# Application to Engineering and Medical Data Using Three-Parameter Exponential Model

Authors: Waleed Almutiry; Amani Abdullah Alahmadi; Ibrahim Elbatal; Ibrahim E. Ragab; Oluwafemi Samson Balogun; Mohammed Elgarhy
Journal: Mobile Information Systems
 (2021)

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## Abstract

This paper is devoted to a new lifetime distribution having three parameters by compound the exponential model and the transmuted Topp-Leone-G. The new proposed model is called the transmuted Topp-Leone exponential model; it is useful in lifetime data and reliability. The new model is very flexible; its pdf can be right skewness, unimodal, and decreasing shaped, but the hrf of the suggested model can be unimodal, constant, and decreasing. Numerous statistical characteristics of the new model, notably the quantile function, moments, incomplete moments, conditional moments, mean residual life, mean inactivity time, and entropy are produced and investigated. The system’s parameters are estimated using the maximum likelihood approach. All estimators should be theoretically convergent, which is supported by a simulation analysis. Finally, two real-world datasets from the engineering and medical disciplines explore the new model’s relevance and adaptability in comparison to the alternatives models such as the beta exponential, the Marshall–Olkin generalized exponential, the exponentiated Weibull, the modified Weibull, and the transmuted Burr type X models.


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## Body

## 1. Introduction

Many experts have offered a wide range of methods for including an extra parameter in distributions, and all these new families have been used to describe data from a wide variety of fields, covering engineering, economics, biological studies, environmental sciences, and so on. These families are created by adding an additional shape parameter to the parent distribution to enhance the capabilities and validity of the data modeling in reality. The statistical literature has demonstrated several new families, the transmuted-G by [1], the logistic-X family by [2], Topp-Leone-G (TL-G) by [3], type II half logistic G [4], odd-Burr G by [5], transmuted exponentiated generalized G by [6], Topp-Leone odd Lindley G by [7], odd Frechet G by [8], truncated inverted Kumaraswamy G [9], and truncated Cauchy power G [10], among others.Reference [11] recently published the TTL-G, a distinct family of continuous distributions with the next distribution function (cdf) and density function (pdf),(1)Fx=1+λ1−1−Gx2θ−λ1−1−Gx22θ,x>0,θ>0,λ<1,(2)fx=2θgx1−Gx1−1−Gx2θ−11+λ−2λ1−1−Gx2θ.The cdf and pdf corresponding to theE model, respectively, are(3)Gx=1−e−αx;x>0,α>0,(4)gx=αe−αx,where α is the scale parameter. Many authors developed the E model recently like [12–14].The major goal of this research is to employ theE model as a baseline model in the TTL-G to create a novel extension of the E model called the TTLE model. Inserting equations (3) and (4) in equations (1) and (2), then the cdf and pdf of the proposed model are(5)FTTLEx=1+λ1−e−2αxθ−λ1−e−2αx2θ,(6)fTTLEx=2θαe−2αx1−e−2αxθ−11+λ−2λ1−e−2αxθ.The random variable (RV) with pdf (6) is now indicated as X∼TTLEλ,θ,α.Figure1 demonstrates plots of the pdf of the TTLE model for number of model parameter values. We can note from Figure 1 that the pdf can be decreasing, right skewed, and unimodal shaped.Figure 1
The TTLE pdf shown against various parameter values.
(a)(b)The remainder of this article is organized as follows: Section2 focuses on the various new models’ statistical characteristics. Section 3 presents reliability measurements such as survival function (sf), hazard rate (hr) function, MREL, and MINT. Section 4 introduces the parameter maximum likelihood estimates by using MATHCAD (14) statistical software. In Section 5, two real-world examples demonstrate the potential and adaptability of the offered model. Results are summarized in Section 6.


## 2. Structural Properties

In this part, we covered the TTLE distribution’s characteristics such as the QuF, Mos, Mo generating function, InMos, CMos, MREL, mean inactivity time (MINT), and entropy (EN).


### 2.1. Quantile Function

The QuF of the TTLE is(7)Qu=F−1u=−12αln1−1+λ−1+λ2−4λu2λ1/θ,where u is a uniform distribution on (0, 1). The Bowley’s skewness (BS) measure is dependent on a quantile, see [15], and it is defined by(8)BS=Q0.75+Q0.25−2Q0.5Q0.75−Q0.25.On the other hand, Moors [16] proposed measure of kurtosis (MK) also based on quantiles and it is(9)MK=Q0.875−Q0.625+Q0.375−Q0.125Q0.75−Q0.25,where Q(.) represents the QuF. Outliers have less of an impact on the measures BS and MK, which exist even for distributions that do not have moments. Figure 2 depicts how BS and MK behave in response to a TTLE distribution. The BS and MK are decreasing from Figure 2.Figure 2
The BS and MK of TTLE distribution.
(a)(b)


### 2.2. Moments

Therth Mo of TTLE model is computed from(10)μr′x=∫0∞xrfxdx=2θα1+λ∫0∞xre−2αx1−e−2αxθ−1dx−4λθα∫0∞xre−2αx1−e−2αx2θ−1dx.Ifw<1 and a>0 is the positive real noninteger, then the binomial series expansion holds.(11)1−wb−1=∑j=0∞−1ja−1jwj,using equation (11), and after some algebraic simplification, the rth Mo can be expressed as(12)μr′x=∑i=0∞−1iθΓr+12αri+1r+11+λθ−1i−2λ2θ−1i.Similarly, the Mo generating function of TTLE model can be deduced from equation (6) as follows:(13)MXt=EetX=∫0∞etxfxdx=∑i=0∞−1i2αθ2αi+1−t1+λθ−1i−2λ2θ−1i.


### 2.3. Conditional Moments

Thesth upper InMo of TTLE distribution can be computed as(14)ϑst=EXs|X>t=∫t∞xsfxdx=∑i=0∞−1iθΓs+1,2αi+1t2αsi+1s+11+λθ−1i−2λ2θ−1i,where Γs,t=∫t∞xs−1e−xdx is the upper in gamma function. Similarly, the sth lower InMo of TTLE model is(15)ϕst=EXs|X<t=∫0∞xsfxdxϕst=∑i=0∞−1iθγs+1,2αi+1t2αsi+1s+11+λθ−1i−2λ2θ−1i,where γs,t=∫0txs−1e−xdx is the lower in gamma function.


### 2.4. Entropy

The Rényi EN is calculated from(16)IRρ=11−ρlog∫0∞fρxdx,ρ>0,ρ≠1,Using equation (6), we have(17)∫0∞fρxdx=2θαρ∫0∞e−2ραx1−e−2αxρθ−11+λ−2λ1−e−2αxθρdx=2θαρ∑j,k=0∞−1j+k2λ1+λjθρ+j−ρk12αρ+k.Thus,(18)IRρ=ρ1−ρlog2θα+log∑j,k=0∞−1j+k2λ1+λjθρ+j−ρk12αρ+k.


## 2.1. Quantile Function

The QuF of the TTLE is(7)Qu=F−1u=−12αln1−1+λ−1+λ2−4λu2λ1/θ,where u is a uniform distribution on (0, 1). The Bowley’s skewness (BS) measure is dependent on a quantile, see [15], and it is defined by(8)BS=Q0.75+Q0.25−2Q0.5Q0.75−Q0.25.On the other hand, Moors [16] proposed measure of kurtosis (MK) also based on quantiles and it is(9)MK=Q0.875−Q0.625+Q0.375−Q0.125Q0.75−Q0.25,where Q(.) represents the QuF. Outliers have less of an impact on the measures BS and MK, which exist even for distributions that do not have moments. Figure 2 depicts how BS and MK behave in response to a TTLE distribution. The BS and MK are decreasing from Figure 2.Figure 2
The BS and MK of TTLE distribution.
(a)(b)


## 2.2. Moments

Therth Mo of TTLE model is computed from(10)μr′x=∫0∞xrfxdx=2θα1+λ∫0∞xre−2αx1−e−2αxθ−1dx−4λθα∫0∞xre−2αx1−e−2αx2θ−1dx.Ifw<1 and a>0 is the positive real noninteger, then the binomial series expansion holds.(11)1−wb−1=∑j=0∞−1ja−1jwj,using equation (11), and after some algebraic simplification, the rth Mo can be expressed as(12)μr′x=∑i=0∞−1iθΓr+12αri+1r+11+λθ−1i−2λ2θ−1i.Similarly, the Mo generating function of TTLE model can be deduced from equation (6) as follows:(13)MXt=EetX=∫0∞etxfxdx=∑i=0∞−1i2αθ2αi+1−t1+λθ−1i−2λ2θ−1i.


## 2.3. Conditional Moments

Thesth upper InMo of TTLE distribution can be computed as(14)ϑst=EXs|X>t=∫t∞xsfxdx=∑i=0∞−1iθΓs+1,2αi+1t2αsi+1s+11+λθ−1i−2λ2θ−1i,where Γs,t=∫t∞xs−1e−xdx is the upper in gamma function. Similarly, the sth lower InMo of TTLE model is(15)ϕst=EXs|X<t=∫0∞xsfxdxϕst=∑i=0∞−1iθγs+1,2αi+1t2αsi+1s+11+λθ−1i−2λ2θ−1i,where γs,t=∫0txs−1e−xdx is the lower in gamma function.


## 2.4. Entropy

The Rényi EN is calculated from(16)IRρ=11−ρlog∫0∞fρxdx,ρ>0,ρ≠1,Using equation (6), we have(17)∫0∞fρxdx=2θαρ∫0∞e−2ραx1−e−2αxρθ−11+λ−2λ1−e−2αxθρdx=2θαρ∑j,k=0∞−1j+k2λ1+λjθρ+j−ρk12αρ+k.Thus,(18)IRρ=ρ1−ρlog2θα+log∑j,k=0∞−1j+k2λ1+λjθρ+j−ρk12αρ+k.


## 3. Reliability Measures

We discussed reliability metrics such as the sf, hr function, MREL, and MINT in this part.The reliability function (or the sf) of TTLE model is derived by(19)F¯TTLEx=1−1−e−2αxθ1+λ−λ1−e−2αxθ.The hr (or failure rate) function represents the time it takes for a unit to fail or component or a device. It is useful in important measures of ageing.For the purposes of distribution, the hr function is deduced as follows:(20)hTTLEx=fxF¯x=2θαe−2αx1−e−2αxθ−11+λ−2λ1−e−2αxθ1−1−e−2αxθ1+λ−λ1−e−2αxθ.Figure3 illustrates the shapes of sf and hr functions for various values of the distribution parameters. The sf of the TTLE distribution is constantly decreasing, as shown in Figure 3(a). On the other hand, this distribution has the unimodal, constant, and decreasing hr shapes (Figure 3(b)).Figure 3
(a) Plots of TTLE sf. (b) Plots of TTLE hr function.
(a)(b)In the estimation of a sf for left-censored lifetimes, the reversed hr (Rhr) function is crucial. The TTLE distribution’s Rhr is calculated by(21)τTTLEx=fxFx=2θαe−2αx1+λ−2λ1−e−2αxθ1−e−2αx1+λ−λ1−e−2αxθ.The MREL function is essential in reliability survival analysis and it is used to model the burn-in and conservation of the component. MREL for TTLE distribution is(22)μt=EX−t|X>t=1F¯t∫t∞xfxdx−t.Then,(23)μt=1F¯t−1iθΓ2,2αi+1t2α2i+121+λθ−1i−2λ2θ−1i−t.The MINT function is also exciting because it indicates the time that has passed since a component failed, assuming that its lifespan is less than or equal tot. That RV is indeed referred to as the time since failure. The MINT is symbolized with(24)mt=EX−t|X<t=t−1Ft∫0txfxdx=t−1Ft∑i=0∞−1iθγ2,2αi+1t2α2i+121+λθ−1i−2λ2θ−1i.Table1 displays the MREL and the MINT for the proposed model at the point t=0.5 for some selected values of distribution parameters (θ, λ), namely, (5.5, −1), (2.5,−0.5), (1, 0), (0.5, 0.5), and (0.01, 1), with fixed parameter α=2. It is observed that the MREL is decreasing and the MINT is increasing when θ decreases and λ increases.Table 1
The MREL and MINT of TTLE distribution.
θ↓λ↑α=2,t=0.5MREL↓MINT↑5.5−10.3437990.0959782.5−0.50.2782790.201663100.2500000.3282590.50.50.2374180.4195100.0110.1191100.499943Figure4 illustrates the behavior of the MREL and MINT for some different levels of distribution parameters. It is seen that the MREL (Figure 4(a)) is decreasing while the MINT (Figure 4(b)) is increasing.Figure 4
(a) Plot of MREL and (b) plot of MINT for selected parameter values.
(a)(b)


## 4. Maximum Likelihood Estimation

MLL estimates (MLLEs) were used to estimate the TTLE distribution’s unknown parameters. Supposex1,x2,…,xn be an n-th random sample from the TTLE distribution given by equation (6). The TTLE distribution’s log-likelihood formula is represented by(25)Ln=nlog2θ+nlogα−2α∑i=1nxi+θ−1∑i=1nlog1−e−2αx+∑i=1nlog1+λ−2λ1−e−2αxθ.The likelihood equations for the TTLE distribution can indeed be managed to obtain by differentiating equation (25) with respect to the parameters λ,θ, and α which are offered by(26)∂Ln∂λ=∑i=1n1−21−e−2αxθ1+λ−2λ1−e−2αxθ,(27)∂Ln∂θ=nθ+∑i=1nlog1−e−2αx+∑i=1n−2λlog1−e−2αx1−e−2αxθ1+λ−2λ1−e−2αxθ,(28)∂Ln∂α=nα−2∑i=1nxi+θ−1∑i=1n2xe−2αx1−e−2αx−∑i=1n4λθxe−2αx1−e−2αxθ−11+λ−2λ1−e−2αxθ.The MLEs, sayλ⌢,θ⌢, and α⌢ of parameters λ,θ, and α, can be acquired by trying to equate and finding solutions of the nonlinear system equations (26)–(28) instantaneously.


### 4.1. Simulation Results

The output of the MLLEs of the parameters of the TTLE distribution is analyzed through Monte-Carlo simulation inside this subsection. The high accuracy of MLEs is explored utilizing bias and mean square errors (MSEs). This study is evaluated based on 1000 replicates. Different samples of sizes 25, 50, 75, and 100 are generated by using equation (7). The parametric values are considered to be λ=0.5,θ=1, and α=2. The average MSEs and the average bias of the simulated estimates εi^ can be computed by the following equations:(29)MSEεn=1N∑i=1Nεi^−ε2,Biasεn=1N∑i=1Nεi^−ε,where ε=λ,θ,α.The estimated values of the distribution parameters, the MSEs, and the bias terms of the simulated estimates are listed in Table2.Table 2
Simulation results for TTLE distribution.
n=25n=50n=75n=100α=2α⌢2.09362.016432.000741.96012MSE0.412600.26570.201580.14632Bias0.093600.016430.000740.0399θ=1θ1.074471.02001.016871.00593MSE0.073740.030270.018320.01483Bias0.074470.019980.016870.00593λ=0.5λ⌢0.529320.529160.528390.52754MSE0.070730.064950.059260.04992Bias0.029320.029160.028390.02754The graphical representations of the simulation process are displayed in Figures5–7. The following conclusions can be drawn from these figures: when the sample size is increased, the MSEs and estimated biases decrease. Furthermore, for large samples, the approximated values of the parameters approach the parametric values.Figure 5
The MSE and bias term of parameterα.
(a)(b)Figure 6
The MSE and bias term of parameterθ.
(a)(b)Figure 7
The MSE and bias term of parameterλ.
(a)(b)


## 4.1. Simulation Results

The output of the MLLEs of the parameters of the TTLE distribution is analyzed through Monte-Carlo simulation inside this subsection. The high accuracy of MLEs is explored utilizing bias and mean square errors (MSEs). This study is evaluated based on 1000 replicates. Different samples of sizes 25, 50, 75, and 100 are generated by using equation (7). The parametric values are considered to be λ=0.5,θ=1, and α=2. The average MSEs and the average bias of the simulated estimates εi^ can be computed by the following equations:(29)MSEεn=1N∑i=1Nεi^−ε2,Biasεn=1N∑i=1Nεi^−ε,where ε=λ,θ,α.The estimated values of the distribution parameters, the MSEs, and the bias terms of the simulated estimates are listed in Table2.Table 2
Simulation results for TTLE distribution.
n=25n=50n=75n=100α=2α⌢2.09362.016432.000741.96012MSE0.412600.26570.201580.14632Bias0.093600.016430.000740.0399θ=1θ1.074471.02001.016871.00593MSE0.073740.030270.018320.01483Bias0.074470.019980.016870.00593λ=0.5λ⌢0.529320.529160.528390.52754MSE0.070730.064950.059260.04992Bias0.029320.029160.028390.02754The graphical representations of the simulation process are displayed in Figures5–7. The following conclusions can be drawn from these figures: when the sample size is increased, the MSEs and estimated biases decrease. Furthermore, for large samples, the approximated values of the parameters approach the parametric values.Figure 5
The MSE and bias term of parameterα.
(a)(b)Figure 6
The MSE and bias term of parameterθ.
(a)(b)Figure 7
The MSE and bias term of parameterλ.
(a)(b)


## 5. Applications

Within this section, the practical use of the TTLE distribution is explored via two sources of data. The first set of data (Guinea Pig data) was published by [17], while the second set of data (Kevlar data) has been presented by [18].Reference [19] demonstrated that the TTT plot is convex (or concave) for decreasing (or increasing) failure rate and straight diagonal for constant failure rate. The empirical failure rate of the data becomes bathtub-shaped if the TTT plot is convex and then concave; otherwise, the failure rate is upside-down bathtub-shaped.The scaled TTT plot for every dataset is shown in Figure8, and it is evident that both sets of data have quite a concave shape, which is indicated that the empirical failure rate of these two datasets is increasing; as a result, the TTLE distribution may be a suitable model for both these datasets. The TTLE distribution is shown to be adaptable using these two examples. The recommended model’s goodness-of-fit indicators are compared to the fits of the relevant lifetime models: the BE by [20], the MOGE by [21], the EW by [22], the MW by [23], and the TBX by [24].Figure 8
The TTT plots for (a) Guinea Pig data and (b) Kevlar data.
(a)(b)We used the MLL technique to estimate the parameters of every model with every set of data. We computed the values of the following information criterion (INC) focusing on the estimations of the unknown parameters: Akaike INC (AINC), Bayesian INC (BINC), and Hannan–Quinn INC (HQINC). The Anderson–Darling (AND), the Cramer-von Mises (CVM) tests, and the Kolmogorov–Smirnov (A1) statistic, as well as the accompanying probability value, are also included (A2). Generally, the optimal model for fitting the data is one with lower values of these measures and a higher A2 of the A1 statistic. Furthermore, for all fitted models, the estimated pdf, estimated sf, and Q-Q plot of the fitted models are taken into account.Table3 exhibits the MLLEs of the distribution parameters and associated standard errors (StEr) for the current proposal model as well as the challenging distributions for the Guinea Pig data. Likewise, we contrasted the fits of the TTLE model to those of the other models listed in Table 4, including the AINC, BINC, and HQINC. Furthermore, the AND, CVM, and A1 statistics, as well as the equivalent A2, are provided in Table 5. Figures 9 and 10 depict the estimated pdf, estimated sf, and Q-Q plots of the fitted models using Guinea Pig data. These tables and figures show that the suggested model outperforms all competitive distributions in terms of fit. As a consequence, the TTLE distribution may be ranked as the top data fitting model.Table 3
The MLLEs and StErs for the Guinea Pig data.
ModelsEstimatesTTLEλ⌢=−0.83494 (0.2045)θ⌢=2.38752 (0.691)α⌢=0.00607 (0.0007)BEa⌢=3.34584 (0.964)b⌢=1.70820 (3.386)α⌢=0.00737 (0.0119)MoGEδ⌢=1.03273 (1.217)θ⌢=3.60839 (1.0456)α⌢=0.01136 (0.0036)EWθ⌢=2.65358 (1.53599)β⌢=1.16045 (0.30896)α⌢=0.00886 (0.0036)MWη⌢=5.86338×10−13 (0.0019)δ⌢=6.32978×10−5 (0.00009)α⌢=1.82535 (0.224648)TBXλ⌢=0.63259 (0.272)μ⌢=1.03896 (0.147)α⌢=0.00417 (0.0006)Table 4
The values of AINC, BINC, and HQINC for the Guinea Pig data.
ModelsAINCBINCHQINCTTLE855.805862.635858.524BE857.479864.309860.198MoGE857.616864.446860.335EW857.312864.142860.031MW860.724867.554863.443TBX858.575865.405861.294Table 5
The values of AND, CVM, A1, and A2 for Guinea Pig data.
ModelsANDCVMA1A2TTLE0.461170.075960.078540.76609BE0.535830.084390.090960.59054MoGE0.520150.080210.093120.56021EW0.544090.086890.089120.616723MW1.018080.169160.104790.40789TBX0.778540.131830.085770.664677Figure 9
(a) Estimated density and (b) estimated survival for the Guinea Pig data.
(a)(b)Figure 10
The Q-Q plot of the fitted distributions for the Guinea Pig data.For the Kevlar data, the MLLEs for the fitted models and their StErs are computed in Table6. The goodness-of-fit measures: AINC, BINC, and HQINC are listed in Table 7. Furthermore, the values of these statistics AND, CVM, A1, and A2 are given in Table 8. Additionally, the plots of estimated density and estimated survival and the Q-Q plot are displayed in Figures 11 and 12. It is evident from Tables 7 and 8 that the TTLE distribution has the weakest goodness-of-fit indices; therefore, this model can be chosen as the better one in fitting this kind of data. In addition, it is clear from Figures 11 and 12 that the TTLE model fits the two real-world data superior than the other competing models.Table 6
The MLLEs and StErs for the Kevlar data.
ModelsMLLEs and StErsTTLEλ⌢=0.84235θ⌢=1.70158α⌢=0.229120.341020.309810.07342BEa=1.67972b⌢=1.50842α⌢=0.484960.317374.537031.32951MoGEδ=1.77828θ⌢=1.44242α⌢=0.821611.463810.491950.00362EWθ=1.44263β⌢=1.10131α⌢=0.609440.643640.262910.21806MWη=0.36638δ⌢=1.32561α⌢=3.69082×10−90.330460.263530.32066TBXλ=0.75459μ⌢=0.62374α⌢=0.259850.210870.076480.03859Table 7
The values of AINC, BINC, and HQINC for the Kevlar data.
ModelsAINCBINCHQINCTTLE249.867256.859252.661BE250.455257.447253.25MoGE250.027257.019252.821EW250.327257.319253.122MW251.049258.042253.844TBX251.669258.661254.463Table 8
The values of AND, CVM, A1, and A2 for the Kevlar data.
ModelsANDCVMA1A2TTLE0.5938970.09768290.08641380.621551BE0.6727930.1119070.0956840.48970MoGE0.6481710.1059020.1000060.432777EW0.6630420.110020.09875710.448846MW0.7970270.1362770.109930.317345TBX0.8287010.1414990.1178920.241425Figure 11
(a) The estimated density and (b) estimated survival for the Kevlar data.
(a)(b)Figure 12
The Q-Q plot of the fitted distributions for the Kevlar data.


## 6. Conclusion

This work proposes and explores a new lifetime distribution, the TTLE, to enhance theE distribution. This model’s hr function may take the shape of an upside-down bathtub (unimodal), and the constant and diminishing diversity of shapes make the model very adaptable. The suggested model’s statistical characteristics and major reliability indicators are deduced. The MLL technique is used to estimate the parameters of the TTLE model. In addition, a numerical result is employed to look into the quality of the estimates. Eventually, real-world data applications are utilized to demonstrate the practicality of TTLE distribution. Also, in the future works, we plan to use this suggested model to study the statistical inference of it using Bayesian estimation under different censored schemes. In the future, other authors can use this model to study its statistical inference using Bayesian and E-Bayesian estimations under complete and different censored schemes.



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