# Application to Biomedical Data: Using the Topp Leone Inverse Lindley Model

Authors: Maha A. Aldahlan
Journal: Journal of Mathematics
 (2022)

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## Abstract

A more flexible two-parameter model named the Topp Leone inverse Lindley model is investigated. Some basic mathematical properties such as quantile, moments, order statistics, and Rényi entropy of the new distribution are considered. Plot analysis for mean, variance, skewness, and kurtosis is performed. The density of the new model can be right skewed and decreasing with unimodal and bimodal shapes. Also, its hazard rate function can be decreasing and upside-down. The maximum likelihood (ML) estimation method is used to estimate the parameters of the distribution. The simulation study is executed to investigate the effectiveness of the estimates. The potential of the distribution is demonstrated through the application of the real biomedical dataset.


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## Body

## 1. Introduction

The Topp Leone (TL-G) class of distributions was introduced by Al-Shomrani et al. [1]. A random variable X has TL-G class of distributions if the distribution function (cdf) and probability density function (pdf) are given by(1)Fx=Gxα2−Gxα=1−G¯x2α,x∈ℝ,and(2)fx=2αgxG¯x1−G¯x2α−1,where G¯x=1−Gx, G(x) is the baseline distribution, and α is a positive shape parameter.The equation (2) can be rewritten as(3)fx=∑i,j=0∞Wi,jgxGxj,where Wi,j=2α−1i+jα−1ii+2j. The quantile function of TL-G is given by(4)Qu=G−11−1−uα,0<u<1.Many distributions have been studied by using TL-G for different baselines. Aryal et al. [2] studied TL Weibull distribution, Abbas et al. [3] introduced and studied TL inverse Weibull, Reyad and Othman [4] studied TL Burr-XII distribution, the TL Nadarajah-Haghighi distribution was proposed by Yousof and Korkmaz [5], the odd Lindley-G was proposed by Gomes et al. [6], Reyad et al. [7] proposed TL generalized inverted Kumaraswamy distribution, TL compound Rayleigh distribution is studied by Rasheed [8], odd Nadarajah-Haghighi-G was proposed by Nascimento et al. [9], Oguntunde et al. [10] introduced TL Lomax distribution, Al-Marzouki et al. [11] studied type II TL power Lomax distribution, Al-Marzouki et al. [12] studied TL odd Fréchet-G family, and Al-Babtain et al. [13] proposed sine TL-G family.The inverse Lindley (IL) distribution was studied by Sharma et al. [14], and it has the following pdf and cdf, respectively,(5)gx=θ21+θ1+xx3e−θ/x,x>0,θ>0,and(6)Gx=1+θ1+θxe−θ/x,x>0,θ>0.The IL distribution has been generalized by many statisticians in last years: Alkarni [15] introduced a more flexible model named extended IL distribution, Sharma et al. [16] proposed the generalized IL distribution, Barco et al. [17] studied the power IL distribution, Dey et al. [18] introduced alpha power transformed IL distribution, Eltehiwy [19] studied extended exponentiated IL distribution, and logarithmic IL model was studied by Eltehiwy [20].The main goal of this article is as follows:(i)
To propose a new two-parameter life time model, so called, Topp Leone inverse Lindley (TLIL) distribution(ii)
The density of the new model can be right skewed and decreasing with unimodal and bimodal shapes. Also, it is hazard rate function can be decreasing and upside-down.(iii)
To study some of its various statistical properties such as quantile function, moments, incomplete moments, order statistics, and Rényi entropy(iv)
To discuss the statistical inference of the TLIL model by using the ML method(v)
To investigate a simulation study to show the behavior of the parameters model(vi)
To give leading fits than some known models with favourable results for the TLIL modelThe new model is very flexible, and we can get the cdf and pdf by inserting (1) and (2) in (5) and (6) as(7)Fx=1−1−1+θ1+θxe−θ/x2α,x>0,θ,α>0,and(8)fx=2αθ21+θ1+xx3e−θ/x1−1+θ1+θxe−θ/x1−1−1+θ1+θxe−θ/x2α−1.The plots of the pdf and cdf of the TLIL distribution are shown in Figures1 and 2.Figure 1
The pdf of the TLIL distribution for different values of parameters.Figure 2
The cdf of the TLIL distribution for different values of parameters.Figure1 shows the pdf can be decreasing, unimodal, and right skewed.The remaining part of this article is organized as follows: the reliability analysis of the new distribution is given in Section2, Section 3 concerns with the linear representation for the new model, the derivation of the mathematical properties of the distribution is proposed in Section 4, the estimation of parameters of the distribution is discussed in Section 5, in Section 6, a simulation study is executed to investigate the performance of the estimates, applications of real-life data to illustrate the potentials of the new distribution is presented in Section 7, and in Section 8, the concluding remarks are presented.


## 2. The Reliability Analysis

Suppose a random variableX follows the TLIL model, then the reliability function R(x), hazard rate function (hrf) h(x), inverse hazard rate function τ(x), and cumulative hazard rate function H(x) for the TLIL distribution are given by(9)Rx=1−1−1−1+θ1+θxe−θ/x2α,hx=2αθ2/1+θ1+x/x3e−θ/x1−1+θ/1+θxe−θ/x1−1−1+θ/1+θxe−θ/x2α−11−1−1−1+θ/1+θxe−θ/x2α,τx=2αθ2/1+θ1+x/x3e−θ/x1−1+θ/1+θxe−θ/x1−1−1+θ/1+θxe−θ/x2,Hx=−ln1−1−1−1+θ1+θxe−θ/x2α.The plots of the pdf and hrf of the TLIL distribution are shown in Figure3.Figure 3
The hrf of the TLIL distribution for different values of parameters.Figure3 shows some possible shapes including monotone decreasing and right-skewed shapes for hrf.


## 3. Important Representation

In this section, representations of the pdf for TLIL distribution are calculated.By using the three equations equation (3), (5), and (6), we can rewrite (8) as(10)fx=∑i,j=0∞Wi,jθ21+θ1+xx3e−θj+1/x1+θ1+θxj.By applying the binomial expansion in the previous equation, we can rewrite it as(11)fx=∑k=0∞Wk1+xxk+3e−θj+1/x,where Wk=∑i,j=0∞Wi,jj/kθk+2/1+θk+1.


## 4. Mathematical Properties

### 4.1. Quantile Function

LetX has a random variable with pdf (8). The quantile function, say Q(u), defined by F(Q(u)) = u is the root of the equation:(12)1+θ1+θQue−θ/Qu=1−1−uα,0<u<1.By multiplying (12) both sides by −1+θe−1+θ, we get(13)−1+θ+θQue−1+θ+θ/Qu=−1+θe−1+θ1−1−uα.By using the Lambert W function which is the solution of the equationWzeWz=z, where z is a complex number, we have(14)W−1−1+θe−1+θ1−1−uα=−1+θ+θQu,and then,(15)Qu=−1+1θ+1θW−1−1+θe−1+θ1−1−uα−1.Corollary 1.
IfX∼TLIL, the median M of X is given by(16)Qu=−1+1θ+1θW−1−1+θe−1+θ1−1−0.5α−1.


### 4.2. Moments

Therth moments of a random variable having TLIL distribution is defined as(17)EXr=∫0∞xrfxdx=∑k=0∞Wk∫0∞xr1+xxk+3e−θj+1/xdx=∑k=0∞Wk∫0∞xr−k−3+xr−k−2e−θj+1/xdx.Lettingy=θj+1/x,x=θj+1/ydx=−θj+1/y2dy and simplifying further,(18)EXr=∑k=0∞Wk∫0∞θj+1yr−k−3+θj+1yr−k−2e−yθj+1y2dyEXr=∑k=0∞Wkθj+1r−k−2∫0∞yk−r+1+θj+1yk−re−ydy∑k=0∞Wkθj+1r−k−2Γk−r+2+θj+1Γk−r+1.The moment generating function of the TLIL model can be calculated by(19)MXt=∑r=0∞trr!EXr=∑r,k=0∞trr!Wkθj+1r−k−2Γk−r+2+θj+1Γk−r+1,r<k+2.Figure4 shows the mean, variance, skewness, and kurtosis curves of the TLIL model for numerous values of α and θ.Figure 4
3D plots of mean, variance, skewness, and kurtosis.


### 4.3. Incomplete Moments

The incomplete moments, sayϖst, is given by(20)ϖst=∫0txsfx;φdx.Using (11), ϖst can be taken the next formula:(21)ϖst=∑k=0∞Wk∫0txs1+xxk+3e−θj+1/xdx.Then, using the lower incomplete gamma function, we obtain(22)ϖst=∑k=0∞Wkθj+1s−k−2νk−s+2,θj+1t−1+θj+1νk−s+1,θj+1t−1,s<k+2,where νs,t=∫0txs−1e−xdx is the lower incomplete gamma function.The Lorenz and Bonferroni curves are given by(23)LFx=∫0txfxdxEX=∑k=0∞Wkθj+1−k−1νk+1,θj+1t−1+θj+1νk,θj+1t−1∑k=0∞Wkθj+1−k−1Γk+1+θj+1Γk,BFx=∫0txfxdxEXFx=LFxFx=∑k=0∞Wkθj+1−k−1νk+1,θj+1t−1+θj+1νk,θj+1t−11−1−1+θ/1+θxe−θ/x2α∑k=0∞Wkθj+1−k−1Γk+1+θj+1Γk.


### 4.4. Order Statistics

SupposeX1<X2<…<Xn is an order sample from TLIL population, the pdf of the ith ordered statistics is given as(24)fxi:n=n!i−1!n−i!fxFxi−11−Fxn−i=n!i−1!n−i!fx∑j=0n−i−1jn−ijFxi+j−1.Substituting (7) and (8) and applying general binomial series expansion (24) become(25)fxi:n=∑j=0n−i2αθ2−1jn−ijn!θ+1i−1!n−i!1+xx3e−θ/x1−1+θ1+θxe−θ/x1−1−1+θ1+θxe−θ/x2αi+j−1.


### 4.5. Rényi Entropy

For a given pdf, the Rényi entropy is defined by(26)IRδ=11−δlog∫0∞fxδdx,δ>0,δ≠1.The functionfxδ is given by(27)fxδ=2αθ21+θδ1+xδx3δe−θδ/x1−1+θ1+θxe−θ/xδ1−1−1+θ1+θxe−θ/x2δα−1.By applying the binomial expansion three times in the previous equation, we get(28)fxδ=2αδθ2δ+k1+θδ+k∑i,j,k=0∞−1i+jδα−1iδ+2ijjkx−k−3δ1+xδe−θj+δ/x.By applying the the binomial expansion to the term1+xδ=∑l=0∞δlxl, we can write the previous equation as(29)fxδ=∑l=0∞tlxl−k−3δe−θj+δ/x,where(30)tl=2αδθ2δ+k1+θδ+k∑i,j,k=0∞−1i+jδα−1iδ+2ijjkδl.Now, we will calculate the integral(31)I=∫0∞fxδdx=∑l=0∞tl∫0∞xl−k−3δe−θj+δ/xdx=∑l=0∞tlθj+δl−k−3δ+1Γk+3δ−l−1.Then, the Rényi entropy is(32)IRδ=11−δlog∑l=0∞tlθj+δl−k−3δ+1Γk+3δ−l−1.


## 4.1. Quantile Function

LetX has a random variable with pdf (8). The quantile function, say Q(u), defined by F(Q(u)) = u is the root of the equation:(12)1+θ1+θQue−θ/Qu=1−1−uα,0<u<1.By multiplying (12) both sides by −1+θe−1+θ, we get(13)−1+θ+θQue−1+θ+θ/Qu=−1+θe−1+θ1−1−uα.By using the Lambert W function which is the solution of the equationWzeWz=z, where z is a complex number, we have(14)W−1−1+θe−1+θ1−1−uα=−1+θ+θQu,and then,(15)Qu=−1+1θ+1θW−1−1+θe−1+θ1−1−uα−1.Corollary 1.
IfX∼TLIL, the median M of X is given by(16)Qu=−1+1θ+1θW−1−1+θe−1+θ1−1−0.5α−1.


## 4.2. Moments

Therth moments of a random variable having TLIL distribution is defined as(17)EXr=∫0∞xrfxdx=∑k=0∞Wk∫0∞xr1+xxk+3e−θj+1/xdx=∑k=0∞Wk∫0∞xr−k−3+xr−k−2e−θj+1/xdx.Lettingy=θj+1/x,x=θj+1/ydx=−θj+1/y2dy and simplifying further,(18)EXr=∑k=0∞Wk∫0∞θj+1yr−k−3+θj+1yr−k−2e−yθj+1y2dyEXr=∑k=0∞Wkθj+1r−k−2∫0∞yk−r+1+θj+1yk−re−ydy∑k=0∞Wkθj+1r−k−2Γk−r+2+θj+1Γk−r+1.The moment generating function of the TLIL model can be calculated by(19)MXt=∑r=0∞trr!EXr=∑r,k=0∞trr!Wkθj+1r−k−2Γk−r+2+θj+1Γk−r+1,r<k+2.Figure4 shows the mean, variance, skewness, and kurtosis curves of the TLIL model for numerous values of α and θ.Figure 4
3D plots of mean, variance, skewness, and kurtosis.


## 4.3. Incomplete Moments

The incomplete moments, sayϖst, is given by(20)ϖst=∫0txsfx;φdx.Using (11), ϖst can be taken the next formula:(21)ϖst=∑k=0∞Wk∫0txs1+xxk+3e−θj+1/xdx.Then, using the lower incomplete gamma function, we obtain(22)ϖst=∑k=0∞Wkθj+1s−k−2νk−s+2,θj+1t−1+θj+1νk−s+1,θj+1t−1,s<k+2,where νs,t=∫0txs−1e−xdx is the lower incomplete gamma function.The Lorenz and Bonferroni curves are given by(23)LFx=∫0txfxdxEX=∑k=0∞Wkθj+1−k−1νk+1,θj+1t−1+θj+1νk,θj+1t−1∑k=0∞Wkθj+1−k−1Γk+1+θj+1Γk,BFx=∫0txfxdxEXFx=LFxFx=∑k=0∞Wkθj+1−k−1νk+1,θj+1t−1+θj+1νk,θj+1t−11−1−1+θ/1+θxe−θ/x2α∑k=0∞Wkθj+1−k−1Γk+1+θj+1Γk.


## 4.4. Order Statistics

SupposeX1<X2<…<Xn is an order sample from TLIL population, the pdf of the ith ordered statistics is given as(24)fxi:n=n!i−1!n−i!fxFxi−11−Fxn−i=n!i−1!n−i!fx∑j=0n−i−1jn−ijFxi+j−1.Substituting (7) and (8) and applying general binomial series expansion (24) become(25)fxi:n=∑j=0n−i2αθ2−1jn−ijn!θ+1i−1!n−i!1+xx3e−θ/x1−1+θ1+θxe−θ/x1−1−1+θ1+θxe−θ/x2αi+j−1.


## 4.5. Rényi Entropy

For a given pdf, the Rényi entropy is defined by(26)IRδ=11−δlog∫0∞fxδdx,δ>0,δ≠1.The functionfxδ is given by(27)fxδ=2αθ21+θδ1+xδx3δe−θδ/x1−1+θ1+θxe−θ/xδ1−1−1+θ1+θxe−θ/x2δα−1.By applying the binomial expansion three times in the previous equation, we get(28)fxδ=2αδθ2δ+k1+θδ+k∑i,j,k=0∞−1i+jδα−1iδ+2ijjkx−k−3δ1+xδe−θj+δ/x.By applying the the binomial expansion to the term1+xδ=∑l=0∞δlxl, we can write the previous equation as(29)fxδ=∑l=0∞tlxl−k−3δe−θj+δ/x,where(30)tl=2αδθ2δ+k1+θδ+k∑i,j,k=0∞−1i+jδα−1iδ+2ijjkδl.Now, we will calculate the integral(31)I=∫0∞fxδdx=∑l=0∞tl∫0∞xl−k−3δe−θj+δ/xdx=∑l=0∞tlθj+δl−k−3δ+1Γk+3δ−l−1.Then, the Rényi entropy is(32)IRδ=11−δlog∑l=0∞tlθj+δl−k−3δ+1Γk+3δ−l−1.


## 5. Maximum Likelihood Estimation

SupposeX1,X2,…,Xn is a random sample of size n from a population having TLIL pdf, the log-likelihood function is given by(33)LogL=nlog2α+2nlogθ−nlog1+θ+∑i=1nlog1+xixi3−θ∑i=1n1xi+∑i=1nlog1−1+θ1+θxie−θ/xi+α−1∑i=1nlog1−1−1+θ1+θxe−θ/x2.The score functions which correspond to equating the first-order partial derivative of the last equation to zero is given by(34)∂LogL∂α=nα+∑i=1nlog1−1−1+θ1+θxe−θ/x2=0,∂LogL∂θ=2nθ−n1+θ−∑i=1n1xi+∑i=1nZiNi−2α−1∑i=1nNiZi1−Ni2=0,where Zi=1/xie−θ/xi1+θ/1+θxi+1/1+θ2 and Ni=1−1+θ/1+θxie−θ/xi. The solutions, say α^ and θ^ of the score functions correspond to the maximum likelihood estimators of TLIL distribution. However, the score functions are nonlinear functions; the numerical values of the maximum likelihood estimates can be obtained using the Newton–Raphson iterative optimisation method.An approximate confidence interval (CI) of model parameters for TLIL distribution is calculated. It is known that the asymptotic distribution of ML estimates of elements of parametersα and θ is given by(35)α^−α,θ^−θ⟶N0,I−1α,θ,where I−1 is the variance covariance matrix of unknown parameters; hence, the approximate (1−ν) 100% two-sided CIs for α and θ are, respectively, given by(36)α^±Zν/2varα^,θ^±Zν/2varθ^,where Zν/2 is the upper ν/2th percentile of the standard normal distribution.


## 6. Simulation

This section provides a simulation study to assess the behavior of the estimators in case of complete samples. Mean square errors (MSEs), biases, lower bound (LB) of CI, upper bound (UB) of CI, and average length (AL) of 90% and 95% are calculated via Mathematica 9. The following algorithm is designed as follows.(i)
5000 random samples of sizen = 50, 100, and 200 are generated from TLIL distribution(ii)
Exact values of parameters are choiced(iii)
The ML estimates, MSEs, biases, LB, UB, and AL for selected values of parameters are calculated(iv)
Numerical outcomes are given in Tables1–5 based on completeTable 1
ML estimates, biases, MSE, LB, UB, and AL of the TLIL model forα = 1.5, θ = 0.5.
nMLBiasMSE90%95%LBUBALLBUBAL501.211−0.2890.129−0.0862.5082.594−0.3352.7563.0910.5740.0740.0120.2390.9100.6720.1740.9750.8001001.401−0.0990.0490.4622.3401.8790.2822.5202.2380.5060.0060.0040.3270.6860.3590.2930.7200.4272001.441−0.0590.0180.7742.1071.3330.6472.2351.5890.5020.0020.0010.3810.6230.2420.3580.6460.288Table 2
ML estimates, biases, MSE, LB, UB, and AL of the TLIL model forα = 1.8, θ = 0.5.
nMLBiasMSE90%95%LBUBALLBUBAL501.194−0.6060.5320.2312.1561.9250.0472.3412.2940.6590.1590.0580.3191.0000.6820.2531.0650.8121001.355−0.4450.2310.5612.1481.5870.4092.3001.8910.5580.0580.0060.3800.7370.3580.3450.7710.4262001.472−0.3280.1090.7392.2061.4670.5982.3471.7480.5570.0570.0040.4140.7000.2860.3870.7270.340Table 3
ML estimates, biases, MSE, LB, UB, and AL of the TLIL model forα = 1.5, θ = 1.2.
nMLBiasMSE90%95%LBUBALLBUBAL501.341−0.1590.0650.0312.6512.620−0.2202.9023.1211.2720.0720.0230.5801.9641.3850.4472.0971.6501001.450−0.0500.0060.4012.4982.0970.2012.6992.4981.189−0.0110.0020.7191.6590.9400.6291.7491.1202001.498−0.0020.0000.7162.2801.5650.5662.4301.8641.2000.0000.0020.8611.5380.6770.7961.6030.807Table 4
ML estimates, biases, MSE, LB, UB, and AL of the TLIL model forα = 1.8, θ = 1.2.
nMLBiasMSE90%95%LBUBALLBUBAL501.296−0.5040.2850.0152.5762.560−0.2302.8213.0511.4200.2200.0580.6292.2121.5820.4782.3631.8851001.362−0.4380.2080.4672.2561.7890.2962.4272.1321.3390.1390.0300.8431.8350.9930.7481.9301.1832001.500−0.3000.0900.9412.0591.1180.8342.1661.3331.2930.0930.0091.0321.5530.5220.9821.6030.622Table 5
ML estimates, biases, MSE, LB, UB, and AL of the TLIL model forα = 1.5, θ = 0.7.
nMLBiasMSE90%95%LBUBALLBUBAL500.993−0.5070.4780.1501.8351.685−0.0111.9962.0071.0340.3340.2070.4391.6291.1900.3251.7421.4171001.185−0.3150.1460.3801.9911.6110.2262.1451.9190.7990.0990.0160.4981.0990.6010.4411.1570.7162001.336−0.1640.0700.6692.0031.3340.5412.1301.5890.7600.0600.0080.5520.9680.4170.5121.0080.496


## 7. Application

The potential of the new model TLIL is illustrated by modelling a real-life dataset. We compare the fits of the TLIL model with five models: IL, generalized IL (GIL), exponentiated generalized IL (EGIL), exponentiated IL (EIL), and inverse Weibull (IW) distributions. The Kolmogorov–Smirnov (KS) statistic is used to determine how well the distributions fit the dataset. The Akaike information criterion (AIC), Bayesian information criterion (BIC), and the negative two log-likelihood values of the distributions are also computed. The least values of the −2LogL, AIC, BIC, and KS statistics have a better fit.Dataset I is taken from Sharma et al. [14] and it represents the survival times of a group of patients suffering from head and neck cancer disease. The patients were treated using radiotherapy.Dataset II is taken from Bjerkedal [21] and it represents the survival times (in days) of 72 guinea pigs infected with virulent tubercle bacilli.The two datasets are concerned with biomedical data. Many works studied biomedical datasets such as the studies by Alanazi et al. [22], Iskanderani et al. [23], and Antor et al. [24]. Some descriptive analyses of the both datasets are given in Table 6.Table 6
Descriptive statistics for the both datasets.
NMeanMedianModeVarSKKUData I201.91.71.70.4711.8624.185Data II721.7681.4951.081.0551.3712.225Tables6–9 present the parameter estimates including the standard errors in parentheses and the goodness-of-fit statistics of the competitive distributions, respectively. The values in Tables 7 and 9 indicate that the TLIL distribution has the least values of the goodness-of-fit statistics which imply that it has a better fit among the competitive models.Table 7
Parameter estimates and the standard error in parentheses for dataset I.
ModelEstimates and MSEsTLIL (α, θ)0.073 (0.043)853.969 (470.089)IL (θ)60.007 (7.754)EIL (α, θ)3.994 (8.202)15.696 (30.438)GIL (α, θ)0.786 (0.072)29.409 (8.237)IW (α, β)0.786 (0.071)70.962 (12.612)EGIL (α, β, θ)0.784 (0.072)5.35 (19.465)6.123 (19.661)Table 8
Summary of goodness-of-fit statistics for dataset I.
Model−2LogLAICBICKSTLIL760.04764.04763.5670.1228IL771.4063773.406775.4670.288EIL771.492775.492780.0450.2884GIL763.2041767.204771.3250.19032IW763.1635767.163771.2840.19030EGIL763.3393769.339775.5210.191Table 9
Parameter estimates and the standard error in parentheses for dataset II.
ModelEstimates and MSEsTLIL (α, θ)0.567 (0.24)3.175 (0.902)IL (θ)1.5767 (0.146)EIL (α, θ)0.6966 (0.993)2.1041 (2.453)GIL (α, θ)1.0713 (0.076)1.5487 (0.147)IW (α, β)1.1731 (0.084)1.0583 (0.113)EGIL (α, β, θ)1.1632 (0.085)0.0534 (0.079)20.6388 (29.624)The graphical fit of the competing distributions on the dataset is shown in Figures5 and 6. The estimated pdf, ecdf, esf, and pp plots plot of TLIL is closest to the empirical plots. Figures 5 and 6 also confirm the goodness-of-fit statistics values given in Tables 8 and 10 that the TLIL has a better fit.Figure 5
The estimated pdf, ecdf, esf, and pp plots for the TLIL model for dataset I.Figure 6
The estimated pdf, ecdf, esf, and pp plots for the TLIL model for dataset II.Table 10
Summary of goodness-of-fit statistics for dataset II.
Model−2LogLAICBICKSTLIL220.527224.527224.2420.179IL239.569241.569243.8460.194EIL239.494243.494247.6150.199GIL238.698242.698247.2510.181IW236.332240.332244.8850.183EGIL236.501242.501249.3310.184


## 8. Conclusion

In this study, a new statistical distribution called the Topp Leone inverse Lindley model is introduced. The density of the new model can be right skewed and decreasing with unimodal and bimodal shapes. Also, its hazard rate function can be decreasing and upside-down. Some of the mathematical properties such as quantile function, ordinary moments, incomplete moments, and pdf of theith ordered statistics are established. The estimation of the parameters of the new distribution using the maximum likelihood method is considered. The simulation study is assessed to investigate the performance of the estimates. The usefulness and potentials of the new distribution are demonstrated by comparing its fit to a real-life dataset with other distributions. The goodness-of-fit statistics indicated that the new distribution has a better fit than other competing distributions.



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*Source: 1985861-2022-02-08.xml*