlnm1+δΓm−j−1i!j!m!βm−ji+1m−2j+12m−2j4γβi+12.


## 4. Renyi Entropy

Entropy is widely used in physics; Renyi entropy is one of the famous measures introduced by Renyi [12]. It is used to measure the unpredictability of a distribution, high entropy exhibits more uncertainty, and low entropy shows minimum uncertainty or another word more informative. Zero value of entropy indicates surety of completely certain information.LetZ~RFWEβ,γ,λ; then, the corresponding Renyi entropy can be obtained as
(35)Hρ=11−ρlog∫0∞gz;β,γ,λρdz,Hρ=11−ρlog∑i=0ρ∑k=0∞∑j=0∞ρi−1kγi2λρ+kρkβρ−i+j2ρ+kjj!k!Γi−j+1γ2ρ+ki−j+1.


## 5. Classical Estimation

In this section, we discussed the estimation of RFWE parameters by using the method of maximum likelihood and asymptotic confidence bounds.


### 5.1. Maximum Likelihood Estimators (MLEs)

LetZ1,Z2,⋯,Zk be independent and identical random sample of size k from RFWE β,γ,λ with observed values z1,z2,⋯,zk; then the likelihood function can be written as
(36)L=∏i=1kgzi;β,γ,λ.Substituting from Equation (7) into Equation (36), we get
(37)L=∏i=1kβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2.The log-likelihood function can be written as(38)lnL=kln2+klnλ+∑i=1klnβ+γzi2+2β∑i=1kkzi−2γ∑i=1k1zi−2λ∑i=1kexpβzi−γzi.The MLEs of the parameters are obtained by differentiating the log-likelihood function with respect to the parametersβ,γ,λ and setting the result to zero.
(39)∂lnL∂β=∑i=1kzi2βzi2+γ+2∑i=1kzi−2λ∑i=1kziexpβzi−γzi,∂lnL∂γ=∑i=1k1βzi2+γ−2∑i=1k1zi+2λ∑i=1kexpβzi−γ/zizi,∂lnL∂λ=kλ−2∑i=1kexpβzi−γzi.There is no closed-form solution to the above equations. Therefore, R software and Mathematica are used to get the numerical solutions.


### 5.2. Asymptotic Confidence Bounds

Now, we obtain the asymptotic confidence interval of the unknown parametersβ,γ,λ. We assume that the MLEs (β,γ,λ) are approximately multivariate normal with mean (β,γ,λ) and covariance matrix I0−1, where I0−1 is the inverse of the observed information matrix defined as
(40)I0−1=−∂2lnL∂β2∂2lnL∂β∂γ∂2lnL∂β∂λ∂2lnL∂γ∂β∂2lnL∂γ2∂2lnL∂γ∂λ∂2lnL∂λ∂β∂2lnL∂λ∂γ∂2lnL∂λ2−1,varβ^covβ^,γ^covβ^,λ^covγ^,β^varγ^covγ^,λ^covλ^,β^covλ^,γ^varλ^.The second partial derivatives included inI0−1 are given as follows:
(41)∂2lnL∂λ2=−kλ2,∂2lnL∂λ∂γ=2∑i=1kexpβzi−γ/zizi,∂2lnL∂β2=∑i=1kzi4βzi2+γ2−2λ∑i=1kzi2expβzi−γzi,∂2lnL∂γ2=−∑i=1k1βzi2+γ2−2λ∑i=1kexpβzi−γ/zizi2,∂2lnL∂β∂γ=−∑i=1kzi2βzi2+γ2+λ∑i=1kexpβzi−γzi,∂2lnL∂λ∂β=−2∑i=1kziexpβzi−γzi.The above expressions are used to derive the1−δ100% confidence intervals for the parameters β,γ, and λ as in the following forms:
(42)β^±zδ/2varβ^,γ^±zδ/2varγ^,λ^±zδ/2varλ^,where zδ/2 is the upper δ/2th percentile of the standard normal distribution.


## 5.1. Maximum Likelihood Estimators (MLEs)

LetZ1,Z2,⋯,Zk be independent and identical random sample of size k from RFWE β,γ,λ with observed values z1,z2,⋯,zk; then the likelihood function can be written as
(36)L=∏i=1kgzi;β,γ,λ.Substituting from Equation (7) into Equation (36), we get
(37)L=∏i=1kβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2.The log-likelihood function can be written as(38)lnL=kln2+klnλ+∑i=1klnβ+γzi2+2β∑i=1kkzi−2γ∑i=1k1zi−2λ∑i=1kexpβzi−γzi.The MLEs of the parameters are obtained by differentiating the log-likelihood function with respect to the parametersβ,γ,λ and setting the result to zero.
(39)∂lnL∂β=∑i=1kzi2βzi2+γ+2∑i=1kzi−2λ∑i=1kziexpβzi−γzi,∂lnL∂γ=∑i=1k1βzi2+γ−2∑i=1k1zi+2λ∑i=1kexpβzi−γ/zizi,∂lnL∂λ=kλ−2∑i=1kexpβzi−γzi.There is no closed-form solution to the above equations. Therefore, R software and Mathematica are used to get the numerical solutions.


## 5.2. Asymptotic Confidence Bounds

Now, we obtain the asymptotic confidence interval of the unknown parametersβ,γ,λ. We assume that the MLEs (β,γ,λ) are approximately multivariate normal with mean (β,γ,λ) and covariance matrix I0−1, where I0−1 is the inverse of the observed information matrix defined as
(40)I0−1=−∂2lnL∂β2∂2lnL∂β∂γ∂2lnL∂β∂λ∂2lnL∂γ∂β∂2lnL∂γ2∂2lnL∂γ∂λ∂2lnL∂λ∂β∂2lnL∂λ∂γ∂2lnL∂λ2−1,varβ^covβ^,γ^covβ^,λ^covγ^,β^varγ^covγ^,λ^covλ^,β^covλ^,γ^varλ^.The second partial derivatives included inI0−1 are given as follows:
(41)∂2lnL∂λ2=−kλ2,∂2lnL∂λ∂γ=2∑i=1kexpβzi−γ/zizi,∂2lnL∂β2=∑i=1kzi4βzi2+γ2−2λ∑i=1kzi2expβzi−γzi,∂2lnL∂γ2=−∑i=1k1βzi2+γ2−2λ∑i=1kexpβzi−γ/zizi2,∂2lnL∂β∂γ=−∑i=1kzi2βzi2+γ2+λ∑i=1kexpβzi−γzi,∂2lnL∂λ∂β=−2∑i=1kziexpβzi−γzi.The above expressions are used to derive the1−δ100% confidence intervals for the parameters β,γ, and λ as in the following forms:
(42)β^±zδ/2varβ^,γ^±zδ/2varγ^,λ^±zδ/2varλ^,where zδ/2 is the upper δ/2th percentile of the standard normal distribution.


## 6. Order Statistics

LetZ1:k,Z2:k,⋯,Zk:k denote the order statistics obtained from a random sample Z1,Z2,⋯,Zk taken from a continuous population with cdf Gz,Φ and pdf gz,Φ as follows:
(43)gi:kz,Φ=1Bi,n−i+1gz,ΦGz,Φi−11−Gz,Φk−i,where Bi,n−i+1 is the beta function.


### 6.1. PDF of Minimum, Median, and Maximum Order Statistics

In this subsection, we will consider the expression for the sample distribution of the minimum, median, and the maximum order statistics when a random sample of sizek is drawn from the RFWE β,γ,λ distribution. These pdfs can be obtained by solving Equation (43).The pdf of the minimum order statistics is as follows:(44)g1:kz=2λkβ+γz2exp2βz−γzexp−λexpβz−γz2k.The pdf of the median order statistics is as follows:(45)gm+1:kz~=2m+1!m!m!2λβ+γz2exp2βz−γzexp−λexpβz−γz2m+1×1−exp−λexpβz−γz2m.Finally, the pdf of the maximum order statistics is as follows:(46)gk:kz=2kλβ+γz2exp2βz−γzexp−λexpβz−γz2×1−exp−λexpβz−γz2k−1.


### 6.2. Joint PDF of the Minimum and Maximum Order Statistics

In this subsection, we will consider the expression for the joint pdf of the minimum and the maximum order statistics when a random sample of sizek is drawn from the RFWE β,γ,λ distribution. These pdfs can be obtained by solving the following equation:
(47)gi:j:kzi,zj=CGzii−1Gzj−Gzij−i−11−Gzjk−jgzigzj,where
(48)C=k!i−1!j−i−1!k−j!.The joint pdf of the minimum and maximum, that is,ith- and jth-order statistics from RFWE distribution, is
(49)gi:j:kzi,zj=4Cλ21−exp−λexpβzi−γzi2i−1×exp−λexpβzi−γzi2−exp−λexpβzj−γzj2j−i−1×exp−λexpβzj−γzj2k−jβ+γzi2exp2βzi−γzi×exp−λexpβzi−γzi2β+γzj2.For special case, let us supposei=1andj=k; the joint pdf of minimum and maximum order statistics is as follows:
(50)g1:k:kz1,zk=kk−14λ2exp−λexpβz1−γz12−exp−λexpβzk−γzk2k−2×β+γz12exp2βz1−γz1exp−λexpβz1−γz12×β+γzk2exp2βzk−γzkexp−λexpβzk−γzk2×exp2βzj−γzjexp−λexpβzj−γzj2.


## 6.1. PDF of Minimum, Median, and Maximum Order Statistics

In this subsection, we will consider the expression for the sample distribution of the minimum, median, and the maximum order statistics when a random sample of sizek is drawn from the RFWE β,γ,λ distribution. These pdfs can be obtained by solving Equation (43).The pdf of the minimum order statistics is as follows:(44)g1:kz=2λkβ+γz2exp2βz−γzexp−λexpβz−γz2k.The pdf of the median order statistics is as follows:(45)gm+1:kz~=2m+1!m!m!2λβ+γz2exp2βz−γzexp−λexpβz−γz2m+1×1−exp−λexpβz−γz2m.Finally, the pdf of the maximum order statistics is as follows:(46)gk:kz=2kλβ+γz2exp2βz−γzexp−λexpβz−γz2×1−exp−λexpβz−γz2k−1.


## 6.2. Joint PDF of the Minimum and Maximum Order Statistics

In this subsection, we will consider the expression for the joint pdf of the minimum and the maximum order statistics when a random sample of sizek is drawn from the RFWE β,γ,λ distribution. These pdfs can be obtained by solving the following equation:
(47)gi:j:kzi,zj=CGzii−1Gzj−Gzij−i−11−Gzjk−jgzigzj,where
(48)C=k!i−1!j−i−1!k−j!.The joint pdf of the minimum and maximum, that is,ith- and jth-order statistics from RFWE distribution, is
(49)gi:j:kzi,zj=4Cλ21−exp−λexpβzi−γzi2i−1×exp−λexpβzi−γzi2−exp−λexpβzj−γzj2j−i−1×exp−λexpβzj−γzj2k−jβ+γzi2exp2βzi−γzi×exp−λexpβzi−γzi2β+γzj2.For special case, let us supposei=1andj=k; the joint pdf of minimum and maximum order statistics is as follows:
(50)g1:k:kz1,zk=kk−14λ2exp−λexpβz1−γz12−exp−λexpβzk−γzk2k−2×β+γz12exp2βz1−γz1exp−λexpβz1−γz12×β+γzk2exp2βzk−γzkexp−λexpβzk−γzk2×exp2βzj−γzjexp−λexpβzj−γzj2.


## 7. Data Analysis

In this section, we use a real data set to show that the RFWE distribution can be a better model, compared with many known distributions such as the FWE distribution, Weibull distribution, exponential distribution, and Rayleigh distribution. Consider that the data obtained by Khan and Jan [13] represents the failure time of electronic devices obtained from power-line voltage spikes during electronic storms. The times are 2.75, 0.13, 1.47, 0.23, 1.81, 0.30, 0.65, 0.10, 3.00, 1.73, 1.06, 3.00, 3.00, 2.12, 3.00, 3.00, 3.00, 0.02, 2.61, 2.93, 0.88, 2.47, 0.28, 1.43, 3.00, 0.23, 3.00, 0.80, 2.45, and 2.66.The MLEs of the unknown parametersβ,γ,λ for the different distributions are given in Table 1. We considered Anderson–Darling (AD) test statistic, Cramer-von-Misses (CM) test statistic, and Kolmogorov–Smirnov (KS) test statistic. Hannan-Quinn information criterion (HQIC), Bayesian information criterion (BIC), Akaike’s Information Criterion (AIC), and Consistent Akaike’s Information Criterion (CAIC) as investigative measures are given in Table 2.Table 1
Goodness of fit results using AD, CM, and KS.
DistributionsMLE’sADCMKSRFWEβ^=0.45, γ^=0.04, λ^=0.1471.140.170.19FEWβ^=0.328, γ^=0.1582.040.320.39Weibullα^=1.264, γ^=1.8201.820.300.21Exponentialλ^=0.5641.900.320.21Rayleighσ^=1.4851.640.260.21Table 2
Goodness of fit results using AIC, BIC, CAIC, and HQIC.
DistributionAICBICCAICHQICRFWE82.7586.9583.6784.09FEW111.31114.11111.75112.20Weibull96.31799.11996.76197.213Exponential96.27097.67196.41296.718Rayleigh103.76105.16103.90104.21


### 7.1. Discussion

From the above tables, the test statistics above mentioned showed smaller values for the RFWE distribution as compared with the rest of the distributions that indicate better performance of the model.


## 7.1. Discussion

From the above tables, the test statistics above mentioned showed smaller values for the RFWE distribution as compared with the rest of the distributions that indicate better performance of the model.


## 8. Simulations

The accuracy and the performance in estimating the model parameters with different sample sizes are investigated by using Equation (17) to generate the data from the RFWE distribution. We take n=20,60,and200, and the true values of the parameters are given in Tables 3 and 4; for each n, we simulate 100 random data sets from the RFWE distribution and for different values of the parameters, average bias (bias), mean square errors (MSE), estimated mean (Est.), and variance (Var) are calculated for different sample sizes.Table 3
True value of parameters with MSE and bias of RFWEβ,γ,λ.
β,γ,λnBias (β^)Bias (γ^)Bias (λ^)MSE (β^)MSE (γ^)MSE (λ^)(0.008, 0.009, 0.3)200.0101320.00598320.245980.00131120.005238680.125635600.0056300.00434960.2370240.00074320.00550310.0947542000.000846-0.0040230.2052160.00000080.000016890.042515(0.008, 0.008, 0.3)200.0056790.00377370.2258850.00063670.003033060.080190600.005270-0.0025860.2152270.00057650.000152900.0703372000.002294-0.0035480.2163390.00022320.000015520.053182(0.0008, 0.0009, 0.3)200.0510850.06586860.6500230.00545120.03455810.6694600.0482040.02344070.4497330.00441140.001260660.3595072000.0298750.02268810.3196490.00183330.001199960.185328(0.0008, 0.0009, 0.4)200.0550870.04414440.4812220.00565200.01505030.457649600.0453610.02100230.2595040.00361510.001042890.179422000.0356120.01234260.1274860.00245610.000366990.015043Table 4
True value of parameters with mean (Est.) and variance (Var).
R-FWExβ,γ,λnβγλEst.VarEst.VarEst.VarR-FWEx (0.008, 0.009, 0.3)200.0180.00120.01490.00520.5450.0657600.0130.00070.01330.00550.5370.03892000.008<0.00010.004<0.00010.5050.0004R-FWEx (0.008, 0.008, 0.3)200.01360.00060.01170.00300.5250.0294600.01320.00050.0054<0.00010.51520.02422000.01020.00020.0044<0.00010.51630.0064R-FWEx (0.0008, 0.0009, 0.3)200.05180.00280.0660.03050.95000.2493600.0490.00210.02430.00070.74970.15882000.03060.00090.02350.00060.6190.0839R-FWEx (0.0008, 0.0009, 0.4)200.05580.00260.04500.01320.88120.2283600.0460.001570.02190.00060.65950.11322000.18570.00970.01660.000020.95010.14413


### 8.1. Discussion

As we increase the sample size, it is clearly noticed from the above tables that the bias and MSE tend to decrease, Est. tends towards the true parameter values, and Var also tends to decrease.


## 8.1. Discussion

As we increase the sample size, it is clearly noticed from the above tables that the bias and MSE tend to decrease, Est. tends towards the true parameter values, and Var also tends to decrease.


## 9. Bayesian Analyses

In this section, we are going to estimate the parameters of the proposed distribution and that of exponential, Rayleigh, Weibull, and FWEx distributions under different prior distributions. The posterior variances are calculated, and the credible intervals are provided for each parameter. To compare our proposed distribution with the others considered, we calculate difference information criteria (DIC). Muhammad et al. [14] used the DIC, and their scheme is better with a minimum number of parameters.The likelihood function of RFWE distribution is(51)L=∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2.First, we consider the uninformative uniform prior distribution for the parameters involved in Equation (51) as
(52)pλ,β,γ∝1.The joint posterior distribution under uniform prior distribution given in Equation (52) is as follows:
(53)pλ,β,γz∝∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2.The constant of proportionality,C, is obtained as
(54)C=∫γ∫β∫λ∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dλdβdγ.The marginal posterior distribution of the parameter given data may be obtained as follows:(55)pγz=1C∫β∫λ∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dλdβ,0≤γ≤∞,(56)pλz=1C∫β∫γ∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dγdβ,0≤λ≤∞,(57)pβz=1C∫λ∫γ∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dγdλ,0≤β≤∞.As the posterior distribution given in Equations (55)–(57) are not analytically tractable, so we use MCMC simulations for the calculation of the Bayes estimates (BE) and other posterior results.Furthermore, we now consider an informative prior distribution, gamma distribution, for the parameters. The joint gamma prior is the simple product of the marginal gamma distribution and is given as(58)pλ,β,γ=b1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3,where, in (58), a1,b1, a2,b2, and a3,b3 are the hyperparameters of the prior distribution of λ,β, and γ, respectively. The joint posterior distribution under Equation (58) is as follows:
(59)pλ,β,γz=1C1b1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3×∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2,where C1 is known as the constant of proportionality and obtained as follows:
(60)C1=∫γ∫β∫λb1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3×∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dλdβdγ.The marginal posterior distribution of the parameter given data, under gamma prior distribution, is obtained by integrating Equation (59) as follows:
(61)pγz=1C1∫β∫λb1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3×∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dλdβ,0≤γ≤∞,pλz=1C1∫β∫γb1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3×∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dβdγ,0≤λ≤∞,pβz=1C1∫λ∫γb1a1Γa1b2a2Γa2b3a3Γa3λa1−1exp−λb1βa2−1exp−βb2γa3−1exp−γb3×∏i=1k2λβ+γzi2exp2βzi−γziexp−λexpβzi−γzi2dγdλ.0≤β≤∞.These posterior distributions are summarized in Tables5 and 6. For numerical calculation, we set the hyperparameters as a1=2,b1=5, a1=3,b1=6, and a1=2,b1=4.Table 5
Posterior results under uniform prior distribution.
DistributionParameterBESD[LCL, UCL]DICRFWEλ0.16120.0733[0.0565, 0.3393]219.5698β0.44380.0791[0.3038, 0.6088]γ0.04300.0201[0.0123, 0.0882]Exponentialλ0.58630.1030[0.4053, 0.8038]248.0342Rayleighθ0.23410.04074[0.1594, 0.3162]266.7312Weibullϕ1.26550.2028[0.8922, 1.6879]256.7191α1.96620.2988[1.4206, 2.6207]FWEβ0.31930.0456[0.2334, 0.4115]293.6861γ0.17930.0515[0.0903, 0.2922]Table 6
Posterior results under gamma prior distribution.
DistributionParameterBES.D[LCL, UCL]DICRFWEλ0.05810.0715[0.0771, 0.3508]205.1394β0.41000.0685[0.2871, 0.5583]γ0.05810.0213[0.0239, 0.1065]Exponentialλ0.54840.0966[0.3734, 0.7625]238.2889Rayleighθ0.23320.0409[0.1576, 0.3194]253.4404Weibullϕ1.04800.1702[0.7487, 1.4153]269.9739α1.54620.2596[1.0749, 2.0872]FWEβ0.32840.0459[0.2380, 0.4164]266.0848γ0.17240.0545[0.0811, 0.2918]


### 9.1. Discussion

We noticed a reduction in the posterior standard deviations (SDs) while changing the prior from noninformative to informative prior distribution in Equation (58), and consequently, the credible intervals, that is, lower credible interval (LCL) and upper credible interval (UCL), become narrower for the all parameters in all of the distributions. Further, the DIC is minimum in gamma prior.So we conclude that the gamma prior is more suitable than the uniform prior. While using the uniform prior distribution for our proposed model, RFWE produced a minimum DIC as compared to the others considered, namely, exponential, Weibull, Rayleigh, and FWE distributions.In fact, this value of 219.5698 is better than the DICs produced by other models under uninformative prior distribution. A decrease in the DIC is observed when we considered gamma prior distribution; and this DIC, 205.1394, is the smallest among all.


## 9.1. Discussion

We noticed a reduction in the posterior standard deviations (SDs) while changing the prior from noninformative to informative prior distribution in Equation (58), and consequently, the credible intervals, that is, lower credible interval (LCL) and upper credible interval (UCL), become narrower for the all parameters in all of the distributions. Further, the DIC is minimum in gamma prior.So we conclude that the gamma prior is more suitable than the uniform prior. While using the uniform prior distribution for our proposed model, RFWE produced a minimum DIC as compared to the others considered, namely, exponential, Weibull, Rayleigh, and FWE distributions.In fact, this value of 219.5698 is better than the DICs produced by other models under uninformative prior distribution. A decrease in the DIC is observed when we considered gamma prior distribution; and this DIC, 205.1394, is the smallest among all.


## 10. Conclusions

We propose a new three-parameter distribution, based on the idea which is to add parameters to the flexible Weibull extension distribution; this new distribution is called the Rayleigh flexible Weibull extension (RFWE) distribution. Its statistical properties are studied. We use the maximum likelihood method for estimating the parameters of the distribution. Finally, the advantage of the RFWE distribution is concluded by an application using real data set and a simulation study is conducted. From the simulation study, as the sample size increases, the bias, MSE, and variance tend to decrease, and the estimated means tend towards the true parameter value.Moreover, it is shown that the RFWE distribution fits better than existing known distributions in the classical approach, while our proposed, RFWE, distribution has minimum DIC under uniform and gamma priors under the Bayesian paradigm. This showed that our proposed model is better than the other consideration. Furthermore, our model performs better when informative gamma prior distributions were considered, better in terms of narrower credible intervals and minimum posterior standard variations. In this study, we used a single real data set and two priors; in the future, one can consider the other uninformative prior, like Jeffrey’s prior, and in informative priors like normal, lognormal can be used for comparison purposes.



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