# Inverted Length-Biased Exponential Model: Statistical Inference and Modeling

Authors: Waleed Almutiry
Journal: Journal of Mathematics
 (2021)

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## Abstract

This research article proposes a new probability distribution, referred to as the inverted length-biased exponential distribution. The hazard rate function (HZRF) and density function (PDF) in the new distribution allow additional flexibility as well as some desired features. It provides a more flexible approach that may be used to represent many forms of real-world data. The quantile function (QuF), moments (MOs), moment generating function (MOGF), mean residual lifespan (MRLS), mean inactivity time (MINT), and probability weighted moments (PRWMOs) are among the mathematical and statistical features of the inverted length-biased exponential distribution. In the case of complete and type II censored samples (TIICS), the maximum likelihood (MLL) strategy can be used to estimate the model parameters. An asymptotic confidence interval (COI) of parameter is constructed at two confidence levels. We perform simulation study to examine the accuracy of estimates depending upon some statistical measures. Simulation results show that there is great agreement between theoretical and empirical studies. We demonstrate the new model’s relevance and adaptability by modeling three lifespan datasets. The proposed model is a better fit than the half logistic inverse Rayleigh (HLOIR), type II Topp–Leone inverse Rayleigh (TIITOLIR), and transmuted inverse Rayleigh (TRIR) distributions. We anticipate that the expanded distribution will attract a broader range of applications in a variety of fields of research.


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## Body

## 1. Introduction

Length-biased exponential (LBE) or moment exponential (ME) distribution is considered as one of the most important univariate and parametric models. It is commonly utilized in the analysis of data collected throughout a lifespan and in problems connected to the modeling of failure processes. There is much to be said for a flexible lifespan distribution model, and this one may be a suitable fit for some sets of failure data. Reference [1] proposed the LBE with the following PDF and distribution function (CDF):(1)gx;α=α2xe−αx;x≥0,α>0,(2)Gx;α=1−1+αxe−αx;x≥0,α>0,where α is the scale parameter. Different values of the shape parameter lead to different shapes of density function.Many authors extended new models from the LBE distribution such as exponentiated ME [2], generalized exponentiated ME [3], and Marshall–Olkin (MO) LBE (MOLBE) distributions [4]. MO Kumaraswamy ME model was discussed in [5].Several univariate continuous distributions have been extensively used in environmental, engineering, financial, and biomedical sciences, among other areas for modeling lifetime data. However, there is still a strong need for a significant improvement of the classical distributions through different techniques for modeling several data lifetime. In this regard, the inverted (or inverse) (I) distribution is one procedure that explores extra properties of the phenomenon which cannot be produced from noninverted distributions. Applications of inverted distributions include econometrics as well as the engineering sciences as well as biology and survey sampling as well as medical research among others. In the literature, several studies related to inverted distributions have been handled by several researchers; for instance, Reference [6] introduced the I Weibull distribution. Reference [7] studied the I Pareto type 1 distribution. Reference [8] investigated the I Pareto type 2 distribution. Reference [9] handled exponentiated I Weibull distribution. Reference [10] provided the I Lindley distribution. Reference [11] suggested the I Kumaraswamy model. Reference [12] presented the I Nadarajah-Haghighi model. Reference [13] studied the I power Lomax model. Reference [14] suggested I exponentiated Lomax model. Reference [15] discussed the Weibull I Lomax model. Reference [16] suggested the power transmuted I Rayleigh model. Reference [17] investigated the I Topp–Leone distribution, and half logistic I Topp–Leone distribution was studied in [18].Our motivation here is (i) introducing a new distribution, referred to as the inverted length-biased exponential (ILBE), (ii) studying some of the main properties, (iii) providing point and interval estimators for the model parameter from complete and censored samples, and (iv) examining its applicability using three real datasets.The inverted LBE (ILBE) distribution is constructed by using the random variableT = 1/X where X follows (2). The ILBE distribution’s CDF is described as(3)Ft;α=1+αte−α/t;t≥0,α>0.The ILBE distribution’s PDF is specified as(4)ft;α=α2t3e−α/t;t≥0,α>0.The survival function (SRF) and HZRF of the ILBE distribution are provided by(5)F¯t;α=1−1+αte−α/t,ht;α=α2e−α/tt31−1+α/te−α/t.Figure1 depicts PDF and HZRF plots for the ILBE distribution. According to Figure 1, the density of the suggested distribution is highly flexible in nature and can take on a number of forms, including positively skewed and unimodal. Through the parameter space, the HZRF can take on many forms, such as decreasing, rising, or upside down.Figure 1
Plots of PDF and HZRF for the ILBE distribution.This paper is organized as follows. In Section2, the basic characteristics of the ILBE distribution are obtained. The MLL estimators for the ILBE model are described in Section 3 and are established on complete and censored samples, accompanied by a simulation analysis. The application to actual data collection is covered in Section 4. Section 5 concludes the paper with some remarks.


## 2. Fundamental Mathematical Properties of ILBE Distribution

Here, we give some essential properties of the ILBE distribution, like QuF, MOs, PRWMOs, incomplete MOs, and inverse MOs.


### 2.1. Quantile Function

A generated random number from the ILBE distribution is obtained by solving the following equation numerically:(6)Qu=α−1−W−1−e−1u,0<u<1,where W−1 denotes the negative branch of the Lambert W function (i.e., the solution of the equation WZeWZ=z. The median, say Q2, is achieved by adjusting u = 0.5 in (6), and the first quartile and third quartile, denoted by Q1 and Q3, are obtained by setting u = 0.25 and 0.75, respectively, in (6). Note that equation (6) is solved numerically by using Mathematica 9.(7)Q1=α−1−W−1−0.25e−1,Q2=α−1−W−1−0.5e−1,Q3=α−1−W−1−0.75e−1.


### 2.2. Moments

Due to its relevance in any statistical study, we shall give then-th MO of the ILBE distribution here. For the ILBE model, the n-th MO of T about the origin is computed as follows:(8)μn′=ETn=∫0∞tnα2t3e−α/tdt=αnΓ2−n,n<2.The following formula may be used to determine the MOGF of the ILBE distribution:(9)Mxt=∑n=0∞tnn!μn′=∑n=0∞tnn!αnΓ2−n,n<2.The incomplete (IN) MO, sayςnx, is(10)ςnt=α2t∫0tn−3e−α/tdt=αnγ2−n,αt,n<2,where, γ.,t is the upper IN gamma function.Further, the conditional MO, sayϖsx, is(11)ϖst=α2∫t∞tα−3e−α/tdt=αnΓ2−n,αt,n<2,where Γ.,t is the lower IN gamma function.For the ILBE distribution, then-th inverse MO is calculated on the basis:(12)τkx=α2∫0∞t−n−3e−α/tdt=α−kΓn+2.Forn = 1, we get the harmonic mean of the ILBE distribution.The Lorenz and Bonferroni curves are obtained as follows.(13)LFt=ς1tET=γ1,αt,(14)BFt=LFtFt=γ1,α/t1+α/te−α/t.


### 2.3. Order Statistics

LetT1,T2,…,Tn be r samples from the ILBE model with order statistics T1,T2,…,Tn. The PDF of Tk of order statistics is given by(15)fTkt=n!k−1!n−k!Fk−1tft1−Ftn−k.The PDF ofTk can be expressed as(16)fTkt=n!α2k−1!n−k!t−31+αtk−1e−kα/t1−1+αte−α/tn−k.Particularly, PDF of the first and largest order statistics can be calculated as(17)fT1t=nα2t−3e−α/t1−1+αte−α/tn−1,(18)fTnt=nα2t−31+αtn−1e−nα/t,respectively.


### 2.4. Mean Residual Life Function

It has an important application of the MOs of residual lifetime function. The MRLS of ILBE distribution is(19)υt=ET−t|T>t=1F¯t∫t∞xfxdt−t=αΓ1,αt1−1+αte−α/t−1−t.The MINT represents the amount of time that has passed after an item has failed, assuming that this failure has occurred. The MINT of ILBE distribution is(20)ϖt=ET−t|T≤t=t−1Ft∫0txfxdx=t−αγ1,αt1+αt−1eα/t.


### 2.5. Probability Weighted Moments

The PRWMOs are often used to investigate additional aspects of the probability distribution. The PRWMOs of the random variableT, denoted by Sr,p, are defined as(21)Sr,p=∫−∞∞trftFtpdt,where r and p are positive integers. Substituting (3) and (4) into (21) yields the PRWMOs of the ILBE distribution as follows:(22)Sr,p=α2∫0∞tr−31+αtpe−p+1α/t=∑j=0ppjαj+2∫0∞tr−3−je−p+1α/tdt.As a result of the simplification, the PRWMOs of the ILBE distribution assume the following structure:(23)Sr,p=∑j=0ppjαrΓj−r+2p+1j−r+2.


## 2.1. Quantile Function

A generated random number from the ILBE distribution is obtained by solving the following equation numerically:(6)Qu=α−1−W−1−e−1u,0<u<1,where W−1 denotes the negative branch of the Lambert W function (i.e., the solution of the equation WZeWZ=z. The median, say Q2, is achieved by adjusting u = 0.5 in (6), and the first quartile and third quartile, denoted by Q1 and Q3, are obtained by setting u = 0.25 and 0.75, respectively, in (6). Note that equation (6) is solved numerically by using Mathematica 9.(7)Q1=α−1−W−1−0.25e−1,Q2=α−1−W−1−0.5e−1,Q3=α−1−W−1−0.75e−1.


## 2.2. Moments

Due to its relevance in any statistical study, we shall give then-th MO of the ILBE distribution here. For the ILBE model, the n-th MO of T about the origin is computed as follows:(8)μn′=ETn=∫0∞tnα2t3e−α/tdt=αnΓ2−n,n<2.The following formula may be used to determine the MOGF of the ILBE distribution:(9)Mxt=∑n=0∞tnn!μn′=∑n=0∞tnn!αnΓ2−n,n<2.The incomplete (IN) MO, sayςnx, is(10)ςnt=α2t∫0tn−3e−α/tdt=αnγ2−n,αt,n<2,where, γ.,t is the upper IN gamma function.Further, the conditional MO, sayϖsx, is(11)ϖst=α2∫t∞tα−3e−α/tdt=αnΓ2−n,αt,n<2,where Γ.,t is the lower IN gamma function.For the ILBE distribution, then-th inverse MO is calculated on the basis:(12)τkx=α2∫0∞t−n−3e−α/tdt=α−kΓn+2.Forn = 1, we get the harmonic mean of the ILBE distribution.The Lorenz and Bonferroni curves are obtained as follows.(13)LFt=ς1tET=γ1,αt,(14)BFt=LFtFt=γ1,α/t1+α/te−α/t.


## 2.3. Order Statistics

LetT1,T2,…,Tn be r samples from the ILBE model with order statistics T1,T2,…,Tn. The PDF of Tk of order statistics is given by(15)fTkt=n!k−1!n−k!Fk−1tft1−Ftn−k.The PDF ofTk can be expressed as(16)fTkt=n!α2k−1!n−k!t−31+αtk−1e−kα/t1−1+αte−α/tn−k.Particularly, PDF of the first and largest order statistics can be calculated as(17)fT1t=nα2t−3e−α/t1−1+αte−α/tn−1,(18)fTnt=nα2t−31+αtn−1e−nα/t,respectively.


## 2.4. Mean Residual Life Function

It has an important application of the MOs of residual lifetime function. The MRLS of ILBE distribution is(19)υt=ET−t|T>t=1F¯t∫t∞xfxdt−t=αΓ1,αt1−1+αte−α/t−1−t.The MINT represents the amount of time that has passed after an item has failed, assuming that this failure has occurred. The MINT of ILBE distribution is(20)ϖt=ET−t|T≤t=t−1Ft∫0txfxdx=t−αγ1,αt1+αt−1eα/t.


## 2.5. Probability Weighted Moments

The PRWMOs are often used to investigate additional aspects of the probability distribution. The PRWMOs of the random variableT, denoted by Sr,p, are defined as(21)Sr,p=∫−∞∞trftFtpdt,where r and p are positive integers. Substituting (3) and (4) into (21) yields the PRWMOs of the ILBE distribution as follows:(22)Sr,p=α2∫0∞tr−31+αtpe−p+1α/t=∑j=0ppjαj+2∫0∞tr−3−je−p+1α/tdt.As a result of the simplification, the PRWMOs of the ILBE distribution assume the following structure:(23)Sr,p=∑j=0ppjαrΓj−r+2p+1j−r+2.


## 3. Statistical Inference

### 3.1. MLL Estimator Based on TIIC

AssumeT1,T2,…,Tn are the recorded TIICS of size r, whose lifetimes have the ILBE distribution with PDF (4), and the experiment is completed once the r-th object fails for just some fixed values of r. The log-likelihood function (LLF), according to TIIC, is provided by(24)lnl2=lnC+2rlnα−3∑i=1rlnti−∑i=1rαti+n−rln1−1+αtre−α/tr,and for the sake of simplification, we abbreviate ti rather than ti. As a result, the partial derivatives of the LLF with regard to the component of the score Uα=∂lnl2/∂α may be computed as follows:(25)Uα=2rα−∑i=1r1ti+n−rαe−α/trtr21−1+α/tre−α/tr.The model parameters’ MLL estimator is produced by numerically solving equation (18) after assigning it to zero. In the case of a complete sample, we acquire the MLL estimators of the model parameters for r = n.


### 3.2. Simulation Results

A simulation is used to evaluate the estimators’ behavior considering a set of parameter choices. Mean square error (℘), bias (ℑ), lower limit (℧) of the COIs, upper bound (ℌ) of the COIs, and average length (ℶ) of 90% and 95% are among the metrics computed. All numerical calculations are made using the R programming (R 4.1.1). The following algorithm are used:(i)
On aggregate, the ILBE distribution produces 1000 random samples with sizes ofn = 100, 200, and 300.(ii)
Values for a few parameters areα = 1.2 and α = 1.5.(iii)
There are three degrees of censorship:r = 60%, 80% (TIIC), and 100% (complete sample).(iv)
℘, ℑ, ℧, ℌ, and ℶ of estimates are computed.Tables1 and 2 include the numerical findings for the complete and TIIC measurements, respectively.Table 1
MLE,℘, ℑ, ℧, ℌ, and ℶ of the ILBE distribution for α = 1.2 under TIIC.
ntr (%)MLE℘ℑ90%95%℧ℌℶ℧ℌℶ100601.60910.40910.21111.30681.91130.60451.24891.96920.7203801.35480.15480.05461.08031.62940.54901.02781.68190.65421001.23160.03160.02630.97001.49310.52310.91991.54320.6233200601.56480.36480.15751.33721.79240.45521.29371.83600.5424801.31950.11950.03191.11251.52660.41421.07281.56630.49351001.19930.00070.01461.00201.39650.39460.96421.43430.4701300601.57770.37770.15511.41541.73990.32451.38441.77100.3866801.33070.13070.02561.18311.47840.29531.15481.50670.35191001.20970.00970.00711.06901.35040.28141.04201.37730.3353Table 2
MLE,℘, ℑ, ℧, ℌ, and ℶ of the ILBE distribution for α = 1.5 under TIIC.
ntr (%)MLE℘ℑ90%95%℧ℌℶ℧ℌℶ100601.98150.48150.28461.60912.35380.74471.53782.42510.8873801.67200.17200.06661.33322.01070.67751.26832.07560.80731001.51960.01960.03131.19691.84230.64541.13511.90410.7690200601.97850.47850.26381.69072.26640.57571.63562.32150.6859801.66950.16950.05441.40751.93150.52401.35731.98170.62431001.51700.01700.02151.26741.76650.49911.21961.81430.5946300601.95620.45620.22731.75502.15730.40231.71652.19580.4793801.64860.14860.03591.46561.83150.36591.43061.86660.43591001.49880.00120.01141.32451.67310.34871.29111.70650.4154From these tables, we conclude the following:(i)
As the sample size grows,℘,ℑ, and ℶ of all estimates decrease.(ii)
℘,ℑ, and ℶ of all estimates decrease as r decreases.(iii)
ℶ of the COIs increases as the confidence levels increase from 90% to 95%.


## 3.1. MLL Estimator Based on TIIC

AssumeT1,T2,…,Tn are the recorded TIICS of size r, whose lifetimes have the ILBE distribution with PDF (4), and the experiment is completed once the r-th object fails for just some fixed values of r. The log-likelihood function (LLF), according to TIIC, is provided by(24)lnl2=lnC+2rlnα−3∑i=1rlnti−∑i=1rαti+n−rln1−1+αtre−α/tr,and for the sake of simplification, we abbreviate ti rather than ti. As a result, the partial derivatives of the LLF with regard to the component of the score Uα=∂lnl2/∂α may be computed as follows:(25)Uα=2rα−∑i=1r1ti+n−rαe−α/trtr21−1+α/tre−α/tr.The model parameters’ MLL estimator is produced by numerically solving equation (18) after assigning it to zero. In the case of a complete sample, we acquire the MLL estimators of the model parameters for r = n.


## 3.2. Simulation Results

A simulation is used to evaluate the estimators’ behavior considering a set of parameter choices. Mean square error (℘), bias (ℑ), lower limit (℧) of the COIs, upper bound (ℌ) of the COIs, and average length (ℶ) of 90% and 95% are among the metrics computed. All numerical calculations are made using the R programming (R 4.1.1). The following algorithm are used:(i)
On aggregate, the ILBE distribution produces 1000 random samples with sizes ofn = 100, 200, and 300.(ii)
Values for a few parameters areα = 1.2 and α = 1.5.(iii)
There are three degrees of censorship:r = 60%, 80% (TIIC), and 100% (complete sample).(iv)
℘, ℑ, ℧, ℌ, and ℶ of estimates are computed.Tables1 and 2 include the numerical findings for the complete and TIIC measurements, respectively.Table 1
MLE,℘, ℑ, ℧, ℌ, and ℶ of the ILBE distribution for α = 1.2 under TIIC.
ntr (%)MLE℘ℑ90%95%℧ℌℶ℧ℌℶ100601.60910.40910.21111.30681.91130.60451.24891.96920.7203801.35480.15480.05461.08031.62940.54901.02781.68190.65421001.23160.03160.02630.97001.49310.52310.91991.54320.6233200601.56480.36480.15751.33721.79240.45521.29371.83600.5424801.31950.11950.03191.11251.52660.41421.07281.56630.49351001.19930.00070.01461.00201.39650.39460.96421.43430.4701300601.57770.37770.15511.41541.73990.32451.38441.77100.3866801.33070.13070.02561.18311.47840.29531.15481.50670.35191001.20970.00970.00711.06901.35040.28141.04201.37730.3353Table 2
MLE,℘, ℑ, ℧, ℌ, and ℶ of the ILBE distribution for α = 1.5 under TIIC.
ntr (%)MLE℘ℑ90%95%℧ℌℶ℧ℌℶ100601.98150.48150.28461.60912.35380.74471.53782.42510.8873801.67200.17200.06661.33322.01070.67751.26832.07560.80731001.51960.01960.03131.19691.84230.64541.13511.90410.7690200601.97850.47850.26381.69072.26640.57571.63562.32150.6859801.66950.16950.05441.40751.93150.52401.35731.98170.62431001.51700.01700.02151.26741.76650.49911.21961.81430.5946300601.95620.45620.22731.75502.15730.40231.71652.19580.4793801.64860.14860.03591.46561.83150.36591.43061.86660.43591001.49880.00120.01141.32451.67310.34871.29111.70650.4154From these tables, we conclude the following:(i)
As the sample size grows,℘,ℑ, and ℶ of all estimates decrease.(ii)
℘,ℑ, and ℶ of all estimates decrease as r decreases.(iii)
ℶ of the COIs increases as the confidence levels increase from 90% to 95%.


## 4. Applications to Real Data

In this part, we demonstrate the ILBE model’s adaptability by examining three real-world datasets. Comparing the fit of the ILBE model with known distributions such as the HLOIR [19], TIITOLIR [20], and TRIR [21] distributions, the ILBE model performs better. The PDFs of competitive models are(26)fHLOIRt=4λα2t−3exp−α/t21−exp−α/t2λ−11+1−exp−α/t2λ2,fTIITOLIRt=4θα2t−3exp−2α/t21−exp−2α/t2θ−1,(27)fTRIRt=2θt−3exp−θt21+λ−2λexp−θt2.In order to make a comparison between various models, some information criteria (INC) like maximized likelihood (𝚲1), Akaike INC (𝚲2), consistent Akaike INC (𝚲3), Bayesian INC (𝚲4), and Hannan–Quinn INC (𝚲5) are used. According to the given data, the optimal model is one with the lowest value of 𝚲1, 𝚲2, 𝚲3, 𝚲4, and 𝚲5.The first dataset [22]: it describes 72 guinea pigs infected with highly pathogenic tubercle bacilli and their survival periods (in days).The second dataset: acquired and documented in [23], the dataset comprises the waiting times (in minutes) of 100 bank clients.The third dataset [24]: it offers 32 observations on the failure time for vertical boring machines.Figures2–4 indicate the fitted PDFs, fitted CDFs of the ILBE distribution, and those of the comparison models (HLOIR, TIITOLIR, and TRIR) for the three datasets.Figure 2
Fitted PDFs and CDFs of comparison models for the first dataset.Figure 3
Fitted PDFs and CDFs of comparison models for the second dataset.Figure 4
Fitted PDFs and CDFs of comparison models for the third dataset.It can be observed from Figures2–4 that the ILBE distribution exhibits good matches, attesting its applicability for the three datasets.Tables3–5 show the ML estimates (MLEs) and standard errors (SEs) for the ILBE model when compared to various known distributions such like HLOIR, TIITOLIR, and TRIR. They also include the relevant measures of fit statistic.Table 3
Numerical results of MLE, SE,∧1, ∧2, ∧3, ∧4, and ∧5 for the first dataset.
ModelMLEs and SE∧1∧2∧4∧5∧3ILBEα2.272 (0.189)224.111226.111225.969227.018226.168HLIRα,λ0.436 (0.05)0.579 (0.07)260.586264.586264.301266.399264.76TIITLIRα,λ0.325 (0.036)0.404 (0.058)280.492284.492284.207286.305284.666TIRα,λ0.352 (0.426)−0.942 (0.351)280.538284.538284.253286.351284.712Table 4
Numerical results of MLE, SE,∧1, ∧2, ∧3, ∧4, and ∧5 for the second dataset.
ModelMLEs and SE∧1∧2∧4∧5∧3ILBEα10.696 (0.7563)664.794666.794666.794667.848666.834HLIRα,λ2.404 (0.226)0.589 (0.06)680.806684.806684.806686.915684.93TIITLIRα,λ1.824 (0.162)0.43 (0.051)700.214704.214704.214706.323704.338TIRα,λ9.978 (1.136)−0.812 (0.085)720.665724.665724.665726.774724.706Table 5
Numerical results of MLE, SE,∧1, ∧2, ∧3, ∧4, and ∧5 for the third dataset.
ModelMLEs and S.E∧1∧2∧4∧5∧3ILBEα4326 (540.7191)567.141569.141568.647569.627569.275HLIRα,λ1237 (184.326)0.866 (0.172)575.242579.242578.252580.214579.656TIITLIRα,λ0.069 (0.041)0.049 (0.0091)716.204720.204719.214721.176720.618TIRα,λ1821000 (346600)−0.859 (0.126)575.303579.303578.313580.275579.717Furthermore, Tables3–5 show that the ILBE distribution is the best match among the other models for the three datasets, since the ILBE distribution has the lowest values of the suggested metrics.


## 5. Conclusions

This paper developed a new one-parameter lifetime distribution, named as inverse length-biased exponential distribution. The new model is quite flexible in nature and can acquire a variety of shapes of density and hazard rate functions. MOs, PRWMOs, inverse MOs, incomplete MOs, MRLS, and MINT are all explored as key characteristics of the new distribution. In both complete and censored samples, the maximum likelihood methodology is developed to calculate the parameters of the new distribution. To investigate the conduct of estimations, a simulation analysis is discussed. Three real-world examples show that the inverse length exponential distribution gives a pretty good fit and may be used as a competitive model to fit real-world data. It is hoped that this distribution would be helpful to scholars in a variety of disciplines. In the future, we plan to use the new proposed model to study the statistical inference of it under different censored schemes, using various methods of estimation to assess the performance of its parameters. Also, researchers can extend and generalized it because this model is very simple and has more flexibility to fitting more datasets.



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