# An Extended Cosine Generalized Family of Distributions for Reliability Modeling: Characteristics and Applications with Simulation Study

Authors: Zafar Mahmood; Taghreed M Jawa; Neveen Sayed-Ahmed; E M Khalil; Abdisalam Hassan Muse; Ahlam H. Tolba
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

An extension of the cosine generalized family is presented in this paper by using the cosine trigonometric function and method of parameter induction concurrently. Prominent characteristics of the proposed family along with useful results are extracted. Moreover, two new subfamilies and several special models are also deduced. A four-parameter model called an Extended Cosine Weibull(ECW) with its mathematical properties is studied deeply. Graphical study reveals that the new model adopts right- and left-skewed, symmetrical, and reversed-J density shapes, while all possible monotone and nonmonotone shapes are exhibited by the hazard rate function. The maximum likelihood technique is exercised for parametric estimation, while estimation performance is accessed via Monte Carlo simulation study graphically and numerically. The superiority of the presented model over several outstanding and competing models is confirmed via three reliability and survival dataset applications.


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## Body

## 1. Introduction with Motivations

This twenty-first century started with setting up and broadening new graphing and analytical instruments for current modern statistics. One novel development is the introduction of extended and generalized distributions utilizing classical one and further certified availing real and lifetime data accessible from easy to complex phenomenon. In this way, improved estimation of model parameters and model goodness of fit is achieved that is effectively applicable in environmental sciences, biological sciences, industry, agricultural biotechnology, engineering, economics, forestry, and many others.An immense research work about the theory and applications of statistical distributions exists in the literature. The first reason is the thrust of statisticians to develop novel and flexible models possessing significant mathematical and graphical characteristics. The struggle, challenging, and endless work of statisticians bore fruit, and many modified, extended, and generalized families of distributions are introduced; for more information, see [2–4]. The literature review explores the second reason that simple and nongeneralized models provide an inadequate fit compared to extended and generalized models, especially in real-life situations. The third and very important fact is that the data behave in a more complex way than what is commonly expected in many disciplines.For a brief review, the reader is referred to the memorable contribution [5] about the transformed- (T-) transformer (X) mechanism of developing new families of statistical distributions, extended work [6] regarding the McDonald-G family, and pivotal work on exponentiated generalized families [7]. Table 1 presents a review of trigonometric function-based models chronologically.Table 1
Chronological review on trigonometric functions based on G families and distributions.
S. no.Introducer(s) (year)Model based on trigonometric function1Raab and Green (1961)Derived a cosine distribution [8]2Nadarajah and Kotz (2006)Derived the beta trigonometric distribution [9]3Chakraborty et al. (2012)Explored the sin-skew logistic distribution [10]4Souza (2015)Suggested new trigonometric classes of probabilistic distributions [1]5Chesneau et al. (2019)Explored the cosine sine (CS) distribution [11]6Souza et al. (2019)Derived mathematical characteristics for cos-G class [12]7Mahmood et al. (2019)Presented a new sine-G family of distributions [13]8Chesneau et al. (2020)Proposed the cosine geometric distribution [14]9Chesneau et al. (2020)Deduced sine Kumaraswamy-G family of distribution [15]10Al-Babtain et al. (2020)Introduced sine Topp–Leone-G family [16]11Ali (2021)Worked on sine power Lomax model [17]12Alkhairy et al. (2021)Introduced the arctangent-X family of distributions [18]13Nagarjuna et al. (2021)Worked on the accuracy of the sine power Lomax model [19]14Nagarjuna et al. (2021)Presented Kumaraswamy generalized power Lomax distribution [20]15Ahmed et al. (2021)Presented generalized power Akshaya distribution with applications [21]16Muse et al. (2021)Introduced the Tan log-logistic distribution [22]17Almetwally (2021)Presented odd Weibull inverse Topp–Leone distribution [23]18Hassan et al. (2021)Presented Kumaraswamy inverted Topp–Leone distribution [24]19Almetwally et al. (2021)Presented Marshall–Olkin alpha Power-X distribution [25–27]20Almongy et al. (2021)Presented extended odd Weibull Rayleigh [28]21Al-Babtain et al. (2021)Presented the flexible burr-XG family [29]A concise summary of the cosine-G family is as follows: letGx (cumulative distribution function (cdf) of baseline), then new cdf, probability density function (pdf), and hazard rate function (hrf), respectively, be Fx=1−cosπ/2Gx, fx=π/2gxsinπ/2Gx, and hx=π/2gxtanπ/2Gx,x∈ℝ. Lehmann (1953) presented Lehmann alternative-1 (shortly LA-1) method, which is a famous method of constructing exponentiated families (EF) of distributions and is also defined by Gupta et al. (1998) as Fx=Gxα.Moreover, the key aspects that become foundations for this study are as follows: (1) A large variety of nontrigonometric G families are added in the statistical literature, only utilizing the algebraic generalizers/generators of distributionsWGx while disregarding trigonometric ones (trigonometric generalizers). (2) The trend in modeling directional and proportional data prompted applied researchers to construct trigonometric function-based statistical models that can be more efficient and effective. (3) The algebraic and trigonometric functions mixed generalizers are not developed and consumed yet.Influenced by foundation points and the produced results regarding accuracy, flexibility, and goodness of fit (GoF), the key motivations for introducing an extended cosine-G family are as follows:(i)
Pursuing the enthusiasm of the cosine-G family and LA-1 mechanism of constructing new G families simultaneously, we develop an extended version of the cosine-G family.(ii)
The proposed family has a lot of advantages, including simplicity, explicit expressions for central functions, free from nonidentifiability issue, and overparametrization.(iii)
The extended cdf can boost flexibility, accuracy, and GoF due to the injection of shape parameters, thus resulting in new flexible and efficient models.(iv)
Adopting the suggested extension, any G class or model can be readily reverted in a subsequent version, which will be trigonometric and doubly exponentiated.(v)
The existing literature attests that nongeneralized, nonexponentiated, and nonextended models provide inadequate GoF.(vi)
The produced extended models are efficient in modeling with monotonic and nonmonotonic hazard functions.Moreover, recent remarkable G families are presented in Table2 in connection with the adopted methodology.Table 2
Some recent G families of distributions.
Introducer(s)Proposed G familyAdopted methodologyHamdani et al. (2021)Type I quasi-Lambert familyLambert function [30]Altun et al. (2021)Additive odd-G familyAdditive model structure [31]Aljarrah et al. (2021)Generator of distributionsQuantile function [32]Salahuddin et al. (2021)New familyBetter GoF and flexibility [33]Korkamaz et al. (2021)New classHjorth’s IDB generator [34]Cordeiro et al. (2020)XGamma familyParameter induction [35]Korkamaz et al. (2019)TL Gen.Odd log-Logistic familyOdd generator [36]Alzaatreh et al. (2019)G-classesT-X methodology [5]Bhatti et al. (2019)On burr III MO familyHazard rate function [37]Korkamaz et al. (2018)New G-classesExtended distribution [35]Alizadeh et al. (2019)Another odd log-logistic logarithmic-GOdd generator [38]Algarni et al. (2021)Type I half logistic Burr-XGExtended T-X methodology [39]The paper layout is as follows: after the introduction with motivations in Section1, an extended cosine-G family with general mathematical properties is presented in Section 2. A special member (ECW) is briefly studied in Section 3. Parametric estimation, simulation study, and applications on two reliability datasets are performed by taking (ECW) as a statistical model in Section 4. Concluding remarks are stated in Section 5.


## 2. The Proposed Family and Significant Characteristics

The construction procedure of the proposed family with significant characteristics is presented in this section.


### 2.1. Construction Procedure

The idea behind the proposed family is to apply the process of exponentiated families (EF) to the former cosine-G family. Indeed, we can write(1)Fx=1−cosπ2Hxβ,where Hx denotes the exponentiated (generalized using LA-1) cdf of Gxα and again Fx=Kxβ, where Kx=1−cosπ/2Hx.


### 2.2. Main Functions

The cdf and pdf of “an extended cosine-G” (shortly an ext cos-G), respectively, are(2)Fx=1−cosπ2Gxαβ,x∈ℝ,(3)fx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxαβ−1.


### 2.3. Subfamilies and New Members of an Ext Cos-G Family

This proposed family possesses three G families (two new and one existing) as subfamilies, which are presented in Table3 and are deduced just by specifying the parametric values.Table 3
Subfamilies of an ext cos-G family.
S. no.αβSubfamilies of distributionsComments11—Generalized cosine-G family (using LA-1 technique)New2—1Cosine exponentiated-G familyNew311Cosine-G family [1]ExistingIn Table4, eight new members are introduced, for example, utilizing the famous statistical distributions on all possible intervals.Table 4
Members of an ext cos-G family and their cdfs.
Baseline modelSupportNew cdfFxParametersTopp–Leone0,11−cosπ/2x2−xαβα,βUniform0,θ1−cosπ/2x/θαβθ,α,βExponential0,∞1−cosπ/21−e−λxαβλ,α,βWeibull0,∞1−cosπ/21−e−λxσαβσ,λ,α,βBurr XII0,∞1−cosπ/21−1+x/sc−kαβs,c,k,α,βLogisticℝ1−cosπ/21+e−x−μ/s−1αβs,μ,α,βGumbelℝ1−cosπ/2e−e−x−μ/σαβσ,μ,α,βNormalℝ1−cosπ/2Φx−μ/σαβσ,μ,α,β


### 2.4. Useful Reliability Functions

The hazard rate function (hrf)hx, survival function (sf) Sx, reversed hazard rate function (rhrf) rx, cumulative hazard rate function (chrf) Hx, mills’ ratio mx, conditional reliability function G¯Gx,α,β|t, and elasticity ex are, respectively, given under(4)hx=π/2αβgxGxα−1sinπ/2Gxα1−cosπ/2Gxαβ−11−1−cosπ/2Gxαβ,Sx=1−1−cosπ2Gxαβ,rx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxα−1,Hx=−ln1−1−cosπ2Gxαβ,mx=1−1−cosπ/2Gxαβπ/2αβgxGxα−1sinπ/2Gxα1−cosπ/2Gxαβ−1,G¯Gx,α,β|t=1−1−cosπ/2Gx+tαβ1−1−cosπ/2Gtαβ,ex=∂∂lnGxln1−cosπ2Gxαβ.


### 2.5. Quantile Function and Quantile Density Function

An additional property ofX is quantile function (qf), which is derived from the direct inverter (2) as(5)QGv=G−12πcos−11−u1/β1/α,where u has a uniform U0,1 distribution, while the quantile density function (qdf) is(6)qdf=2u1−β/β2/πcos−11−u1/β1−α/απαβ1−1−u1/β2.


### 2.6. Analytical Points of the Density and Hazard Functions

The solutionx0 of the nonlinear equation f′x0=0 presents the analytical points of density, which are(7)12παβGx0α−1g′x0sin12πGx0α1−cos12πGx0αβ−1+14π2α2βGx02α−2gx02cos12πGx0α1−cos12πGx0αβ−1+14π2α2β−1βGx02α−2gx02sin212πGx0α1−cos12πGx0αβ−2+12πα−1αβGx0α−2gx02sin12πGx0α1−cos12πGx0αβ−1=0.Similarly, the analytical points of the hazard function are obtained by the solutionx∗ of h′x∗=0.(8)12παβGx∗α−1g′x∗sin12πGx∗α1−cos12πGx∗αβ−1+14π2α2βGx∗2α−2gx∗2cos12πGx∗α1−cos12πGx∗αβ−1+14π2α2β−1βGx∗2α−2gx2sin212πGx∗α1−cos12πGx∗αβ−2+12πα−1αβGx∗α−2gx∗2sin12πGx∗α1−cos12πGx∗αβ−1−12παβGx∗α−1gx∗sin12πGx∗α1−cos12πGx∗αβ−1=0.


### 2.7. Useful Expansions

It is a good idea to have a distribution function power series representation in terms of the baseline distribution’s positive integer powers. Using the series expansions for the cosine function,Fx can be expressed as a linear sum of power cdfs. For example, θ is an integer then Gxθ. Thus, the cdf (2) allows the expansion(9)Fx=1−cosπ2Gxαβ=∑i=0∞−1iβicosπ2Gxαi.Via Mathematica 11.0,cosxi=∑j=0∞bjix2j, where b0i=1 , b1i=−i/2 , b2i=i3i−2/24, and so on. Thus,(10)cosπ2Gxαi=∑k=0∞bkiπ2Gxα2k.Finally, the linear representation for cdf is(11)Fx=∑k=0∞Wi,kH2kαx,where H2kαx is the exponentiated cdf with 2kα as power parameter and(12)Wi,k=∑i=0∞−1ibkiβiπ22k.Similarly, the expansion for (3) is(13)fx=∑k=0∞vi,kh2kαx,where h2kαx is the exponentiated density with 2kα as power parameter and(14)vi,k=∑i=0∞−1ibkiβiπ22k,b0i=1,b1i=−i2,b2i=i3i−224,etc.And(15)hα2kx=2kαgxGx2kα−1.Equation (13) proves that the new density is a true linear combination of exp-G densities. Therefore, utilizing those exp-G properties, various properties of the new family can be extracted easily.


### 2.8. Moments, Incomplete Moments, and Moment Generating Function

Ther-th moment of X is expressed as μr′=EXr=∫0∞xrfxdx. By using (13) and assuming that all the sum and integral terms exist, we get(16)μr′=∑k=0∞vi,k2kα∫0∞xrGx2kα−1gxdx=∑k=0∞vi,kEYkr,where Yk expresses the exp-G distribution with 2kα as power parameter.They-th incomplete moment of X is(17)Ixy=∑k=0∞vi,kEIkr,where Ikry=∫0yxrhα2kxdx.The corresponding moment generating function (mgf) ofX is(18)Mt=∑k=0∞vi,k2kα∫0∞etxgxGx2kα−1dx=∑k=0∞vi,kEetx.


### 2.9. Probability Weighted Moments (PWMs)

Ther,s-th PWM of X (for r≥1,s≥0) is properly defined by(19)ρr,s=EXrFXs=∫0∞xrFxsfxdx(Greenwood, Landwehr, Matalas, and Wallis, 1979). Then,(20)Fxs=1−cosπ2Gxαsβ.Substituting (3) and (20) into (19) and then applying binomial series expansion, PWM=ρr,q becomes(21)ρr,q=∑i=0∞−1iπ2αββs+1−1i∫−∞∞xrgxGxα−1sinπ2Gxαcosπ2Gxαidx.Insert(22)sinπ2Gxα=∑j=0∞−1j2−2j+1π2j+1Gxα2j+12j+1!.And(23)cosπ2Gxαi=∑k=0∞bkiπ2Gxα2k.In (21), we get(24)ρr,q=∑i,j,k=0∞αβckβs+1−1i−1i+j2−2j+1−1π2j+22j+1!∫−∞∞xrgxGx2αj+k+1−1dx,ρr,q=∑i,j,k=0∞Ui,j,k∫−∞∞xr2αj+k+1gxGx2αj+k+1−1dx,where(25)Ui,j,k=αβckβs+1−1i−1i+j2−2j+2π2j+22j+1!2αj+k+1.In case of generalized distributions, the PWMs are helpful to find the quantiles and estimators of concerned parameters. Moreover, low variance values with no severe biases are additional properties of PWMs. The obtained estimators through PWMs can be efficiently and favorably set side by side with relevant MLEs. Research may be conducted in the future on both methods for ext cos-G family and derived models.


### 2.10. Stochastic Ordering

Now we derive a useful result on the stochastic ordering using an ext cos-G family with common parametersα and β. See [40] for further details on stochastic ordering. In the following theorem, an ext cos-G family of distributions is ordered with respect to the strongest “likelihood ratio” ordering, which expresses the flexibility of two parameters for the ext cos-G family of distributions α,β.Theorem.
LetX follow an ext cos-G α1,β1 and Y follow an ext cos-G α2,β2. If α1=α2=α and   β1≥β2, then X⩽lrY.Proof.
We have the following likelihood ratio:(26)fXxfYx=1/2πα1β1Gxα1−1gxsin1/2πGxα11−cos1/2πGxα1β1−11/2πα2β2Gxα2−1gxsin1/2πGxα21−cos1/2πGxα2β2−1.
Puttingα1=α2=α, we get(27)fXxfYx=β1β21−cos12πGxαβ1−β2,ddxfXxfYx=β1β2β1−β21−cos12πGxαβ1−β2−112παGxα−1gxsin12πGxα.
Ifβ1<β2, then d/dxfXx/fYx<0, which completes the proof and implies that X⩽lrY and similarly X⩽lrY,X⩽hrY,X⩽mrlY,X⩽stY. This property is a widget used to observe the constitutional characteristics of complex stochastic structure.


### 2.11. Order Statistics

In a number of fields of statistical methods and theory, especially in applications of survival testing studies, order statistics naturally arise. For more details, please consult the book of [41]. Here, we presented the order statistics of an ext cos-G family of distributions. Suppose X1,…,Xn be a random sample from an ext cos-G family; then i-th order statistic density is(28)fi:nx=KfxFi−1x1−Fxn−i=K∑j=0n−i−1jn−ijfxFxj+i−1,where K=n!/i−1!n−i!. Pursuing similar algebraic developments as done in PWM, the above density can be written as(29)fi:nx=∑j,l,m,p=0∞wj,l,m,ph2αl+m+1x,whereh2αl+m+1x represents the exp-G density with 2αl+m+1 as power parameter (for j,l,m,p≥0)(30)wj,l,m,p=kαβcmn−ijβi+j−1p−1j+l+p2−2l+1π2l+12αl+m+1.And(31)h2αl+m+1x=2αl+m+1gxGx2αl+m+1−1.The main result of this subsection is (29).


### 2.12. Rényi Entropy

The Rényi entropy is formerly expressed as(32)Iδf=11−δlogIδ,where Iδf=∫ℜfδxdx,δ>0 and δ≠1.(33)fδx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxαβ−1δ,fδx=π2δαδβδgxδGxδα−1sinπ2Gxαδ1−cosπ2Gxαδβ−1.After applying the power series expansion, we get(34)1−cosπ2Gxαδβ−1=∑j=0∞−1j∑k=0∞δβ−1jCkGx2kα.Finally, the Rényi entropy becomes(35)Iδf=11−δlog∑j,k=0∞Dj,k,δ∫−∞∞gxδGxδ2k+α−1sinπ2Gxαδ,where Dj,k,δ=π/2δαδβδ−1jδβ−1jCk.


## 2.1. Construction Procedure

The idea behind the proposed family is to apply the process of exponentiated families (EF) to the former cosine-G family. Indeed, we can write(1)Fx=1−cosπ2Hxβ,where Hx denotes the exponentiated (generalized using LA-1) cdf of Gxα and again Fx=Kxβ, where Kx=1−cosπ/2Hx.


## 2.2. Main Functions

The cdf and pdf of “an extended cosine-G” (shortly an ext cos-G), respectively, are(2)Fx=1−cosπ2Gxαβ,x∈ℝ,(3)fx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxαβ−1.


## 2.3. Subfamilies and New Members of an Ext Cos-G Family

This proposed family possesses three G families (two new and one existing) as subfamilies, which are presented in Table3 and are deduced just by specifying the parametric values.Table 3
Subfamilies of an ext cos-G family.
S. no.αβSubfamilies of distributionsComments11—Generalized cosine-G family (using LA-1 technique)New2—1Cosine exponentiated-G familyNew311Cosine-G family [1]ExistingIn Table4, eight new members are introduced, for example, utilizing the famous statistical distributions on all possible intervals.Table 4
Members of an ext cos-G family and their cdfs.
Baseline modelSupportNew cdfFxParametersTopp–Leone0,11−cosπ/2x2−xαβα,βUniform0,θ1−cosπ/2x/θαβθ,α,βExponential0,∞1−cosπ/21−e−λxαβλ,α,βWeibull0,∞1−cosπ/21−e−λxσαβσ,λ,α,βBurr XII0,∞1−cosπ/21−1+x/sc−kαβs,c,k,α,βLogisticℝ1−cosπ/21+e−x−μ/s−1αβs,μ,α,βGumbelℝ1−cosπ/2e−e−x−μ/σαβσ,μ,α,βNormalℝ1−cosπ/2Φx−μ/σαβσ,μ,α,β


## 2.4. Useful Reliability Functions

The hazard rate function (hrf)hx, survival function (sf) Sx, reversed hazard rate function (rhrf) rx, cumulative hazard rate function (chrf) Hx, mills’ ratio mx, conditional reliability function G¯Gx,α,β|t, and elasticity ex are, respectively, given under(4)hx=π/2αβgxGxα−1sinπ/2Gxα1−cosπ/2Gxαβ−11−1−cosπ/2Gxαβ,Sx=1−1−cosπ2Gxαβ,rx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxα−1,Hx=−ln1−1−cosπ2Gxαβ,mx=1−1−cosπ/2Gxαβπ/2αβgxGxα−1sinπ/2Gxα1−cosπ/2Gxαβ−1,G¯Gx,α,β|t=1−1−cosπ/2Gx+tαβ1−1−cosπ/2Gtαβ,ex=∂∂lnGxln1−cosπ2Gxαβ.


## 2.5. Quantile Function and Quantile Density Function

An additional property ofX is quantile function (qf), which is derived from the direct inverter (2) as(5)QGv=G−12πcos−11−u1/β1/α,where u has a uniform U0,1 distribution, while the quantile density function (qdf) is(6)qdf=2u1−β/β2/πcos−11−u1/β1−α/απαβ1−1−u1/β2.


## 2.6. Analytical Points of the Density and Hazard Functions

The solutionx0 of the nonlinear equation f′x0=0 presents the analytical points of density, which are(7)12παβGx0α−1g′x0sin12πGx0α1−cos12πGx0αβ−1+14π2α2βGx02α−2gx02cos12πGx0α1−cos12πGx0αβ−1+14π2α2β−1βGx02α−2gx02sin212πGx0α1−cos12πGx0αβ−2+12πα−1αβGx0α−2gx02sin12πGx0α1−cos12πGx0αβ−1=0.Similarly, the analytical points of the hazard function are obtained by the solutionx∗ of h′x∗=0.(8)12παβGx∗α−1g′x∗sin12πGx∗α1−cos12πGx∗αβ−1+14π2α2βGx∗2α−2gx∗2cos12πGx∗α1−cos12πGx∗αβ−1+14π2α2β−1βGx∗2α−2gx2sin212πGx∗α1−cos12πGx∗αβ−2+12πα−1αβGx∗α−2gx∗2sin12πGx∗α1−cos12πGx∗αβ−1−12παβGx∗α−1gx∗sin12πGx∗α1−cos12πGx∗αβ−1=0.


## 2.7. Useful Expansions

It is a good idea to have a distribution function power series representation in terms of the baseline distribution’s positive integer powers. Using the series expansions for the cosine function,Fx can be expressed as a linear sum of power cdfs. For example, θ is an integer then Gxθ. Thus, the cdf (2) allows the expansion(9)Fx=1−cosπ2Gxαβ=∑i=0∞−1iβicosπ2Gxαi.Via Mathematica 11.0,cosxi=∑j=0∞bjix2j, where b0i=1 , b1i=−i/2 , b2i=i3i−2/24, and so on. Thus,(10)cosπ2Gxαi=∑k=0∞bkiπ2Gxα2k.Finally, the linear representation for cdf is(11)Fx=∑k=0∞Wi,kH2kαx,where H2kαx is the exponentiated cdf with 2kα as power parameter and(12)Wi,k=∑i=0∞−1ibkiβiπ22k.Similarly, the expansion for (3) is(13)fx=∑k=0∞vi,kh2kαx,where h2kαx is the exponentiated density with 2kα as power parameter and(14)vi,k=∑i=0∞−1ibkiβiπ22k,b0i=1,b1i=−i2,b2i=i3i−224,etc.And(15)hα2kx=2kαgxGx2kα−1.Equation (13) proves that the new density is a true linear combination of exp-G densities. Therefore, utilizing those exp-G properties, various properties of the new family can be extracted easily.


## 2.8. Moments, Incomplete Moments, and Moment Generating Function

Ther-th moment of X is expressed as μr′=EXr=∫0∞xrfxdx. By using (13) and assuming that all the sum and integral terms exist, we get(16)μr′=∑k=0∞vi,k2kα∫0∞xrGx2kα−1gxdx=∑k=0∞vi,kEYkr,where Yk expresses the exp-G distribution with 2kα as power parameter.They-th incomplete moment of X is(17)Ixy=∑k=0∞vi,kEIkr,where Ikry=∫0yxrhα2kxdx.The corresponding moment generating function (mgf) ofX is(18)Mt=∑k=0∞vi,k2kα∫0∞etxgxGx2kα−1dx=∑k=0∞vi,kEetx.


## 2.9. Probability Weighted Moments (PWMs)

Ther,s-th PWM of X (for r≥1,s≥0) is properly defined by(19)ρr,s=EXrFXs=∫0∞xrFxsfxdx(Greenwood, Landwehr, Matalas, and Wallis, 1979). Then,(20)Fxs=1−cosπ2Gxαsβ.Substituting (3) and (20) into (19) and then applying binomial series expansion, PWM=ρr,q becomes(21)ρr,q=∑i=0∞−1iπ2αββs+1−1i∫−∞∞xrgxGxα−1sinπ2Gxαcosπ2Gxαidx.Insert(22)sinπ2Gxα=∑j=0∞−1j2−2j+1π2j+1Gxα2j+12j+1!.And(23)cosπ2Gxαi=∑k=0∞bkiπ2Gxα2k.In (21), we get(24)ρr,q=∑i,j,k=0∞αβckβs+1−1i−1i+j2−2j+1−1π2j+22j+1!∫−∞∞xrgxGx2αj+k+1−1dx,ρr,q=∑i,j,k=0∞Ui,j,k∫−∞∞xr2αj+k+1gxGx2αj+k+1−1dx,where(25)Ui,j,k=αβckβs+1−1i−1i+j2−2j+2π2j+22j+1!2αj+k+1.In case of generalized distributions, the PWMs are helpful to find the quantiles and estimators of concerned parameters. Moreover, low variance values with no severe biases are additional properties of PWMs. The obtained estimators through PWMs can be efficiently and favorably set side by side with relevant MLEs. Research may be conducted in the future on both methods for ext cos-G family and derived models.


## 2.10. Stochastic Ordering

Now we derive a useful result on the stochastic ordering using an ext cos-G family with common parametersα and β. See [40] for further details on stochastic ordering. In the following theorem, an ext cos-G family of distributions is ordered with respect to the strongest “likelihood ratio” ordering, which expresses the flexibility of two parameters for the ext cos-G family of distributions α,β.Theorem.
LetX follow an ext cos-G α1,β1 and Y follow an ext cos-G α2,β2. If α1=α2=α and   β1≥β2, then X⩽lrY.Proof.
We have the following likelihood ratio:(26)fXxfYx=1/2πα1β1Gxα1−1gxsin1/2πGxα11−cos1/2πGxα1β1−11/2πα2β2Gxα2−1gxsin1/2πGxα21−cos1/2πGxα2β2−1.
Puttingα1=α2=α, we get(27)fXxfYx=β1β21−cos12πGxαβ1−β2,ddxfXxfYx=β1β2β1−β21−cos12πGxαβ1−β2−112παGxα−1gxsin12πGxα.
Ifβ1<β2, then d/dxfXx/fYx<0, which completes the proof and implies that X⩽lrY and similarly X⩽lrY,X⩽hrY,X⩽mrlY,X⩽stY. This property is a widget used to observe the constitutional characteristics of complex stochastic structure.


## 2.11. Order Statistics

In a number of fields of statistical methods and theory, especially in applications of survival testing studies, order statistics naturally arise. For more details, please consult the book of [41]. Here, we presented the order statistics of an ext cos-G family of distributions. Suppose X1,…,Xn be a random sample from an ext cos-G family; then i-th order statistic density is(28)fi:nx=KfxFi−1x1−Fxn−i=K∑j=0n−i−1jn−ijfxFxj+i−1,where K=n!/i−1!n−i!. Pursuing similar algebraic developments as done in PWM, the above density can be written as(29)fi:nx=∑j,l,m,p=0∞wj,l,m,ph2αl+m+1x,whereh2αl+m+1x represents the exp-G density with 2αl+m+1 as power parameter (for j,l,m,p≥0)(30)wj,l,m,p=kαβcmn−ijβi+j−1p−1j+l+p2−2l+1π2l+12αl+m+1.And(31)h2αl+m+1x=2αl+m+1gxGx2αl+m+1−1.The main result of this subsection is (29).


## 2.12. Rényi Entropy

The Rényi entropy is formerly expressed as(32)Iδf=11−δlogIδ,where Iδf=∫ℜfδxdx,δ>0 and δ≠1.(33)fδx=π2αβgxGxα−1sinπ2Gxα1−cosπ2Gxαβ−1δ,fδx=π2δαδβδgxδGxδα−1sinπ2Gxαδ1−cosπ2Gxαδβ−1.After applying the power series expansion, we get(34)1−cosπ2Gxαδβ−1=∑j=0∞−1j∑k=0∞δβ−1jCkGx2kα.Finally, the Rényi entropy becomes(35)Iδf=11−δlog∑j,k=0∞Dj,k,δ∫−∞∞gxδGxδ2k+α−1sinπ2Gxαδ,where Dj,k,δ=π/2δαδβδ−1jδβ−1jCk.


## 3. An Extended Cosine Weibull (ECW) Distribution

In this section, the ECW model with its properties is presented.


### 3.1. Main Functions of ECW with Graphs

Now, using Weibull distribution (as baseline) having the cdfGx=1−e−λxσ, λ,σ,x>0, we presented a new model having right skew and heavy tail characteristics and extended flexibility compared to the former Weibull model. Hereafter, this will be called an Extended Cosine Weibull (ECW) α,β,λ,σ distribution. By using (2), the main functions of the newly presented distribution are(36)Fx=1−cosπ21−e−λxσαβ,x>0,(37)fx=π2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ21−e−λxσα1−cosπ21−e−λxσαβ−1,(38)hx=π/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−11−1−cosπ/21−e−λxσαβ.Figure1 displays the exhibiting density shapes of the proposed model.Figure 1
Some labeled density curves of the ECW.Figure2 displays the exhibiting hazard rate function shapes of the proposed model.Figure 2
Some labeled hazard function curves of the ECW.


### 3.2. Useful Reliability Functions of ECW

The important reliability functions are(39)Sx=1−1−cosπ21−e−λxσαβ,rx=π/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−11−cosπ/21−e−λxσαβ,Hx=−lnSx=−ln1−1−cosπ21−e−λxσαβ,mx=1−1−cosπ/21−e−λxσαβπ/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−1,G¯x,α,β,λ,σ|t=G¯x+t,α,β,λ,σG¯t,α,β,λ,σ=1−1−cosπ/21−e−λx+tσαβ1−1−cosπ/21−e−λtσαβ,ex=∂∂lnx1−cosπ/21−e−λxσαβ.


### 3.3. Residual and Reverse Residual Life

In reliability theory and risk management, the residual life offers wide applications. LetX denote the lifetime of a unit and X≥0 with PX=1; then similar to residual lifetime random variable, Xt=t−X|X≤t, for a fixed t>0, named as “time since failure.”LetRtx denote the residual lifetime of the ECW random variable X, which is(40)Rtx=1−1−cosπ/21−e−λx+tσαβ1−1−cosπ/21−e−λxσαβ.In addition, the reversed hazard rate functionR¯tx has attracted the attention of several practitioners in the recent past, which is derived as(41)R¯tx=1−1−cosπ/21−e−λx−tσαβ1−1−cosπ/21−e−λxσαβ.


### 3.4. Quantile Function, Quantile Density Function, Median, Skewness, and Kurtosis

By virtue of (36), the qf of ECW distribution is(42)QFu=−1λlog1−2πcos−11−u1/β1/α1/σ.U is a uniform random variable. Moreover, the quantile density function (qdf) of ECW distribution is(43)Q′u=2u1−β/βπ/2cos−11−u1/β1−α/α−1/λlog1−π/2cos−11−u1/β1/α1−σ/σπλβσα1−1−u1/β21−π/2cos−11−u1/β1/α.The median of ECW distribution is(44)Median=QF0.5=−1λlog1−2/πcos−11−0.51/β1/α1/σ.Table5 describes the numerical values of the first quartile Q1, second quartile Q2, third quartile Q3, skewness (Bowley), and kurtosis (Moors) of ECW distribution for selected values for α, β, λ, and σ.Table 5
First quartile, median, third quartile, Bowley skewness, and Moors kurtosis of ECWα,β,λ,σ distribution for the following selected parameters’ values in order α,β,λ,σ: (i) 2.5,0.5,1.5,0.7, (ii) 2.5,1.0,0.5,1.2, (iii) 2.5,2.0,1.5,1.5, (iv) 3.0,2.5,2.0,2.0, and (v) 6.0,1.0,5.0,3.0.
Q1/4MedQ3/4BM(i)0.4090.8341.5760.2721.394(ii)2.2473.0414.0670.1281.273(iii)1.1701.4131.7080.0971.258(iv)1.0651.2051.3680.0761.249(v)0.7500.8170.8920.0531.249The numerical results in Tables5 and 6 are interpreted as follows:(i)
Whenβ,λ,σ are put constant and α is increased, then skewness and kurtosis ⟶0.(ii)
Whenα,λ,σ are put constant and β is increased, skewness and kurtosis ⟶0.(iii)
Whenα,β,σ are put constant and λ is increased, no significant effect is observed on skewness and kurtosis.(iv)
Whenα,β,λ are put constant and σ is increased, skewness and kurtosis ⟶0.Table 6
Some moments of ECWα,β,λ,σ distribution for the following selected parameters’ values in order α,β,λ,σ: (i) 2.5,0.5,1.5,0.7, (ii) 2.5,1.0,0.5,1.2, (iii) 2.5,2.0,1.5,1.5, (iv) 3.0,2.5,2.0,2.0, and (v) 6.0,1.0,5.0,3.0.
EXEX2EX3EX4VXσXCVCSCK(i)0.5051.0114.86743.5950.7560.8701.7235.46261.384(ii)0.4000.3130.3690.5850.1520.3900.9762.0489.366(iii)0.8680.8120.8150.8730.0590.2420.2790.5713.483(iv)0.9130.8620.8400.8450.0280.1670.1820.4573.328(v)0.6330.4440.3460.3020.0430.2070.3261.2415.739


### 3.5. MacGillivary’s Skewness

MacGillivary (1986) developed a graphical method to access the skewness of the model using quantile function, which is defined as follows:(45)δp=δ1α,β,σ,λδ2α,β,σ,λ=Q1−p+Qp−2Q1/2Q1−p−Qp,wherepε0,1 and Q. are the qf stated in (42).(46)δ1α,β,σ,λ=−1λlog1−2πcos−1p1/β1/α1/σ+−1λlog1−2πcos−11−p1/β1/α1/σ−2−1λlog1−2πcos−10.51/β1/α1/σ.(47)δ2α,β,σ,λ=−1λlog1−2πcos−1p1/β1/α1/σ+−1λlog1−2πcos−11−p1/β1/α1/σ.MacGillivary measure of skewnessδp is purely based on qf and hence can efficiently describe the parametric effect on the skewness. Figure 3 (left) shows that if α,λ,σ are put constant and β is increased, then skewness ⟶0, while Figure 3 (right) shows that if α,λ,β are put constant and σ is increased, then skewness ⟶0.Figure 3
MacGillivary skewness graphs for specific parametric values.


### 3.6. Skewness and Kurtosis via 3D Graphs

Recently, the trend in graphical representation has been changed, and now visual representation gets popularity and is preferred on numerical and tabular representation. The 3D graphs presented in the following clearly attest to the change in kurtosis and skewness after altering the parametric values forα and β.In Figure4, it is clear that as α and β are increased, the skewness is decreased (symmetry is increased), and in the same pattern, the kurtosis is reduced (normality is increased).Figure 4
3D graphs for the skewness (left) and the kurtosis (right) using different parametric values.


## 3.1. Main Functions of ECW with Graphs

Now, using Weibull distribution (as baseline) having the cdfGx=1−e−λxσ, λ,σ,x>0, we presented a new model having right skew and heavy tail characteristics and extended flexibility compared to the former Weibull model. Hereafter, this will be called an Extended Cosine Weibull (ECW) α,β,λ,σ distribution. By using (2), the main functions of the newly presented distribution are(36)Fx=1−cosπ21−e−λxσαβ,x>0,(37)fx=π2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ21−e−λxσα1−cosπ21−e−λxσαβ−1,(38)hx=π/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−11−1−cosπ/21−e−λxσαβ.Figure1 displays the exhibiting density shapes of the proposed model.Figure 1
Some labeled density curves of the ECW.Figure2 displays the exhibiting hazard rate function shapes of the proposed model.Figure 2
Some labeled hazard function curves of the ECW.


## 3.2. Useful Reliability Functions of ECW

The important reliability functions are(39)Sx=1−1−cosπ21−e−λxσαβ,rx=π/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−11−cosπ/21−e−λxσαβ,Hx=−lnSx=−ln1−1−cosπ21−e−λxσαβ,mx=1−1−cosπ/21−e−λxσαβπ/2αβλσxσ−1e−λxσ1−e−λxσα−1sinπ/21−e−λxσα1−cosπ/21−e−λxσαβ−1,G¯x,α,β,λ,σ|t=G¯x+t,α,β,λ,σG¯t,α,β,λ,σ=1−1−cosπ/21−e−λx+tσαβ1−1−cosπ/21−e−λtσαβ,ex=∂∂lnx1−cosπ/21−e−λxσαβ.


## 3.3. Residual and Reverse Residual Life

In reliability theory and risk management, the residual life offers wide applications. LetX denote the lifetime of a unit and X≥0 with PX=1; then similar to residual lifetime random variable, Xt=t−X|X≤t, for a fixed t>0, named as “time since failure.”LetRtx denote the residual lifetime of the ECW random variable X, which is(40)Rtx=1−1−cosπ/21−e−λx+tσαβ1−1−cosπ/21−e−λxσαβ.In addition, the reversed hazard rate functionR¯tx has attracted the attention of several practitioners in the recent past, which is derived as(41)R¯tx=1−1−cosπ/21−e−λx−tσαβ1−1−cosπ/21−e−λxσαβ.


## 3.4. Quantile Function, Quantile Density Function, Median, Skewness, and Kurtosis

By virtue of (36), the qf of ECW distribution is(42)QFu=−1λlog1−2πcos−11−u1/β1/α1/σ.U is a uniform random variable. Moreover, the quantile density function (qdf) of ECW distribution is(43)Q′u=2u1−β/βπ/2cos−11−u1/β1−α/α−1/λlog1−π/2cos−11−u1/β1/α1−σ/σπλβσα1−1−u1/β21−π/2cos−11−u1/β1/α.The median of ECW distribution is(44)Median=QF0.5=−1λlog1−2/πcos−11−0.51/β1/α1/σ.Table5 describes the numerical values of the first quartile Q1, second quartile Q2, third quartile Q3, skewness (Bowley), and kurtosis (Moors) of ECW distribution for selected values for α, β, λ, and σ.Table 5
First quartile, median, third quartile, Bowley skewness, and Moors kurtosis of ECWα,β,λ,σ distribution for the following selected parameters’ values in order α,β,λ,σ: (i) 2.5,0.5,1.5,0.7, (ii) 2.5,1.0,0.5,1.2, (iii) 2.5,2.0,1.5,1.5, (iv) 3.0,2.5,2.0,2.0, and (v) 6.0,1.0,5.0,3.0.
Q1/4MedQ3/4BM(i)0.4090.8341.5760.2721.394(ii)2.2473.0414.0670.1281.273(iii)1.1701.4131.7080.0971.258(iv)1.0651.2051.3680.0761.249(v)0.7500.8170.8920.0531.249The numerical results in Tables5 and 6 are interpreted as follows:(i)
Whenβ,λ,σ are put constant and α is increased, then skewness and kurtosis ⟶0.(ii)
Whenα,λ,σ are put constant and β is increased, skewness and kurtosis ⟶0.(iii)
Whenα,β,σ are put constant and λ is increased, no significant effect is observed on skewness and kurtosis.(iv)
Whenα,β,λ are put constant and σ is increased, skewness and kurtosis ⟶0.Table 6
Some moments of ECWα,β,λ,σ distribution for the following selected parameters’ values in order α,β,λ,σ: (i) 2.5,0.5,1.5,0.7, (ii) 2.5,1.0,0.5,1.2, (iii) 2.5,2.0,1.5,1.5, (iv) 3.0,2.5,2.0,2.0, and (v) 6.0,1.0,5.0,3.0.
EXEX2EX3EX4VXσXCVCSCK(i)0.5051.0114.86743.5950.7560.8701.7235.46261.384(ii)0.4000.3130.3690.5850.1520.3900.9762.0489.366(iii)0.8680.8120.8150.8730.0590.2420.2790.5713.483(iv)0.9130.8620.8400.8450.0280.1670.1820.4573.328(v)0.6330.4440.3460.3020.0430.2070.3261.2415.739


## 3.5. MacGillivary’s Skewness

MacGillivary (1986) developed a graphical method to access the skewness of the model using quantile function, which is defined as follows:(45)δp=δ1α,β,σ,λδ2α,β,σ,λ=Q1−p+Qp−2Q1/2Q1−p−Qp,wherepε0,1 and Q. are the qf stated in (42).(46)δ1α,β,σ,λ=−1λlog1−2πcos−1p1/β1/α1/σ+−1λlog1−2πcos−11−p1/β1/α1/σ−2−1λlog1−2πcos−10.51/β1/α1/σ.(47)δ2α,β,σ,λ=−1λlog1−2πcos−1p1/β1/α1/σ+−1λlog1−2πcos−11−p1/β1/α1/σ.MacGillivary measure of skewnessδp is purely based on qf and hence can efficiently describe the parametric effect on the skewness. Figure 3 (left) shows that if α,λ,σ are put constant and β is increased, then skewness ⟶0, while Figure 3 (right) shows that if α,λ,β are put constant and σ is increased, then skewness ⟶0.Figure 3
MacGillivary skewness graphs for specific parametric values.


## 3.6. Skewness and Kurtosis via 3D Graphs

Recently, the trend in graphical representation has been changed, and now visual representation gets popularity and is preferred on numerical and tabular representation. The 3D graphs presented in the following clearly attest to the change in kurtosis and skewness after altering the parametric values forα and β.In Figure4, it is clear that as α and β are increased, the skewness is decreased (symmetry is increased), and in the same pattern, the kurtosis is reduced (normality is increased).Figure 4
3D graphs for the skewness (left) and the kurtosis (right) using different parametric values.


## 4. Estimation and Inference of ECW

Letx1,…,xn be a random sample of size n, which follow the ECW model. Then the log-likelihood function based on (37) is(48)ℓα,β,λ,σ=nlogα+nlogβ+nlogλ+nlogσ+β+1nlog1−cos12π1−e−λxσα+nlogsin12π1−e−λxσα−nλxσ+nσ−1loge−λxσeλxσ−1−nσ−1logx+nlogπ2.The maximum likelihood estimators (MLEs) are the solution of the nonlinear equations:∂ℓα,β,λ,σ/∂α=0, ∂ℓα,β,λ,σ/∂β=0, ∂ℓα,β,λ,σ/∂λ=0, and ∂ℓα,β,λ,σ/∂σ=0 with(49)∂ℓα,β,λ,σ∂α=nα+12πn1−e−λxσαlog1−e−λxσβ+1cot14π1−e−λxσα+cot12π1−e−λxσα,∂ℓα,β,λ,σ∂β=n1β+logsin214π1−e−λxσα+log2,∂ℓα,β,λ,σ∂λ=nπαλxσ1−e−λxσαtan1/4π1−e−λxσα2β+3cot21/4π1−e−λxσα−14λeλxσ−1+4λσxσ+eλxσ1−λxσ−14λeλxσ−1,∂ℓα,β,λ,σ∂σ=n1σ+log1−e−λxσ−logx+λnxσlogx4σ+πα1−e−λxσαtan1/4π1−e−λxσα4eλxσ−1∗2β+3cot21/4π1−e−λxσα−1−4eλxσ4eλxσ−1.The MLEsα^,β^,λ^,σ^ can be obtained by setting the above equations to zero and solving them simultaneously.


### 4.1. Simulation Study

To check the convergence of the MLEs, a Monte Carlo simulation study is carried out atN=1,000 repetition times with n=50,100,200, 300,500,1000 sample sizes and the following parametric combinations:  I: α=2.0,β=1.5,  λ=0.5, and σ=1.2, II: α=2.5,β=3.5,  λ=1.5, and σ=2.2, III: α=1.5,β=4.0,  λ=2.5, and σ=2.0, and IV: α=3.5,β=4.0, λ=2.0, and σ=3.0. The empirical biases, mean squared errors (MSEs), coverage probabilities (CP) of the 95% two-sided confidence intervals for some sample sizes, the lower bound (L.bound) of these intervals, and their upper bounds (U.bound) for the model ECW parameters can be found in Tables 7 and 8.Table 7
The bias, MSE, CP, L.bound, and U.bound obtained from simulation of ECW model.
nInitialBiasMSECPL.boundU.boundInitialBiasMSECPL.boundU.bound50α2.05.514621.4430.9388.695103.7232.56.748393.7070.94138.835157.332β1.52.62160.2210.9850.66558.9083.53.295456.4360.91178.791192.380λ0.50.3730.9990.881.3043.0501.50.1811.8460.882.0125.375σ1.20.0980.4540.980.1802.6182.20.4742.1701.000.4365.605100α2.01.96177.4400.9420.84928.7702.55.372250.0290.9661.02076.764β1.52.67521.2930.9736.44644.7953.52.565188.5060.9084.61596.745λ0.50.2100.3920.900.5251.9441.50.2271.1970.920.8514.305σ1.20.0280.1680.960.3812.0772.20.1380.7030.990.5864.128200α2.01.16423.5130.9211.34117.6692.53.37189.9620.9025.70137.443β1.52.41820.1130.9426.63434.4713.52.12430.7530.8849.61060.858λ0.50.1270.1430.950.1561.4081.50.1770.5570.960.1813.413σ1.2−0.0120.0720.970.6141.7632.20.0190.3060.981.0133.425300α2.00.91914.1530.939.10014.9382.53.66087.3610.9521.76234.082β1.52.13015.4850.9420.10127.3613.51.57922.7050.8435.64145.799λ0.50.1010.0880.960.0691.2221.50.1390.3770.970.2983.018σ1.2−0.0220.0460.960.7101.6472.2−0.0050.1860.981.2193.171500α2.00.5798.9750.906.47511.6322.52.41642.4030.9214.74724.579β1.52.19616.4610.9317.32724.7193.51.57823.1980.8729.97740.134λ0.50.0670.0440.970.1081.0351.50.1070.2070.960.5542.679σ1.2−0.0180.0250.970.8141.5492.2−0.0210.1200.971.4142.9451000α2.00.5586.6500.945.00110.1162.52.15634.2040.9510.18919.500β1.51.70112.2640.9311.15217.5543.51.39119.2680.8521.25631.039λ0.50.0500.0220.980.2180.8841.50.0770.0990.980.8132.341σ1.2−0.0220.0130.970.9121.4452.2−0.0310.0550.971.6202.718Table 8
The bias, MSE, CP, L.bound, and U.bound obtained from simulation of ECW model.
50α1.54.090129.9840.9264.87376.0543.510.3791159.9820.93277.221304.979β4.01.91799.6120.90124.588136.4234.04.041803.3600.90233.217249.300λ2.50.1182.0900.931.2226.4482.00.0892.4080.892.4416.620σ2.00.4221.9151.000.3604.9973.00.7123.8241.000.6727.870100α1.53.28681.1990.8931.86341.4353.511.3641155.2760.90197.695227.423β4.01.64860.9780.8870.67181.9684.03.941659.5010.88170.913186.795λ2.50.2321.3330.950.3105.4582.00.2652.0660.921.0915.619σ2.00.1070.5600.980.5653.6613.00.2361.4820.990.7405.832200α1.52.45741.9890.9216.48324.3983.56.050303.3600.9350.92870.028β4.01.44828.2880.8648.32359.2204.02.07761.7630.8857.58069.734λ2.50.1590.6880.980.8304.5102.00.1690.8740.960.2204.343σ2.00.0150.2300.960.9653.0653.00.0600.5820.991.3214.803300α1.52.15028.8570.9312.67319.9723.55.660191.6440.9434.47752.798β4.01.26727.0040.8538.65749.1914.01.36326.2290.8641.52852.254λ2.50.1160.3880.981.1484.0872.00.1430.5920.960.4393.902σ2.00.0000.1410.971.1482.8533.00.0340.4320.971.6084.460500α1.51.90621.8950.919.05415.8663.54.850149.7370.9325.29041.989β4.00.87720.2260.8228.66238.4174.01.19123.1630.8432.99743.380λ2.50.0790.2290.971.4373.7222.00.0790.3500.970.6293.536σ2.0−0.0070.0910.961.3292.6573.0−0.0300.2440.981.8744.0661000α1.51.56014.9690.976.63612.7573.53.80890.1840.9816.40331.018β4.01.19823.4100.8124.17834.5744.01.11022.7870.8223.48333.704λ2.50.0660.1150.981.7173.4142.00.0680.1460.981.1133.027σ2.0−0.0230.0430.981.4922.4613.0−0.0250.1090.982.1823.769Specifically, concerning the theory, we see that the biases and MSEs decline with expanding sample size. Additionally, the CPs of the confidence intervals are very near to the 95% nominal levels. Accordingly, the MLEs and their asymptotic outcomes can be utilized for estimating and constructing confidence intervals for the model parameters.In Figure5, the graphs (left and middle) clearly depict that biases and MSE(s) are approaching zero, while the graphs (right) show that CP is overall increasing.Figure 5
Graphs for bias (left), MSE (middle), and coverage probability (right).


### 4.2. Analysis of ECW Model

Considering ECWα,β,λ,σ distribution as a statistical model, the MLEs are calculated, and significant goodness-of-fit criterion results, like the log-likelihood function checked at the MLEs ℓ^, Cramér–von Mises W∗, Anderson-Darling A∗, p-values, and Kolmogorov–Smirnov (K–S), are computed for comparison with configured models. Reference [42] provided good details about A∗ and W∗ statistics. Here, we compare the ECW distribution with six models, namely, Exponentiated Kumaraswamy Weibull (EKumW), Beta Weibull (BW), Topp–Leone Weibull (TLW), Logistic Weibull (LW), Cosine Weibull (CW), and Weibull (W).


#### 4.2.1. First Application: Cancer Patients’ Survival Times Data

The first data represent 121 breast cancer patients’ survival times during a specific period from 1929 to 1938. The data source is [43] and Ramos et al. (2013) and Tahir et al. (2014) studied these datasets. The observations are listed as follows:0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0, 126.0, 127.0, 129.0, 129.0, 139.0, and 154.0.The summary statistics aremin.x=0.30, max.x=154, Q1=17.50, median=40.00, mean=46.33, Q3=60.00, SD=35.28, range=153.7, skewness=1.03, and kurtosis=0.35.Figure6(a) presents the histogram of the first data, showing a heavy right tail, while the TTT plot is shown in Figure 6(b), which is concave only. The boxplot is displayed in Figure 6(c), while the kernel density is shown in Figure 6(d), and the conclusion is that the first data is right-skewed. These graphics motivate the use of a model with a heavy right tail, such as the ECW model.Figure 6
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for cancer patients’ survival times data.
(a)(b)(c)(d)Table9 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for cancer patients’ survival times data.Table 9
MLEs and their standard errors (in parentheses) for cancer patients’ survival times data.
DistributionαβabcECW0.006 91.260 22.668 80.116—(0.000 7)(0.004 8)(0.004 8)(0.022 5)—EKumW0.005 61.375 31.057 85.458 30.834 5(0.001 9)(0.542 6)(0.697 9)(4.277 3)(0.427 8)BW0.996 70.776 90.008 51.281 7—(0.446 7)(1.390 7)(0.004 7)(0.340 1)—TLW0.006 11.162 21.185 2——(0.002 1)(0.072 8)(0.173 9)——LW0.665 00.093 62.789 8——(0.941 1)(0.313 8)(3.954 7)——CW0.042 60.909 2———(0.011 0)(0.062 8)———W0.006 31.295 3———(0.001 8)(0.068 4)———Table10 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.Table 10
The statisticsℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.
Distributionℓ^A∗W∗K–SP-valueECW579.150 00.394 50.051 70.058 80.796 9EKumW579.909 50.429 70.057 10.059 10.790 3BW579.159 30.398 30.053 30.065 40.678 3TLW579.348 90.394 90.053 60.070 20.589 9LW587.599 61.258 00.209 70.093 60.238 7CW581.537 20.558 70.086 00.087 30.313 9W579.032 10.400 00.053 50.064 10.702 0Figure7(a) presents the plots of the estimated densities, Figure 7(b) shows the plots of the estimated distribution functions, and Figure 7(c) provides the P–P plots for cancer patients’ survival times data.Figure 7
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for cancer patients’ survival times data.
(a)(b)(c)


### 4.3. Second Application: Electrical Appliance Failure Data

The second data presented by Lawless (2002) regarding the number of 1000s of cycles to failure for 60 electrical appliances in a life test and the values are as follows:0.014, 0.034, 0.059, 0.061, 0.069, 0.080, 0.123, 0.142, 0.165, 0.210, 0.381, 0.464, 0.479, 0.556, 0.574, 0.839, 0.917, 0.969, 0.991, 1.064, 1.088, 1.091, 1.174, 1.270, 1.275, 1.355, 1.397, 1.477, 1.578, 1.649, 1.702, 1.893, 1.932, 2.001, 2.161, 2.292, 2.326, 2.337, 2.628, 2.785, 2.811, 2.886, 2.993, 3.122, 3.248, 3.715, 3.790, 3.857, 3.912, 4.100, 4.106, 4.116, 4.315, 4.510, 4.580, 5.267, 5.299, 5.583, 6.065, and 9.701.The summary statistics are as follows:min.x=0.014, max.x=9.7010, Q1=0.7728, median=1.6755, mean=2.1930, Q3=3.3647, SD=1.92, range=9.69, skewness=1.23, and kurtosis=2.05.Figure8(a) presents the histogram of second data, showing a right tail, while the TTT plot is shown in Figure 8(b), which is first convex and then concave. The boxplot is displayed in Figure 8(c), while the kernel density is shown in Figure 8(d). So, the second data is right-skewed. These graphics motivate the use of a model like ECW with a right tail.Figure 8
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for electrical appliance failure data.
(a)(b)(c)(d)Table11 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for electrical appliance failure data.Table 11
MLEs and their standard errors (in parentheses) for electrical appliance failure data.
DistributionαβλσECW0.052 21.992 61.312 70.139 4—(0.013 5)(0.079 9)(0.065 9)(0.022 2)—EKumW0.122 11.348 02.100 97.119 90.258 6(0.340 1)(2.517 6)(6.299 1)(46.411 6)(0.424 9)BW0.531 70.153 31.563 41.384 7—(0.113 1)(0.021 7)(0.002 6)(0.002 6)—TLW0.031 01.953 20.373 3——(0.039 9)(0.619 9)(0.161 6)——LW0.568 00.811 22.274 2——(27.244 3)(8.139 8)(109.068 5)——CW0.885 70.661 1———(0.102 5)(0.068 2)———W0.045 51.000 8———(0.081 4)(0.106 6)———Table12 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for electrical appliance failure data.Table 12
The statisticsℓ^, A∗, W∗, K–S, and P-value for electrical appliance failure data.
Distributionℓ^A∗W∗K–SP-valueECW105.062 80.280 80.033 10.061 40.967 0EKumW105.122 50.285 40.033 70.062 30.962 2BW105.727 50.281 50.033 30.067 60.928 8TLW105.158 80.289 60.034 20.062 60.960 7LW115.199 12.035 90.328 50.118 80.338 0CW110.376 51.341 00.210 40.104 10.501 0W107.115 30.715 40.103 60.077 70.834 2Figure9(a) presents the plots of the estimated densities, Figure 9(b) shows the plots of the estimated distribution functions, and Figure 9(c) provides the P–P plots for electrical appliance failure data.Figure 9
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for electrical appliance failure data.
(a)(b)(c)


### 4.4. Third Application: Failure Data of 50 Components

The third data are an uncensored dataset from Murty, Xie, and Jiang (2004) used in the industry, representing the failure time (in weeks) of 50 components put into use at time. The data values are as follows:0.013, 0.065, 0.111, 0.111, 0.163, 0.309, 0.426, 0.535, 0.684, 0.747, 0.997, 1.284, 1.304, 1.647, 1.829, 2.336, 2.838, 3.269, 3.977, 3.981, 4.520, 4.789, 4.849, 5.202, 5.291, 5.349, 5.911, 6.018, 6.427, 6.456, 6.572, 7.023, 7.087, 7.291, 7.787, 8.596, 9.388, 10.261, 10.713, 11.658, 13.006, 13.388, 13.842, 17.152, 17.283, 19.418, 23.471, 24.777, 32.795, and 48.105.The summary statistics are as follows:min.x=0.013, max.x=48.105, Q1=1.390, median=5.320, mean=7.821, Q3=10.04, SD=7.82, range=48.09, skewness=2.24, and kurtosis=6.04.Figure10(a) presents the histogram of the third dataset, showing a right tail, while the TTT plot is shown in Figure 10(b), which is convex only. The boxplot is displayed in Figure 10(c), while the kernel density is shown in Figure 10(d), and the conclusion is that the third dataset is right-skewed. These graphics motivate the use of a model with a right tail, such as the ECW model.Figure 10
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for failure data of 50 components.
(a)(b)(c)(d)Table13 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for failure data of 50 components.Table 13
MLEs and their standard errors (in parentheses) for failure data of 50 components.
DistributionαβλσECW0.055 31.138 71.378 80.207 4—(0.089 0)(0.433 5)(2.695 6)(0.371 0)—EKumW0.059 10.533 54.887 18.098 50.276 7(0.301 3)(2.666 4)(38.306 4)(81.724 1)(0.556 9)BW0.559 93.454 20.014 11.165 0—(0.301 9)(7.557 9)(0.026 3)(0.426 7)—TLW0.602 00.034 91.096 2——(0.308 9)(0.048 1)(0.374 7)——LW0.788 61.374 90.144 3——(0.102 2)(0.063 5)(0.034 3)——CW0.518 90.543 9———(0.085 1)(0.060 3)———W0.211 40.800 2———(0.055 4)(0.090 6)———Table14 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for failure data of 50 components.Table 14
The statisticsℓ^, A∗, W∗, K–S, and P-value for failure data of 50 components.
Distributionℓ^A∗W∗K–SP-valueECW150.167 20.289 30.056 80.094 80.759 4EKumW150.279 40.322 10.064 10.101 50.681 0BW150.239 60.299 10.058 70.095 60.750 4TLW150.259 40.305 10.060 00.096 70.737 1LW155.997 61.382 20.257 30.132 50.343 3CW152.552 10.823 20.159 30.140 40.277 2W150.676 80.427 20.085 60.111 80.559 1Figure11(a) presents the plots of the estimated densities, Figure 11(b) shows the plots of the estimated distribution functions, and Figure 11(c) provides the P–P plots for failure data of 50 components.Figure 11
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for failure data of 50 components.
(a)(b)(c)


## 4.1. Simulation Study

To check the convergence of the MLEs, a Monte Carlo simulation study is carried out atN=1,000 repetition times with n=50,100,200, 300,500,1000 sample sizes and the following parametric combinations:  I: α=2.0,β=1.5,  λ=0.5, and σ=1.2, II: α=2.5,β=3.5,  λ=1.5, and σ=2.2, III: α=1.5,β=4.0,  λ=2.5, and σ=2.0, and IV: α=3.5,β=4.0, λ=2.0, and σ=3.0. The empirical biases, mean squared errors (MSEs), coverage probabilities (CP) of the 95% two-sided confidence intervals for some sample sizes, the lower bound (L.bound) of these intervals, and their upper bounds (U.bound) for the model ECW parameters can be found in Tables 7 and 8.Table 7
The bias, MSE, CP, L.bound, and U.bound obtained from simulation of ECW model.
nInitialBiasMSECPL.boundU.boundInitialBiasMSECPL.boundU.bound50α2.05.514621.4430.9388.695103.7232.56.748393.7070.94138.835157.332β1.52.62160.2210.9850.66558.9083.53.295456.4360.91178.791192.380λ0.50.3730.9990.881.3043.0501.50.1811.8460.882.0125.375σ1.20.0980.4540.980.1802.6182.20.4742.1701.000.4365.605100α2.01.96177.4400.9420.84928.7702.55.372250.0290.9661.02076.764β1.52.67521.2930.9736.44644.7953.52.565188.5060.9084.61596.745λ0.50.2100.3920.900.5251.9441.50.2271.1970.920.8514.305σ1.20.0280.1680.960.3812.0772.20.1380.7030.990.5864.128200α2.01.16423.5130.9211.34117.6692.53.37189.9620.9025.70137.443β1.52.41820.1130.9426.63434.4713.52.12430.7530.8849.61060.858λ0.50.1270.1430.950.1561.4081.50.1770.5570.960.1813.413σ1.2−0.0120.0720.970.6141.7632.20.0190.3060.981.0133.425300α2.00.91914.1530.939.10014.9382.53.66087.3610.9521.76234.082β1.52.13015.4850.9420.10127.3613.51.57922.7050.8435.64145.799λ0.50.1010.0880.960.0691.2221.50.1390.3770.970.2983.018σ1.2−0.0220.0460.960.7101.6472.2−0.0050.1860.981.2193.171500α2.00.5798.9750.906.47511.6322.52.41642.4030.9214.74724.579β1.52.19616.4610.9317.32724.7193.51.57823.1980.8729.97740.134λ0.50.0670.0440.970.1081.0351.50.1070.2070.960.5542.679σ1.2−0.0180.0250.970.8141.5492.2−0.0210.1200.971.4142.9451000α2.00.5586.6500.945.00110.1162.52.15634.2040.9510.18919.500β1.51.70112.2640.9311.15217.5543.51.39119.2680.8521.25631.039λ0.50.0500.0220.980.2180.8841.50.0770.0990.980.8132.341σ1.2−0.0220.0130.970.9121.4452.2−0.0310.0550.971.6202.718Table 8
The bias, MSE, CP, L.bound, and U.bound obtained from simulation of ECW model.
50α1.54.090129.9840.9264.87376.0543.510.3791159.9820.93277.221304.979β4.01.91799.6120.90124.588136.4234.04.041803.3600.90233.217249.300λ2.50.1182.0900.931.2226.4482.00.0892.4080.892.4416.620σ2.00.4221.9151.000.3604.9973.00.7123.8241.000.6727.870100α1.53.28681.1990.8931.86341.4353.511.3641155.2760.90197.695227.423β4.01.64860.9780.8870.67181.9684.03.941659.5010.88170.913186.795λ2.50.2321.3330.950.3105.4582.00.2652.0660.921.0915.619σ2.00.1070.5600.980.5653.6613.00.2361.4820.990.7405.832200α1.52.45741.9890.9216.48324.3983.56.050303.3600.9350.92870.028β4.01.44828.2880.8648.32359.2204.02.07761.7630.8857.58069.734λ2.50.1590.6880.980.8304.5102.00.1690.8740.960.2204.343σ2.00.0150.2300.960.9653.0653.00.0600.5820.991.3214.803300α1.52.15028.8570.9312.67319.9723.55.660191.6440.9434.47752.798β4.01.26727.0040.8538.65749.1914.01.36326.2290.8641.52852.254λ2.50.1160.3880.981.1484.0872.00.1430.5920.960.4393.902σ2.00.0000.1410.971.1482.8533.00.0340.4320.971.6084.460500α1.51.90621.8950.919.05415.8663.54.850149.7370.9325.29041.989β4.00.87720.2260.8228.66238.4174.01.19123.1630.8432.99743.380λ2.50.0790.2290.971.4373.7222.00.0790.3500.970.6293.536σ2.0−0.0070.0910.961.3292.6573.0−0.0300.2440.981.8744.0661000α1.51.56014.9690.976.63612.7573.53.80890.1840.9816.40331.018β4.01.19823.4100.8124.17834.5744.01.11022.7870.8223.48333.704λ2.50.0660.1150.981.7173.4142.00.0680.1460.981.1133.027σ2.0−0.0230.0430.981.4922.4613.0−0.0250.1090.982.1823.769Specifically, concerning the theory, we see that the biases and MSEs decline with expanding sample size. Additionally, the CPs of the confidence intervals are very near to the 95% nominal levels. Accordingly, the MLEs and their asymptotic outcomes can be utilized for estimating and constructing confidence intervals for the model parameters.In Figure5, the graphs (left and middle) clearly depict that biases and MSE(s) are approaching zero, while the graphs (right) show that CP is overall increasing.Figure 5
Graphs for bias (left), MSE (middle), and coverage probability (right).


## 4.2. Analysis of ECW Model

Considering ECWα,β,λ,σ distribution as a statistical model, the MLEs are calculated, and significant goodness-of-fit criterion results, like the log-likelihood function checked at the MLEs ℓ^, Cramér–von Mises W∗, Anderson-Darling A∗, p-values, and Kolmogorov–Smirnov (K–S), are computed for comparison with configured models. Reference [42] provided good details about A∗ and W∗ statistics. Here, we compare the ECW distribution with six models, namely, Exponentiated Kumaraswamy Weibull (EKumW), Beta Weibull (BW), Topp–Leone Weibull (TLW), Logistic Weibull (LW), Cosine Weibull (CW), and Weibull (W).


### 4.2.1. First Application: Cancer Patients’ Survival Times Data

The first data represent 121 breast cancer patients’ survival times during a specific period from 1929 to 1938. The data source is [43] and Ramos et al. (2013) and Tahir et al. (2014) studied these datasets. The observations are listed as follows:0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0, 126.0, 127.0, 129.0, 129.0, 139.0, and 154.0.The summary statistics aremin.x=0.30, max.x=154, Q1=17.50, median=40.00, mean=46.33, Q3=60.00, SD=35.28, range=153.7, skewness=1.03, and kurtosis=0.35.Figure6(a) presents the histogram of the first data, showing a heavy right tail, while the TTT plot is shown in Figure 6(b), which is concave only. The boxplot is displayed in Figure 6(c), while the kernel density is shown in Figure 6(d), and the conclusion is that the first data is right-skewed. These graphics motivate the use of a model with a heavy right tail, such as the ECW model.Figure 6
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for cancer patients’ survival times data.
(a)(b)(c)(d)Table9 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for cancer patients’ survival times data.Table 9
MLEs and their standard errors (in parentheses) for cancer patients’ survival times data.
DistributionαβabcECW0.006 91.260 22.668 80.116—(0.000 7)(0.004 8)(0.004 8)(0.022 5)—EKumW0.005 61.375 31.057 85.458 30.834 5(0.001 9)(0.542 6)(0.697 9)(4.277 3)(0.427 8)BW0.996 70.776 90.008 51.281 7—(0.446 7)(1.390 7)(0.004 7)(0.340 1)—TLW0.006 11.162 21.185 2——(0.002 1)(0.072 8)(0.173 9)——LW0.665 00.093 62.789 8——(0.941 1)(0.313 8)(3.954 7)——CW0.042 60.909 2———(0.011 0)(0.062 8)———W0.006 31.295 3———(0.001 8)(0.068 4)———Table10 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.Table 10
The statisticsℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.
Distributionℓ^A∗W∗K–SP-valueECW579.150 00.394 50.051 70.058 80.796 9EKumW579.909 50.429 70.057 10.059 10.790 3BW579.159 30.398 30.053 30.065 40.678 3TLW579.348 90.394 90.053 60.070 20.589 9LW587.599 61.258 00.209 70.093 60.238 7CW581.537 20.558 70.086 00.087 30.313 9W579.032 10.400 00.053 50.064 10.702 0Figure7(a) presents the plots of the estimated densities, Figure 7(b) shows the plots of the estimated distribution functions, and Figure 7(c) provides the P–P plots for cancer patients’ survival times data.Figure 7
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for cancer patients’ survival times data.
(a)(b)(c)


## 4.2.1. First Application: Cancer Patients’ Survival Times Data

The first data represent 121 breast cancer patients’ survival times during a specific period from 1929 to 1938. The data source is [43] and Ramos et al. (2013) and Tahir et al. (2014) studied these datasets. The observations are listed as follows:0.3, 0.3, 4.0, 5.0, 5.6, 6.2, 6.3, 6.6, 6.8, 7.4, 7.5, 8.4, 8.4, 10.3, 11.0, 11.8, 12.2, 12.3, 13.5, 14.4, 14.4, 14.8, 15.5, 15.7, 16.2, 16.3, 16.5, 16.8, 17.2, 17.3, 17.5, 17.9, 19.8, 20.4, 20.9, 21.0, 21.0, 21.1, 23.0, 23.4, 23.6, 24.0, 24.0, 27.9, 28.2, 29.1, 30.0, 31.0, 31.0, 32.0, 35.0, 35.0, 37.0, 37.0, 37.0, 38.0, 38.0, 38.0, 39.0, 39.0, 40.0, 40.0, 40.0, 41.0, 41.0, 41.0, 42.0, 43.0, 43.0, 43.0, 44.0, 45.0, 45.0, 46.0, 46.0, 47.0, 48.0, 49.0, 51.0, 51.0, 51.0, 52.0, 54.0, 55.0, 56.0, 57.0, 58.0, 59.0, 60.0, 60.0, 60.0, 61.0, 62.0, 65.0, 65.0, 67.0, 67.0, 68.0, 69.0, 78.0, 80.0, 83.0, 88.0, 89.0, 90.0, 93.0, 96.0, 103.0, 105.0, 109.0, 109.0, 111.0, 115.0, 117.0, 125.0, 126.0, 127.0, 129.0, 129.0, 139.0, and 154.0.The summary statistics aremin.x=0.30, max.x=154, Q1=17.50, median=40.00, mean=46.33, Q3=60.00, SD=35.28, range=153.7, skewness=1.03, and kurtosis=0.35.Figure6(a) presents the histogram of the first data, showing a heavy right tail, while the TTT plot is shown in Figure 6(b), which is concave only. The boxplot is displayed in Figure 6(c), while the kernel density is shown in Figure 6(d), and the conclusion is that the first data is right-skewed. These graphics motivate the use of a model with a heavy right tail, such as the ECW model.Figure 6
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for cancer patients’ survival times data.
(a)(b)(c)(d)Table9 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for cancer patients’ survival times data.Table 9
MLEs and their standard errors (in parentheses) for cancer patients’ survival times data.
DistributionαβabcECW0.006 91.260 22.668 80.116—(0.000 7)(0.004 8)(0.004 8)(0.022 5)—EKumW0.005 61.375 31.057 85.458 30.834 5(0.001 9)(0.542 6)(0.697 9)(4.277 3)(0.427 8)BW0.996 70.776 90.008 51.281 7—(0.446 7)(1.390 7)(0.004 7)(0.340 1)—TLW0.006 11.162 21.185 2——(0.002 1)(0.072 8)(0.173 9)——LW0.665 00.093 62.789 8——(0.941 1)(0.313 8)(3.954 7)——CW0.042 60.909 2———(0.011 0)(0.062 8)———W0.006 31.295 3———(0.001 8)(0.068 4)———Table10 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.Table 10
The statisticsℓ^, A∗, W∗, K–S, and P-value for cancer patients’ survival times data.
Distributionℓ^A∗W∗K–SP-valueECW579.150 00.394 50.051 70.058 80.796 9EKumW579.909 50.429 70.057 10.059 10.790 3BW579.159 30.398 30.053 30.065 40.678 3TLW579.348 90.394 90.053 60.070 20.589 9LW587.599 61.258 00.209 70.093 60.238 7CW581.537 20.558 70.086 00.087 30.313 9W579.032 10.400 00.053 50.064 10.702 0Figure7(a) presents the plots of the estimated densities, Figure 7(b) shows the plots of the estimated distribution functions, and Figure 7(c) provides the P–P plots for cancer patients’ survival times data.Figure 7
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for cancer patients’ survival times data.
(a)(b)(c)


## 4.3. Second Application: Electrical Appliance Failure Data

The second data presented by Lawless (2002) regarding the number of 1000s of cycles to failure for 60 electrical appliances in a life test and the values are as follows:0.014, 0.034, 0.059, 0.061, 0.069, 0.080, 0.123, 0.142, 0.165, 0.210, 0.381, 0.464, 0.479, 0.556, 0.574, 0.839, 0.917, 0.969, 0.991, 1.064, 1.088, 1.091, 1.174, 1.270, 1.275, 1.355, 1.397, 1.477, 1.578, 1.649, 1.702, 1.893, 1.932, 2.001, 2.161, 2.292, 2.326, 2.337, 2.628, 2.785, 2.811, 2.886, 2.993, 3.122, 3.248, 3.715, 3.790, 3.857, 3.912, 4.100, 4.106, 4.116, 4.315, 4.510, 4.580, 5.267, 5.299, 5.583, 6.065, and 9.701.The summary statistics are as follows:min.x=0.014, max.x=9.7010, Q1=0.7728, median=1.6755, mean=2.1930, Q3=3.3647, SD=1.92, range=9.69, skewness=1.23, and kurtosis=2.05.Figure8(a) presents the histogram of second data, showing a right tail, while the TTT plot is shown in Figure 8(b), which is first convex and then concave. The boxplot is displayed in Figure 8(c), while the kernel density is shown in Figure 8(d). So, the second data is right-skewed. These graphics motivate the use of a model like ECW with a right tail.Figure 8
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for electrical appliance failure data.
(a)(b)(c)(d)Table11 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for electrical appliance failure data.Table 11
MLEs and their standard errors (in parentheses) for electrical appliance failure data.
DistributionαβλσECW0.052 21.992 61.312 70.139 4—(0.013 5)(0.079 9)(0.065 9)(0.022 2)—EKumW0.122 11.348 02.100 97.119 90.258 6(0.340 1)(2.517 6)(6.299 1)(46.411 6)(0.424 9)BW0.531 70.153 31.563 41.384 7—(0.113 1)(0.021 7)(0.002 6)(0.002 6)—TLW0.031 01.953 20.373 3——(0.039 9)(0.619 9)(0.161 6)——LW0.568 00.811 22.274 2——(27.244 3)(8.139 8)(109.068 5)——CW0.885 70.661 1———(0.102 5)(0.068 2)———W0.045 51.000 8———(0.081 4)(0.106 6)———Table12 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for electrical appliance failure data.Table 12
The statisticsℓ^, A∗, W∗, K–S, and P-value for electrical appliance failure data.
Distributionℓ^A∗W∗K–SP-valueECW105.062 80.280 80.033 10.061 40.967 0EKumW105.122 50.285 40.033 70.062 30.962 2BW105.727 50.281 50.033 30.067 60.928 8TLW105.158 80.289 60.034 20.062 60.960 7LW115.199 12.035 90.328 50.118 80.338 0CW110.376 51.341 00.210 40.104 10.501 0W107.115 30.715 40.103 60.077 70.834 2Figure9(a) presents the plots of the estimated densities, Figure 9(b) shows the plots of the estimated distribution functions, and Figure 9(c) provides the P–P plots for electrical appliance failure data.Figure 9
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for electrical appliance failure data.
(a)(b)(c)


## 4.4. Third Application: Failure Data of 50 Components

The third data are an uncensored dataset from Murty, Xie, and Jiang (2004) used in the industry, representing the failure time (in weeks) of 50 components put into use at time. The data values are as follows:0.013, 0.065, 0.111, 0.111, 0.163, 0.309, 0.426, 0.535, 0.684, 0.747, 0.997, 1.284, 1.304, 1.647, 1.829, 2.336, 2.838, 3.269, 3.977, 3.981, 4.520, 4.789, 4.849, 5.202, 5.291, 5.349, 5.911, 6.018, 6.427, 6.456, 6.572, 7.023, 7.087, 7.291, 7.787, 8.596, 9.388, 10.261, 10.713, 11.658, 13.006, 13.388, 13.842, 17.152, 17.283, 19.418, 23.471, 24.777, 32.795, and 48.105.The summary statistics are as follows:min.x=0.013, max.x=48.105, Q1=1.390, median=5.320, mean=7.821, Q3=10.04, SD=7.82, range=48.09, skewness=2.24, and kurtosis=6.04.Figure10(a) presents the histogram of the third dataset, showing a right tail, while the TTT plot is shown in Figure 10(b), which is convex only. The boxplot is displayed in Figure 10(c), while the kernel density is shown in Figure 10(d), and the conclusion is that the third dataset is right-skewed. These graphics motivate the use of a model with a right tail, such as the ECW model.Figure 10
(a) Histogram. (b) TTT plot. (c) Boxplot. (d) Kernel density for failure data of 50 components.
(a)(b)(c)(d)Table13 lists the MLEs and their corresponding standard errors (in parentheses) of the model parameters for failure data of 50 components.Table 13
MLEs and their standard errors (in parentheses) for failure data of 50 components.
DistributionαβλσECW0.055 31.138 71.378 80.207 4—(0.089 0)(0.433 5)(2.695 6)(0.371 0)—EKumW0.059 10.533 54.887 18.098 50.276 7(0.301 3)(2.666 4)(38.306 4)(81.724 1)(0.556 9)BW0.559 93.454 20.014 11.165 0—(0.301 9)(7.557 9)(0.026 3)(0.426 7)—TLW0.602 00.034 91.096 2——(0.308 9)(0.048 1)(0.374 7)——LW0.788 61.374 90.144 3——(0.102 2)(0.063 5)(0.034 3)——CW0.518 90.543 9———(0.085 1)(0.060 3)———W0.211 40.800 2———(0.055 4)(0.090 6)———Table14 lists the statistics ℓ^, A∗, W∗, K–S, and P-value for failure data of 50 components.Table 14
The statisticsℓ^, A∗, W∗, K–S, and P-value for failure data of 50 components.
Distributionℓ^A∗W∗K–SP-valueECW150.167 20.289 30.056 80.094 80.759 4EKumW150.279 40.322 10.064 10.101 50.681 0BW150.239 60.299 10.058 70.095 60.750 4TLW150.259 40.305 10.060 00.096 70.737 1LW155.997 61.382 20.257 30.132 50.343 3CW152.552 10.823 20.159 30.140 40.277 2W150.676 80.427 20.085 60.111 80.559 1Figure11(a) presents the plots of the estimated densities, Figure 11(b) shows the plots of the estimated distribution functions, and Figure 11(c) provides the P–P plots for failure data of 50 components.Figure 11
(a) Plots of the estimated pdfs. (b) Plots of the estimated cdfs. (c) P–P plots for failure data of 50 components.
(a)(b)(c)


## 5. Concluding Remarks

An extended version of the cosine-G family of distributions is introduced, utilizing parameter induction methodology and trigonometric generalizers concurrently. This family is proved to be more flexible and better fitting in many practical situations, especially survival and reliability modeling. Moreover, a four-parameter model (ECW) is derived with symmetrical, right-skewed, left-skewed, reversed-J, and bimodal reversed-J density shapes and increasing, decreasing, bathtub, and upside-down bathtub hazard rate shapes. Several mathematical, numerical, and structural properties such as linear representation for cdf and pdf, moments and weighted moments, quantile function, order statistics, entropies, and stochastic ordering are also derived.The parametric estimation is exercised via the maximum likelihood method, and the adequacy of the estimates is attested by the Monte Carlo simulation study. We use a variety of well-known goodness-of-fit statistics to test which distribution is best suited to survival and failure datasets. We conclude that this proposed distribution consistently provides better fits than other existing and competing models even with more number of parameters. We hope that the proposed family and derived models will draw broader applications in a number of areas such as reliability and survival research, hydrology, geology, and several other fields [44–50].



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