# A New Lifetime Distribution: Properties, Copulas, Applications, and Different Classical Estimation Methods

Authors: Naif Alotaibi
Journal: Complexity
 (2021)

---

## Abstract

A new continuous version of the inverse flexible Weibull model is proposed and studied. Some of its properties such as quantile function, moments and generating functions, incomplete moments, mean deviation, Lorenz and Bonferroni curves, the mean residual life function, the mean inactivity time, and the strong mean inactivity time are derived. The failure rate of the new model can be “increasing-constant,” “bathtub-constant,” “bathtub,” “constant,” “J-HRF,” “upside down bathtub,” “increasing,” “upside down-increasing-constant,” and “upside down.” Different copulas are used for deriving many bivariate and multivariate type extensions. Different non-Bayesian well-known estimation methods under uncensored scheme are considered and discussed such as the maximum likelihood estimation, Anderson Darling estimation, ordinary least square estimation, Cramér-von-Mises estimation, weighted least square estimation, and right tail Anderson Darling estimation methods. Simulation studies are performed for comparing these estimation methods. Finally, two real datasets are analyzed to illustrate the importance of the new model.


---

## Body

## 1. Introduction

The Weibull model [1] is a very useful distribution in modeling real data exhibiting monotonic hazard rate function (HRF). But it cannot be used in modeling and studying data which have nonmonotonic HRF such as the “bathtub shape (U-HRF).” For avoiding this drawback, Bebbington et al. [2] have defined a new two-parameter distribution which is an extension of the Weibull distribution referred to as a flexible Weibull (FW) extension distribution; it has a failure function that can be “decreasing,” “increasing,” or “bathtub-shaped.” Analogously, El-Goharyet et al. [3] derived the two-parameter inverse flexible Weibull (IFW) model which is the reciprocal of a random variable (RV) which has FW model. Several mathematical properties of this distribution such as the mode, moments, and moment generating function (MGF) have been discussed. El-Gohary et al. [3] proved that the hazard rate function (RRF) of the IFW model can be “upside down constant,” the cumulative distribution function (CDF) of IFW distribution is given by(1)Gα,θz=exp−expαz−θz|z>0,where the two parameters α>0 and θ>0 control the shape of the distribution. The corresponding probability density function (PDF) is(2)gα,θz=θ+αz2expαz−θzexp−expαz−θz|z>0.Many authors introduced new families of continuous distributions that extend well-known existing ones and, at the same time, provide great flexibility in modeling data in practice, for example, the Marshall-Olkin-generated family [4], the Kumaraswamy-G [5], the Burr X generator of distributions [6], the Topp-Leone odd log-logistic family [7], the transmuted Topp-Leone family [8], the Burr XII system of densities [9], and the odd log-logistic Topp-Leone family [10], among others. Recently, Elgarhy et al. [11] represented a new flexible class of continuous distributions with an extra positive parameter called the type II Topp-Leone generated (TIITL-G) family of distributions. Due to [11], the CDF of the TIITL-G family is given by(3)Fλ,ψ¯z=1−1−Gψ¯2zλ.The PDF is defined by(4)fλ,ψ¯z=2λgψ¯zGψ¯z1−Gψ¯2zλ−1,where λ>0 is a shape parameter, gψ¯z=dGψ¯z/dz is the baseline PDF, and Gψ¯z is a baseline CDF.The remainder of the paper is organized as follows: in Section2, we define the CDF, PDF, and HRF of TIITLIFW model and provide a simple expansion of the PDF. Simple type copula is derived in Section 3. Various mathematical properties are discussed in Section 4. Non-Bayesian estimation methods under uncensored schemes are given in Section 5. Section 6 presents a comparison under the non-Bayesian estimation methods using uncensored schemes via a simulation study. Section 7 presents a comparison under uncensorship with some competitive models. Concluding remarks are contained in Section 8.


## 2. The New Model

Using (1) in (3), the CDF of the TIITLIFW distribution can be written as(5)FV¯z|V¯=α,θ,λ=1−1−exp−2expαz−θzλ.The corresponding PDF is given by(6)fV¯z=2λθ+αz2exp−2expα/z−θzexpα/z−θz1−exp−2expα/z−θz1−λ.The HRF can be expressed as(7)hV¯z=2λexp−2expα/z−θz1−exp−2expα/z−θzθ+αz2expαz−θz.Figure1 shows some plots of the PDF of the TIITLIFW distribution for some different values of the parameters. Figure 2 shows some plots of the HRF of the TIITLIFW distribution for some different parameter values. Based on Figure 1, we conclude that the new PDF can have many right skewed heavy tail shapes. Based on Figure 2, it is observed that the new HRF can be “increasing-constant,” “bathtub-constant,” “bathtub,” “constant,” “J-HRF,” “upside down bathtub,” “increasing,” “upside down-increasing-constant,” and “upside down”Figure 1
PDF plots of the TIITLIFW model for some different values of the parameters. (a)α=1,θ=1,λ=1. (b) α=0.0001,θ=4,λ=0.15. (c) α=0.0001,θ=2,λ=0.1. (d) α=5,θ=0.05,λ=0.05. (e) α=2,θ=2,λ=10. (f) α=0.1,θ=5,λ=0.15.
(a)(b)(c)(d)(e)(f)Figure 2
HRFs of the TIITLIFW model for some different values of the parameters. (a)α=1,θ=1,λ=1. (b) α=0.1,θ=2.5,λ=0.1. (c) α=0.001,θ=2,λ=0.1. (d) α=5,θ=0.5,λ=0.5. (e) α=2,θ=2,λ=10. (f) α=0.1,θ=5,λ=0.15. (g) α=0.0001,θ=4,λ=0.15. (h) α=0.25,θ=5,λ=0.25. (i) α=0.25,θ=0.25,λ=0.25.
(a)(b)(c)(d)(e)(f)(g)(h)(i)The PDF of the TIITLIFW distribution can be written as(8)fV¯z=∑i,j,k=0∞∇i,j,kθz−k+αz−k−2exp−j+1θz.Proof.
Supposingw1/w2<1,w3>0 is a real noninteger, we have the power series expansion(9)1−w1w2w3=∑i=0∞−w1w2iΓ1+w3i!Γ1+w3−i,using the power series (9) in equation (8), and the fact that 0<exp−2expα/z−θz<1, we get(10)fV¯z=∑i=0∞−1i2λΓλi!Γλ−iθ+αz2expαz−θzexp−2i+1expαz−θz,expanding exp−2i+1expα/z−θz using Taylor series(11)exp−2i+1expαz−θz=∑j=0∞−1jj!2i+1jexpjαz−θz,again, using a series expansion of expj+1α/z, and after some algebras, the PDF can be written as(12)fV¯z=∑i,j,k=0∞∇i,j,kθz−k+αz−k−2exp−j+1θz,where(13)∇i,j,k=2j+1αk−1i+ji+1jj+1kΓλ+1i!j!k!Γλ−i.


## 3. Copula

### 3.1. Bivariate TIITLIFW (BTIITLIFW) via Morgenstern Family

The CDF of the Morgenstern family of two random RVsX1,X2 can be derived as(14)ℍρx1,x2|ρ≤1=F1x1F2x21+ρF¯1x1F¯2x2,where(15)F¯1x1=1−F1x1,F¯2x2=1−F2x2,setting(16)FV1¯x|V1¯=α1,θ1,λ1=1−1−exp−2expα1x1−θ1x1λ1,FV2¯x|V2¯=α2,θ2,λ2=1−1−exp−2expα2x2−θ2x2λ2,then we have(17)ℍρ,V1¯,V2¯x1,x2=1−1−exp−2expα1x1−θ1x1λ1×1−1−exp−2expα2x2−θ2x2λ2×1+ρ1−exp−2expα1x1−θ1x1λ1×1−exp−2expα2x2−θ2x2λ2.


### 3.2. Via Clayton Copula

The BTIITLIFW type extension: the weighted version of the Clayton copula can be expressed as(18)ℍu,v=u−ρ+v−φ1+φ2−1−1/ρ|ρ≥0.Let us assume thatX ∼ TIITLIFW V1¯ and Y ∼ TIITLIFW V2¯. Then, setting(19)u=uV1¯x=1−1−exp−2expα1x−θ1xλ1,v=uV2¯y=1−1−exp−2expα2y−θ2yλ2,the associated CDF of the BTIITLIFW type distribution can be written as(20)ℍx,y=1−1−exp−2expα1x−θ1xλ1−ρ+1−1−exp−2expα2y−θ2yλ2−ρ−1−1/ρ.A straightforwardM-dimensional extension from the above will be(21)ℍx1,x2,…,xM=∑i=1M1−1−exp−2expαixi−θixiλi−ρ−M+1−1/ρ.


### 3.3. Via Modified Farlie-Gumbel-Morgenstern (FGM) Copula

The joint CDF of the modified FGM copula is given asℍρu,w=uw1+ρAuVw|ρ∈−1,1 or ℍρu,w=uw+ρA˙uV˙w|ρ∈−1,1, where A˙u=uAu, and V˙w=wVw, where Au and Vw are being two continuous functions on the interval 0,1 where 0=A0=A1=V0=V1. Let(22)c1,u=inf∂∂uA˙u|k1u<0,c1,u=sup∂∂uA˙u|k1u<0,d2,w=inf∂∂wV˙w|k2w>0,d2,w=sup∂∂wV˙w|k2w>0.Then,1≤minc1.uc1,u,d2,wd2,w≤∞, where(23)∂∂uAu=Au+u∂∂uAu,k1u=u:u∈0,1|∂A˙u∂uexists,k2w=w:w∈0,1|∂V˙w∂wexists.Type I:Recalling the following functional form for bothAu and Vw. Then, the BTIITLIFW-FGM (Type-I) can be derived from(24)ℍρu,w=uw+ρA˙uV˙w|ρ∈−1,1,where(25)A˙u=u1−FV¯1u,V˙w=wFV¯2w.Therefore,(26)ℍρu,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2+ρ1−1−exp−2expα1u−θ1uλ11−1−exp−2expα2w−θ2wλ2|ρ∈−1,1.Type II:LetAu∗ and Vw∗ be two functional forms, satisfying all the conditions mentioned above, where(27)Au∗|ρ1>0=1−u1−ρ1uρ1,Vw∗|ρ2>0=1−w1−ρ2wρ2.Then, the corresponding BTIITLIFW-FGM(Type-II) can be derived from(28)ℍρ,ρ1,ρ2u,w=uw1+ρAu∗Vw∗.Then,(29)ℍρ,ρ1,ρ2u,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2×1+ρ1−1−exp−2expα1u−θ1uλ11−ρ11−exp−2expα1u−θ1uλ1ρ1×1−1−exp−2expα2w−θ2wλ21−ρ21−exp−2expα2w−θ2wλ2ρ2.Type-III:LetA¨u|u∈0,1 and V¨w|w∈0,1 be two functional forms, satisfying all the abovementioned conditions where(30)A¨u=ulogu˙+1|u˙+u=1,V¨w=wlogw˙+1|w˙+w=1.In this case, one can also derive a closed form expression for the associated CDF of the BTIITLIFW-FGM (Type-III) from(31)ℍρu,w=uwρA¨uV¨u+1,as follows(32)ℍρu,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2×1+ρ1−exp−2expα1u−θ1uλ1log2−1−exp−2expα1u−θ1uλ1×1−exp−2expα2w−θ2wλ2log2−1−exp−2expα2w−θ2wλ2.


### 3.4. Via Renyi’s Entropy

Letm∈0,1=FV¯1x1 and w∈0,1=FV¯2x2. Then, the Renyi’s entropy copula can be expressed as(33)ℍu,w=x2u+x1w−x1x2.Then, the associated BTIITLIFW can be directly derived fromCm,w=CFP¯1x1,FP¯2x2 as(34)ℍu,w=x21−exp−2expα1x1−θ1x1λ1+x11−exp−2expα2x1−θ2x1λ2−x1x2.


## 3.1. Bivariate TIITLIFW (BTIITLIFW) via Morgenstern Family

The CDF of the Morgenstern family of two random RVsX1,X2 can be derived as(14)ℍρx1,x2|ρ≤1=F1x1F2x21+ρF¯1x1F¯2x2,where(15)F¯1x1=1−F1x1,F¯2x2=1−F2x2,setting(16)FV1¯x|V1¯=α1,θ1,λ1=1−1−exp−2expα1x1−θ1x1λ1,FV2¯x|V2¯=α2,θ2,λ2=1−1−exp−2expα2x2−θ2x2λ2,then we have(17)ℍρ,V1¯,V2¯x1,x2=1−1−exp−2expα1x1−θ1x1λ1×1−1−exp−2expα2x2−θ2x2λ2×1+ρ1−exp−2expα1x1−θ1x1λ1×1−exp−2expα2x2−θ2x2λ2.


## 3.2. Via Clayton Copula

The BTIITLIFW type extension: the weighted version of the Clayton copula can be expressed as(18)ℍu,v=u−ρ+v−φ1+φ2−1−1/ρ|ρ≥0.Let us assume thatX ∼ TIITLIFW V1¯ and Y ∼ TIITLIFW V2¯. Then, setting(19)u=uV1¯x=1−1−exp−2expα1x−θ1xλ1,v=uV2¯y=1−1−exp−2expα2y−θ2yλ2,the associated CDF of the BTIITLIFW type distribution can be written as(20)ℍx,y=1−1−exp−2expα1x−θ1xλ1−ρ+1−1−exp−2expα2y−θ2yλ2−ρ−1−1/ρ.A straightforwardM-dimensional extension from the above will be(21)ℍx1,x2,…,xM=∑i=1M1−1−exp−2expαixi−θixiλi−ρ−M+1−1/ρ.


## 3.3. Via Modified Farlie-Gumbel-Morgenstern (FGM) Copula

The joint CDF of the modified FGM copula is given asℍρu,w=uw1+ρAuVw|ρ∈−1,1 or ℍρu,w=uw+ρA˙uV˙w|ρ∈−1,1, where A˙u=uAu, and V˙w=wVw, where Au and Vw are being two continuous functions on the interval 0,1 where 0=A0=A1=V0=V1. Let(22)c1,u=inf∂∂uA˙u|k1u<0,c1,u=sup∂∂uA˙u|k1u<0,d2,w=inf∂∂wV˙w|k2w>0,d2,w=sup∂∂wV˙w|k2w>0.Then,1≤minc1.uc1,u,d2,wd2,w≤∞, where(23)∂∂uAu=Au+u∂∂uAu,k1u=u:u∈0,1|∂A˙u∂uexists,k2w=w:w∈0,1|∂V˙w∂wexists.Type I:Recalling the following functional form for bothAu and Vw. Then, the BTIITLIFW-FGM (Type-I) can be derived from(24)ℍρu,w=uw+ρA˙uV˙w|ρ∈−1,1,where(25)A˙u=u1−FV¯1u,V˙w=wFV¯2w.Therefore,(26)ℍρu,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2+ρ1−1−exp−2expα1u−θ1uλ11−1−exp−2expα2w−θ2wλ2|ρ∈−1,1.Type II:LetAu∗ and Vw∗ be two functional forms, satisfying all the conditions mentioned above, where(27)Au∗|ρ1>0=1−u1−ρ1uρ1,Vw∗|ρ2>0=1−w1−ρ2wρ2.Then, the corresponding BTIITLIFW-FGM(Type-II) can be derived from(28)ℍρ,ρ1,ρ2u,w=uw1+ρAu∗Vw∗.Then,(29)ℍρ,ρ1,ρ2u,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2×1+ρ1−1−exp−2expα1u−θ1uλ11−ρ11−exp−2expα1u−θ1uλ1ρ1×1−1−exp−2expα2w−θ2wλ21−ρ21−exp−2expα2w−θ2wλ2ρ2.Type-III:LetA¨u|u∈0,1 and V¨w|w∈0,1 be two functional forms, satisfying all the abovementioned conditions where(30)A¨u=ulogu˙+1|u˙+u=1,V¨w=wlogw˙+1|w˙+w=1.In this case, one can also derive a closed form expression for the associated CDF of the BTIITLIFW-FGM (Type-III) from(31)ℍρu,w=uwρA¨uV¨u+1,as follows(32)ℍρu,w=1−exp−2expα1u−θ1uλ11−exp−2expα2w−θ2wλ2×1+ρ1−exp−2expα1u−θ1uλ1log2−1−exp−2expα1u−θ1uλ1×1−exp−2expα2w−θ2wλ2log2−1−exp−2expα2w−θ2wλ2.


## 3.4. Via Renyi’s Entropy

Letm∈0,1=FV¯1x1 and w∈0,1=FV¯2x2. Then, the Renyi’s entropy copula can be expressed as(33)ℍu,w=x2u+x1w−x1x2.Then, the associated BTIITLIFW can be directly derived fromCm,w=CFP¯1x1,FP¯2x2 as(34)ℍu,w=x21−exp−2expα1x1−θ1x1λ1+x11−exp−2expα2x1−θ2x1λ2−x1x2.


## 4. Statistical and Reliability Measures

### 4.1. Quantile Function

For RVZ has CDF of the TIITLIFW distribution, the quantile function Zq of the TIITLIFW distribution is given by the following equation:(35)Zq=12θ−dq+d2q+4θα,0<q<1,where(36)dq=log−12log1−1−q1/λ.


### 4.2. Moments and Generating Functions

Therth moment of the TIITLIFW distribution is obtained using the formula(37)μr′z=∫0∞zrfzdz,hence using equation (7) we obtain(38)μr′z=∑i,j,k=0∞∇i,j,k∫0∞θzr−k+αzr−k−2exp−j+1θzdz.Settingy=j+1θz, it follows that(39)μr′z=∑i,j,k=0∞∇i,j,kθΓr−k+1θ∗r−k+1+αΓr−k−1θ∗r−k−1|θ∗=j+1θ,where(40)Γ1+∇=∫0∞z∇exp−zdzis a gamma function. In particular, if r=1 and r=2, we obtain the mean and variance of the TIITLIFW distribution. The MGF of the TIITLIFW is given by(41)MZt=EexptZ=∑i,j,k=0∞∇i,j,kθΓ1−kj+1θ−t1−k+αΓ−k−1j+1θ−t−k−1.The mathematical form of the Galton skewness and Moors kurtosis of TIITLIFW distribution can be computed using the quantile function and well-known relationships. The first four moments, skewness, and kurtosis of the TIITLIFW distribution for different values of parameters are represented in Table1. Table 1 shows that the skewness is always positive, and kurtosis is always greater than three.Table 1
Moments, skewness, and kurtosis of the TIITLIFW distribution.
αθλμ₁′μ₂′μ₃′μ₄′SkewnessKurtosis1111.931.0501.868.61.7407.871340.610.0100.00090.00050.8804.372720.580.0040.00039.90×10−51.0745.2321050.450.0019.6×10−61.85×10−60.4893.513181.650.0450.0060.0070.5683.6133100.960.0048.9×10−56.09×10−50.3123.2447170.720.00069.2×10−79.96×10−70.0663.02410200.610.00031.5×10−102.00×10−73.55×10−53.002Figure3 shows two plots of the skewness of the TIITLIFW distribution. Figure 4 presents two plots of the kurtosis of the TIITLIFW distribution. Based on Figures 3 and 4, we conclude that the TIITLIFW model can have many useful skewness and kurtosis shapes.Figure 3
Plots of the skewness of the TIITLIFW distribution. (a) Skewness forλ=1.1. (b) Skewness for λ=7.
(a)(b)Figure 4
Plots of the kurtosis of the TIITLIFW distribution. (a) Kurtosis forλ=1.1. (b) Kurtosis for λ=7.
(a)(b)


### 4.3. Incomplete Moments

Thesth lower and upper incomplete moments of Z are defined by(42)υs,Zu=EZs|Z<u=∫0ufzzsdz,Os,Zu=EZs|Z>u=∫u∞zsfzzsdz,respectively, for any real = s>0. The sth lower incomplete moment of the TIITLIFW distribution is(43)υs,Zt=∫0tzsfzdz=∑i,j,k=0∞∇i,j,k∫0tθzs−k+αzs−k−2exp−j+1θzdz=∑i,j,k=0∞∇i,j,kθγs−k+1,j+1θtθ∗s−k+1+αγs−k−1,j+1θtθ∗s−k−1,where(44)γs1,s2=∫0s2zs1−1exp−zdzis the lower incomplete gamma function. Similarly, the sth upper incomplete moment of the TIITLIFW distribution is(45)Os,Zt=∫t∞zsfzdz=∑i,j,k=0∞∇i,j,k∫t∞θzs−k+αzs−k−2exp−j+1θzdz=∑i,j,k=0∞∇i,j,kθΓs−k+1,j+1θtθ∗s−k+1+αΓs−k−1,j+1θtθ∗s−k−1,where(46)Γ1+s,t=∫t∞zsexp−zdzis the upper incomplete gamma function.


### 4.4. Mean Deviation, Lorenz, and Bonferroni Curves

For RVZ with PDF, fz, distribution function Fz, mean μ=EZ, and M=MedianZ, the mean deviation about the mean and median, respectively, is given by(47)δ1,μz=∫0∞z−μfzdz=2μFμ−2μ+2O1μ,δ2,Mz=∫0∞z−Mfzdz=2MFM−M−μ+2O1M,where(48)O1,Zd=∫d∞zfzdz=∑i,j,k=0∞∇i,j,kθΓ−k+2,k+1θdθ∗−k+2+αΓ−k,j+1θdθ∗−k.The Lorenz curve for a positive RVZ is defined as(49)Lp=1μ∫0qzfzdz=1μυ1,Zq.Then, we have(50)Lp=1μ∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θqθ∗−k+2+αγ−k,j+1θqθ∗−k,where q=G−1p. Also, Bonferroni curve is defined by(51)Bp=1μp∫0qzfzdz=υ1,Zqμp.Then,(52)Bp=1μp∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θqθ∗−k+2+αγ−k,j+1θqθ∗−k.


### 4.5. The Mean and Strong Mean Inactivity Times Functions

The mean inactivity time (MIT) can be derived from(53)mt=Et−Z|Z≤t=t−1Ft∫0tzfzdz.Then, the MIT can be derived as(54)mt=t−1Ft∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θtθ∗−k+2+αγ−k,j+1θtθ∗−k.The strong mean inactivity time (SMIT) is a new reliability function given by(55)St=1Ft∫0t2zFzdz=t−1Ft∫0tz2fzdz.Therefore, the SMIT can be expressed as(56)St=t−1Ft∑i,j,k=0∞∇i,j,kθγ−k+3,j+1θtθ∗−k+3+αγ1−k,j+1θtθ∗1−k.


## 4.1. Quantile Function

For RVZ has CDF of the TIITLIFW distribution, the quantile function Zq of the TIITLIFW distribution is given by the following equation:(35)Zq=12θ−dq+d2q+4θα,0<q<1,where(36)dq=log−12log1−1−q1/λ.


## 4.2. Moments and Generating Functions

Therth moment of the TIITLIFW distribution is obtained using the formula(37)μr′z=∫0∞zrfzdz,hence using equation (7) we obtain(38)μr′z=∑i,j,k=0∞∇i,j,k∫0∞θzr−k+αzr−k−2exp−j+1θzdz.Settingy=j+1θz, it follows that(39)μr′z=∑i,j,k=0∞∇i,j,kθΓr−k+1θ∗r−k+1+αΓr−k−1θ∗r−k−1|θ∗=j+1θ,where(40)Γ1+∇=∫0∞z∇exp−zdzis a gamma function. In particular, if r=1 and r=2, we obtain the mean and variance of the TIITLIFW distribution. The MGF of the TIITLIFW is given by(41)MZt=EexptZ=∑i,j,k=0∞∇i,j,kθΓ1−kj+1θ−t1−k+αΓ−k−1j+1θ−t−k−1.The mathematical form of the Galton skewness and Moors kurtosis of TIITLIFW distribution can be computed using the quantile function and well-known relationships. The first four moments, skewness, and kurtosis of the TIITLIFW distribution for different values of parameters are represented in Table1. Table 1 shows that the skewness is always positive, and kurtosis is always greater than three.Table 1
Moments, skewness, and kurtosis of the TIITLIFW distribution.
αθλμ₁′μ₂′μ₃′μ₄′SkewnessKurtosis1111.931.0501.868.61.7407.871340.610.0100.00090.00050.8804.372720.580.0040.00039.90×10−51.0745.2321050.450.0019.6×10−61.85×10−60.4893.513181.650.0450.0060.0070.5683.6133100.960.0048.9×10−56.09×10−50.3123.2447170.720.00069.2×10−79.96×10−70.0663.02410200.610.00031.5×10−102.00×10−73.55×10−53.002Figure3 shows two plots of the skewness of the TIITLIFW distribution. Figure 4 presents two plots of the kurtosis of the TIITLIFW distribution. Based on Figures 3 and 4, we conclude that the TIITLIFW model can have many useful skewness and kurtosis shapes.Figure 3
Plots of the skewness of the TIITLIFW distribution. (a) Skewness forλ=1.1. (b) Skewness for λ=7.
(a)(b)Figure 4
Plots of the kurtosis of the TIITLIFW distribution. (a) Kurtosis forλ=1.1. (b) Kurtosis for λ=7.
(a)(b)


## 4.3. Incomplete Moments

Thesth lower and upper incomplete moments of Z are defined by(42)υs,Zu=EZs|Z<u=∫0ufzzsdz,Os,Zu=EZs|Z>u=∫u∞zsfzzsdz,respectively, for any real = s>0. The sth lower incomplete moment of the TIITLIFW distribution is(43)υs,Zt=∫0tzsfzdz=∑i,j,k=0∞∇i,j,k∫0tθzs−k+αzs−k−2exp−j+1θzdz=∑i,j,k=0∞∇i,j,kθγs−k+1,j+1θtθ∗s−k+1+αγs−k−1,j+1θtθ∗s−k−1,where(44)γs1,s2=∫0s2zs1−1exp−zdzis the lower incomplete gamma function. Similarly, the sth upper incomplete moment of the TIITLIFW distribution is(45)Os,Zt=∫t∞zsfzdz=∑i,j,k=0∞∇i,j,k∫t∞θzs−k+αzs−k−2exp−j+1θzdz=∑i,j,k=0∞∇i,j,kθΓs−k+1,j+1θtθ∗s−k+1+αΓs−k−1,j+1θtθ∗s−k−1,where(46)Γ1+s,t=∫t∞zsexp−zdzis the upper incomplete gamma function.


## 4.4. Mean Deviation, Lorenz, and Bonferroni Curves

For RVZ with PDF, fz, distribution function Fz, mean μ=EZ, and M=MedianZ, the mean deviation about the mean and median, respectively, is given by(47)δ1,μz=∫0∞z−μfzdz=2μFμ−2μ+2O1μ,δ2,Mz=∫0∞z−Mfzdz=2MFM−M−μ+2O1M,where(48)O1,Zd=∫d∞zfzdz=∑i,j,k=0∞∇i,j,kθΓ−k+2,k+1θdθ∗−k+2+αΓ−k,j+1θdθ∗−k.The Lorenz curve for a positive RVZ is defined as(49)Lp=1μ∫0qzfzdz=1μυ1,Zq.Then, we have(50)Lp=1μ∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θqθ∗−k+2+αγ−k,j+1θqθ∗−k,where q=G−1p. Also, Bonferroni curve is defined by(51)Bp=1μp∫0qzfzdz=υ1,Zqμp.Then,(52)Bp=1μp∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θqθ∗−k+2+αγ−k,j+1θqθ∗−k.


## 4.5. The Mean and Strong Mean Inactivity Times Functions

The mean inactivity time (MIT) can be derived from(53)mt=Et−Z|Z≤t=t−1Ft∫0tzfzdz.Then, the MIT can be derived as(54)mt=t−1Ft∑i,j,k=0∞∇i,j,kθγ−k+2,j+1θtθ∗−k+2+αγ−k,j+1θtθ∗−k.The strong mean inactivity time (SMIT) is a new reliability function given by(55)St=1Ft∫0t2zFzdz=t−1Ft∫0tz2fzdz.Therefore, the SMIT can be expressed as(56)St=t−1Ft∑i,j,k=0∞∇i,j,kθγ−k+3,j+1θtθ∗−k+3+αγ1−k,j+1θtθ∗1−k.


## 5. Non-Bayesian Estimation Methods under Uncensored Schemes

### 5.1. The MLE Method

LetZ1,Z2,…,Zn be a random sample of size n from TIITLIFW. The log-likelihood function for the vector of parameters V¯ can be written as(57)logℒ=nlog2λ+∑i=1nlogθ+αzi:n2−α∑i=1nzi:n−1−θ∑i=1nzi:n−2∑i=1nexpαzi:n−θzi:n+λ−1∑i=1nlog1−exp−2expαzi:n−θzi:n.The associated score function is given by(58)UnV¯=∂logℒ∂λ,∂logℒ∂α,∂logℒ∂θT.Thelogℒ can be maximized by solving the following nonlinear likelihood equations UnV¯|α=∂logℒ/∂α,UnV¯|θ=∂logℒ/∂θ and UnV¯|λ=∂logℒ/∂λ. Then,(59)UnV¯|α=1α+θzi:n2−∑i=1nzi:n−1−2∑i=1nzi:n−1expαzi:n−θzi:n+2λ−1∑i=1nexp−2expα/zi:n−θzi:nexpα/zi:n−θzi:nzi:n−11−exp−2expα/zi:n−θzi:n,UnV¯|θ=∑i=1nzi2α+θzi:n2−∑i=1nzi:n−2∑i=1nzi:nexpαzi:n−θzi:n+2λ−1∑i=1nzi:nexp−2expα/zi:n−θzi:nexpα/zi:n−θzi:n1−exp−2expα/zi:n−θzi:n,UnV¯|λ=nλ+∑i=1nlog1−exp−2expαzi:n−θzi:n.The maximum likelihood estimation (MLE) ofV¯, say V⌢¯, is obtained by solving the system of nonlinear equations(60)UnV¯|α=UnV¯|θ=UnV¯|λ=0.


### 5.2. The CVME Method

The CVMEs of the parametersα,θ, and λ [12] are obtained by minimizing the following expression with respect to the parameters α,θ, and λ, respectively, where(61)CVMEV¯=112n−1+∑i=1nFV¯zi:n−pi,n2,where(62)pi,n=2i−12nrefers to the empirical estimate of the CDF at zi:n computed from a certain sample and(63)CVMEV¯=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,n2.Then, CVMEs of the parametersα,θ, and λ are obtained by solving the following nonlinear system:(64)∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDαzi:n;V¯=0,∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDθzi:n;V¯=0,∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDλzi:n;V¯=0,where(65)Dαzi:n;V¯=∂FV¯zi:n∂β,Dθzi:n;V¯=∂FV¯zi:n∂b,Dλzi:n;V¯=∂FV¯zi:n∂a.


### 5.3. The OLSE and WLSE Method

LetFV¯zi:n denote the CDF of the TIITLIFW model and z1:n<z2:n<⋯<zn:n be the n ordered RS. The OLSEs [13] are obtained by minimizing(66)OLSEV¯=∑i=1nFV¯zi:n−qi,n2,where(67)qi,n=in+1.Then,(68)OLSEV¯=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,n2.The OLSEs are obtained by solving(69)0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDαzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDθzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDλzi:n;V¯,where Dαzi:n;V¯, Dθzi:n;V¯, and Dλzi:n;V¯ are as defined above. The WLSE is obtained by minimizing the function WLSE V¯ with respect to α,θ, and λ. Then,(70)WLSEV¯=∑i=1nτi,nFV¯zi:n−qi,n2,where(71)τi,n=1+n22+ni1+n−i.The WLSEs are obtained by solving(72)0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDαzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDθzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDλzi:n;V¯.


### 5.4. The ADE Method

The ADEs are obtained by minimizing the function(73)ADEzi:n,z−i+1+n:nV¯=−n−n−1∑i=1n2i−1×logFV¯zi:n+logFV¯¯,where(74)FV¯¯z−i+1+n:n=1−FV¯z−i+1+n:n.Then, the parameter estimates are derived by solving the nonlinear equations(75)0=∂ADEzi:n,z−i+1+n:nV¯∂α,0=∂ADEzi:n,z−i+1+n:nV¯∂θ,0=∂ADEzi:n,z−i+1+n:nV¯∂λ.


### 5.5. The ADE (R-T) Method

The ADEs (R-T) are obtained by minimizing(76)ADER−Tzi:n,z−i+1+n:nV¯=12n−2∑i=1nFV¯zi:n−1n∑i=1n2i−1logFV¯¯z−i+1+n:n.Then, the estimates follow by solving the nonlinear equations(77)0=∂ADER−Tzi:n,z−i+1+n:nV¯∂α,0=∂ADER−Tzi:n,z−i+1+n:nV¯∂θ,0=∂ADER−Tzi:n,z−i+1+n:nV¯∂λ.


## 5.1. The MLE Method

LetZ1,Z2,…,Zn be a random sample of size n from TIITLIFW. The log-likelihood function for the vector of parameters V¯ can be written as(57)logℒ=nlog2λ+∑i=1nlogθ+αzi:n2−α∑i=1nzi:n−1−θ∑i=1nzi:n−2∑i=1nexpαzi:n−θzi:n+λ−1∑i=1nlog1−exp−2expαzi:n−θzi:n.The associated score function is given by(58)UnV¯=∂logℒ∂λ,∂logℒ∂α,∂logℒ∂θT.Thelogℒ can be maximized by solving the following nonlinear likelihood equations UnV¯|α=∂logℒ/∂α,UnV¯|θ=∂logℒ/∂θ and UnV¯|λ=∂logℒ/∂λ. Then,(59)UnV¯|α=1α+θzi:n2−∑i=1nzi:n−1−2∑i=1nzi:n−1expαzi:n−θzi:n+2λ−1∑i=1nexp−2expα/zi:n−θzi:nexpα/zi:n−θzi:nzi:n−11−exp−2expα/zi:n−θzi:n,UnV¯|θ=∑i=1nzi2α+θzi:n2−∑i=1nzi:n−2∑i=1nzi:nexpαzi:n−θzi:n+2λ−1∑i=1nzi:nexp−2expα/zi:n−θzi:nexpα/zi:n−θzi:n1−exp−2expα/zi:n−θzi:n,UnV¯|λ=nλ+∑i=1nlog1−exp−2expαzi:n−θzi:n.The maximum likelihood estimation (MLE) ofV¯, say V⌢¯, is obtained by solving the system of nonlinear equations(60)UnV¯|α=UnV¯|θ=UnV¯|λ=0.


## 5.2. The CVME Method

The CVMEs of the parametersα,θ, and λ [12] are obtained by minimizing the following expression with respect to the parameters α,θ, and λ, respectively, where(61)CVMEV¯=112n−1+∑i=1nFV¯zi:n−pi,n2,where(62)pi,n=2i−12nrefers to the empirical estimate of the CDF at zi:n computed from a certain sample and(63)CVMEV¯=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,n2.Then, CVMEs of the parametersα,θ, and λ are obtained by solving the following nonlinear system:(64)∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDαzi:n;V¯=0,∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDθzi:n;V¯=0,∑i=1n1−1−exp−2expαzi:n−θzi:nλ−pi,nDλzi:n;V¯=0,where(65)Dαzi:n;V¯=∂FV¯zi:n∂β,Dθzi:n;V¯=∂FV¯zi:n∂b,Dλzi:n;V¯=∂FV¯zi:n∂a.


## 5.3. The OLSE and WLSE Method

LetFV¯zi:n denote the CDF of the TIITLIFW model and z1:n<z2:n<⋯<zn:n be the n ordered RS. The OLSEs [13] are obtained by minimizing(66)OLSEV¯=∑i=1nFV¯zi:n−qi,n2,where(67)qi,n=in+1.Then,(68)OLSEV¯=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,n2.The OLSEs are obtained by solving(69)0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDαzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDθzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nDλzi:n;V¯,where Dαzi:n;V¯, Dθzi:n;V¯, and Dλzi:n;V¯ are as defined above. The WLSE is obtained by minimizing the function WLSE V¯ with respect to α,θ, and λ. Then,(70)WLSEV¯=∑i=1nτi,nFV¯zi:n−qi,n2,where(71)τi,n=1+n22+ni1+n−i.The WLSEs are obtained by solving(72)0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDαzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDθzi:n;V¯,0=∑i=1n1−1−exp−2expαzi:n−θzi:nλ−qi,nτi,nDλzi:n;V¯.


## 5.4. The ADE Method

The ADEs are obtained by minimizing the function(73)ADEzi:n,z−i+1+n:nV¯=−n−n−1∑i=1n2i−1×logFV¯zi:n+logFV¯¯,where(74)FV¯¯z−i+1+n:n=1−FV¯z−i+1+n:n.Then, the parameter estimates are derived by solving the nonlinear equations(75)0=∂ADEzi:n,z−i+1+n:nV¯∂α,0=∂ADEzi:n,z−i+1+n:nV¯∂θ,0=∂ADEzi:n,z−i+1+n:nV¯∂λ.


## 5.5. The ADE (R-T) Method

The ADEs (R-T) are obtained by minimizing(76)ADER−Tzi:n,z−i+1+n:nV¯=12n−2∑i=1nFV¯zi:n−1n∑i=1n2i−1logFV¯¯z−i+1+n:n.Then, the estimates follow by solving the nonlinear equations(77)0=∂ADER−Tzi:n,z−i+1+n:nV¯∂α,0=∂ADER−Tzi:n,z−i+1+n:nV¯∂θ,0=∂ADER−Tzi:n,z−i+1+n:nV¯∂λ.


## 6. Comparing the Non-Bayesian Estimation Methods under Uncensored Schemes via a Simulation Study

A numerical simulation study is conducted to compare the non-Bayesian estimation methods. The simulation study is based onN=1000 which generated datasets from the TIITLIFW distribution where n=20,60,100,200, and 500 and α=2,θ=3, and λ=1.1. The comparison is performed based on the bias BIASV¯ and root mean - standard error RMSEV¯.From Table2, we note the following:1) TheBIASV¯ tends to 0 as n increases and tends to ∞ which means that all estimators are nonbiased.2) TheRMSEV¯ tends to 0 as n increases and tends to ∞ which means incidence of consistency property.Table 2
Simulation study where the parametersα=2,θ=3, and λ=1.1.
nαθλBIASRMSEBIASRMSEBIASRMSEMLE203.9606646617.3712925306.50297605812.278605855.99443441019.60201039CVM1.6205841714.0697299582.2132707306.52976607612.5999318030.87329914OLS2.7101784605.6090865275.16707615510.577460964.26978127311.26286107WLS0.6069866311.3288896441.2401193352.8013317540.3030937511.238119094ADE0.8779951101.077412768−1.634642151.7619635130.4381869180.623443953ADE (R-T)0.8054060790.893762293−1.643685311.7097710200.4678850670.699341007MLE600.4904907301.3084286450.7958295772.4864672500.6825383592.697087211CVM0.2253506901.2150839790.2018135322.2825949602.6604819537.613754145OLS0.6905191242.0904604671.1952120753.9414274791.7658699276.040806374WLS0.6017640481.2471098251.2897256792.611933294-0.006444290.873971382ADE0.3737856970.452084104−2.167354112.1814781210.4231263880.611877218ADE (R-T)0.4571397280.511691568−2.127605562.1397454080.420177690.547818729MLE1000.0181727750.507215114−0.0253706131.0373064450.3197740710.782082919CVM0.5380248861.567154110.84517982202.8086146242.72148886410.63346516OLS0.2443210920.9086337230.44969647901.7556029450.8564440633.727821407WLS0.3721817731.0939685440.65479179902.0931928930.2964320691.102596681ADE0.2999289980.363326475−2.2947066872.3033368840.5535741100.880204347ADE (R-T)0.2993687040.357439216−2.2938116862.3014409940.3221066290.456459819MLE2000.0594901600.3795637070.0862404900.7993016980.1007500150.513932285CVM0.2061375660.7557632410.3438708961.4800815180.4595692372.626198546OLS0.1895690920.7059691350.3036838781.3591362000.6349675563.901570961WLS0.5312257620.9914401591.0254891591.903738414-0.141929110.631061741ADE0.1504728780.210401078−2.433517472.4373393040.3803664260.721144948ADE (R-T)0.1710834700.238382214−2.411883942.4164409800.2144222380.556727524MLE5000.0519556670.2421203490.1069917200.5144496590.00883300000.288297422CVM0.0167978520.343498639−0.009340920.6797923000.14960969900.459788656OLS-0.014988100.294877728−0.037302240.6095485520.13119532100.481402194WLS0.2784990220.5719079740.5372597801.106740303-0.109500150.406177306ADE0.0927223010.159113157−2.486914282.489745756-0.134884360.375397440ADE (R-T)0.0756200570.146911912−2.506398672.508592698-0.103857640.461934338


## 7. Comparing Models under Uncensorship

We illustrate the flexibility and the performance of the TIITLIFW distribution as compared to some alternative models using two real data applications. The goodness-of-fit statistics for this distribution are compared with other competitive distributions. The MLEs of the distribution parameters are determined numerically. To compare the distributions, we consider the measures of goodness-of-fit, such as Akaike information criterion (C1), consistent Akaike information criterion (C2), and Bayesian information criterion (C3) statistic. The better distribution to fit the data corresponds to smaller values of these statistics.We consider two uncensored datasets for comparing competitive models. The first data present the remission times (in months) of a random sample of 128 bladder cancer patients [14]: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.69, 4.23, 5.41, 0.90, 2.69, 4.18, 5.34, 7.59, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 5.71, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 10.66, 15.96, 36.66, 1.05, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, and 22.69. The second data present the lifetimes of 38 devices provided by [14]: 0.1, 1, 1, 1, 2, 3, 6, 40, 45, 46, 47, 50, 55, 0.2, 7, 11, 12, 18, 18, 18, 18, 18, 21, 32, 36, 1, 82, 83, 84, 84, 84, 85, 85, 1, 60, 63, and 86, 86. According to [15], the total time in test (TTT) plots, box plots, quantile-quantile (QQ) plots, and kernel density estimation (KDE) plots are shown in Figure 5 and Figures 6(a) and 6(c) for bladder cancer data and lifetimes data, respectively. Based on Figures 6(a) and 6(c), the HRF of the bladder cancer data is “upside down” or “reversed U-shape.” Based on Figures 6(a) and 6(c), the HRF of the lifetimes data is “U-shape.” The box plots (middle panels) are presented along with its corresponding normal quantile-quantile plot (right panels) in Figures 5 and 6 for discovering the outliers and normality.Figure 5
The TTT plot (a), box plot (b), QQ plot (c), and KDE plot (d) for cancer data.
(a)(b)(c)(d)Figure 6
The TTT plot (a), box plot (b), QQ plot (c), and KDE plot (d) for the lifetimes data.
(a)(b)(c)(d)The following competitive models are considered in the comparison: the exponentiated IFW (Exp-IFW) [12], IFW [3], exponentiated generalized IW (ExpG-IFW) [16], generalized IW (G-IW) [17], IW [18], and [2].Tables3 and 4 present the MLEs for the bladder cancer data and lifetimes data. Tables 5 and 6 show the statistics criteria for the bladder cancer data and lifetime data. From Tables 5 and 6, it is clear that the TIITLIFW distribution provides the best fits for the two datasets. Figures 7 and 8 show the estimated PDFs (EPDFs) (left panel) and the estimated HRF (EHRFs) (right panel) for bladder cancer data and lifetimes data, respectively. Figures 9 and 10 show the profile of the log-likelihood function for bladder cancer data and lifetimes data, respectively. From Figures 7 and 8, we conclude that the new model can achieve a good fit.Table 3
The MLEs for bladder cancer data.
ModelEstimatesαθλβTIITLIFW0.0700.4680.26Exp-IFW0.0800.1692.47IFW0.1260.143ExpG-IFW1.0060.5001.052G-IW0.7500.3401.79IW16.140.464FW0.0540.915Table 4
The MLEs for lifetimes data.
ModelEstimatesαθλβTIITLIFW0.1080.0251.198Exp-IFW0.0990.0302.187IFW0.1650.024ExpG-IFW1.0080.6102.140.75G-IW0.5960.2741.27IW1.0430.397FW0.0120.70Table 5
Statistics for bladder cancer data.
Model−2log (ℒ)C₁C₂C₃TIITLIFW413.85833.695833.889842.251Exp-IFW423.46852.909853.104861.470IFW453.61911.220911.310916.920ExpG-IFW488.05984.090984.420995.500G-IW495.18996.360996.5601004.92IW500.121004.251004.331009.94FW525.531055.071055.161060.77Table 6
Statistics for lifetimes data.
Model−2log (ℒ)C₁C₂C₃TIITLIFW233.51473.027473.549478.763Exp-IFW233.52473.029473.551478.765IFW242.57488.914489.169492.738ExpG-IFW250.81505.620505.880509.448G-IW254.92517.839518.727525.487IW287.48580.951581.473586.687FW281.07566.140566.396569.964Figure 7
The EPDF (a) and the EHRF (b) for bladder cancer data.
(a)(b)Figure 8
The EPDF (a) and the EHRF (b) for lifetimes data.
(a)(b)Figure 9
The profile of the log-likelihood function for bladder cancer data.
(a)(b)(c)Figure 10
The profile of the log-likelihood function for lifetimes data.
(a)(b)(c)


## 8. Concluding Remarks

A three-parameter lifetime distribution, so-called the TIITLIFW distribution, is introduced as an extension of the inverse flexible Weibull distribution. Some explicit expressions for mathematical quantities of the TIITLIFW distribution are derived. The hazard rate function allows constant, decreasing, increasing, upside down bathtub, or bathtub-shaped forms. We consider six different estimation methods to estimate the parameters of the TIITLIFW distribution. The performance of these proposed estimation methods is conducted via some simulations. A real data application proves that the TIITLIFW model provides consistently better fits compared to some other well-known competitive models.



---

*Source: 6657172-2021-01-21.xml*