Inverted Length-Biased Exponential Model: Statistical

Research ArticleInverted Length-Biased Exponential Model Statistical Inferenceand Modeling

Waleed Almutiry

Department of Mathematics College of Science and Arts in Ar Rass Qassim University Ar Rass Saudi Arabia

Correspondence should be addressed to Waleed Almutiry wkmtieriequedusa

Received 17 September 2021 Revised 29 September 2021 Accepted 11 October 2021 Published 26 October 2021

Academic Editor naeem jan

Copyright copy 2021 Waleed Almutiry )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

)is research article proposes a new probability distribution referred to as the inverted length-biased exponential distribution)e hazard rate function (HZRF) and density function (PDF) in the new distribution allow additional flexibility as well as somedesired features It provides a more flexible approach that may be used to represent many forms of real-world data )e quantilefunction (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 invertedlength-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 isconstructed at two confidence levels We perform simulation study to examine the accuracy of estimates depending upon somestatistical measures Simulation results show that there is great agreement between theoretical and empirical studies Wedemonstrate the new modelrsquos relevance and adaptability by modeling three lifespan datasets )e proposed model is a better fitthan the half logistic inverse Rayleigh (HLOIR) type II ToppndashLeone inverse Rayleigh (TIITOLIR) and transmuted inverseRayleigh (TRIR) distributionsWe anticipate that the expanded distribution will attract a broader range of applications in a varietyof fields of research

1 Introduction

Length-biased exponential (LBE) or moment exponential(ME) distribution is considered as one of the most importantunivariate and parametric models It is commonly utilized inthe analysis of data collected throughout a lifespan and inproblems connected to the modeling of failure processes)ere is much to be said for a flexible lifespan distributionmodel and this one may be a suitable fit for some sets offailure data Reference [1] proposed the LBE with the fol-lowing PDF and distribution function (CDF)

g(x α) α2xeminus αx xge 0 αgt 0 (1)

G(x α) 1 minus (1 + αx)eminus αx

xge 0 αgt 0 (2)

where α is the scale parameter Different values of the shapeparameter lead to different shapes of density function

Many authors extended new models from the LBEdistribution such as exponentiated ME [2] generalizedexponentiated ME [3] and MarshallndashOlkin (MO) LBE(MOLBE) distributions [4] MO Kumaraswamy ME modelwas discussed in [5]

Several univariate continuous distributions have beenextensively used in environmental engineering financialand biomedical sciences among other areas for modelinglifetime data However there is still a strong need for asignificant improvement of the classical distributionsthrough different techniques for modeling several datalifetime In this regard the inverted (or inverse) (I) distri-bution is one procedure that explores extra properties of thephenomenon which cannot be produced from noninverteddistributions Applications of inverted distributions includeeconometrics as well as the engineering sciences as well asbiology and survey sampling as well as medical researchamong others In the literature several studies related to

HindawiJournal of MathematicsVolume 2021 Article ID 1980480 8 pageshttpsdoiorg10115520211980480

inverted distributions have been handled by several re-searchers for instance Reference [6] introduced the IWeibull distribution Reference [7] studied the I Pareto type1 distribution Reference [8] investigated the I Pareto type 2distribution Reference [9] handled exponentiated I Weibulldistribution Reference [10] provided the I Lindley distri-bution Reference [11] suggested the I KumaraswamymodelReference [12] presented the I Nadarajah-Haghighi modelReference [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 Refer-ence [17] investigated the I ToppndashLeone distribution andhalf logistic I ToppndashLeone distribution was studied in [18]

Our motivation here is (i) introducing a new distribu-tion 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 pa-rameter from complete and censored samples and (iv)examining its applicability using three real datasets

)e inverted LBE (ILBE) distribution is constructed byusing the random variable T1X where X follows (2) )eILBE distributionrsquos CDF is described as

F(t α) 1 +αt

1113874 1113875eminus αt

tge 0 αgt 0 (3)

)e ILBE distributionrsquos PDF is specified as

f(t α) α2

t3 e

minus αt tge 0 αgt 0 (4)

)e survival function (SRF) and HZRF of the ILBEdistribution are provided by

F(t α) 1 minus 1 +αt

1113874 1113875eminus αt

h(t α) α2eminus αt

t3 1 minus (1 + αt)eminus αt1113960 1113961

(5)

Figure 1 depicts PDF and HZRF plots for the ILBEdistribution According to Figure 1 the density of thesuggested distribution is highly flexible in nature and cantake on a number of forms including positively skewed andunimodal )rough the parameter space the HZRF can takeon many forms such as decreasing rising or upside down

)is paper is organized as follows In Section 2 the basiccharacteristics of the ILBE distribution are obtained )eMLL estimators for the ILBE model are described in Section3 and are established on complete and censored samplesaccompanied by a simulation analysis )e application toactual data collection is covered in Section 4 Section 5concludes the paper with some remarks

2 Fundamental Mathematical Properties ofILBE Distribution

Here we give some essential properties of the ILBE dis-tribution like QuF MOs PRWMOs incomplete MOs andinverse MOs

21 Quantile Function A generated random number fromthe ILBE distribution is obtained by solving the followingequation numerically

Q(u) α

minus 1 minus Wminus 1 minus eminus 1

u1113872 1113873 0lt ult 1 (6)

where Wminus 1 denotes the negative branch of the Lambert Wfunction (ie the solution of the equation W(Z)eW(Z) z

)e median say Q2 is achieved by adjusting u 05 in (6)and the first quartile and third quartile denoted by Q1 andQ3 are obtained by setting u 025 and 075 respectively in(6) Note that equation (6) is solved numerically by usingMathematica 9

Q1 α

minus 1 minus Wminus 1 minus 025eminus 1

1113872 1113873

Q2 α


1113872 1113873

Q3 α


1113872 1113873

(7)

22 Moments Due to its relevance in any statistical studywe shall give the n-th MO of the ILBE distribution here Forthe ILBE model the n-th MO of T about the origin iscomputed as follows

μnprime E T

n( 1113857 1113946

infin

0tn α

2

t3 e

minus αtdt αnΓ(2 minus n) nlt 2 (8)

)e following formula may be used to determine theMOGF of the ILBE distribution

Mx(t) 1113944

infin

n0

tn

nμnprime 1113944

infin

n0

tn

nαnΓ(2 minus n) nlt 2 (9)

)e incomplete (IN) MO say ςn(x) is

ςn(t) α2t 1113946 0tnminus 3

eminus αtdt αn

c 2 minus nαt

1113874 1113875 nlt 2 (10)

where c( t) is the upper IN gamma functionFurther the conditional MO say ϖs(x) is

ϖs(t) α2 1113946infin

ttαminus 3

eminus αtdt αnΓ 2 minus n

αt

1113874 1113875 nlt 2 (11)

where Γ( t) is the lower IN gamma functionFor the ILBE distribution the n-th inverse MO is cal-

culated on the basis

τk(x) α2 1113946infin

0tminus nminus 3

eminus αtdt αminus kΓ(n + 2) (12)

For n 1 we get the harmonic mean of the ILBEdistribution

)e Lorenz and Bonferroni curves are obtained asfollows

2 Journal of Mathematics

LF(t) ς1(t)

E(T) c 1

αt

1113874 1113875 (13)

BF(t) LF(t)

F(t)

c(1 (αt))(1 +(αt))eminus αt (14)

23 Order Statistics Let T1 T2 Tn be r samples fromthe ILBE model with order statistics T(1) T(2) T(n))e PDF of T(k) of order statistics is given by

fT(k)(t)

n

(k minus 1)(n minus k)F

kminus 1(t)f(t)(1 minus F(t))

nminus k

(15)

)e PDF of T(k) can be expressed as

fT(k)(t)

nα2

(k minus 1)(n minus k)tminus 3 1 +

αt

1113874 1113875kminus 1

eminus kαt 1 minus 1 +

αt

1113874 1113875eminus αt

1113874 1113875nminus k

(16)

Particularly PDF of the first and largest order statisticscan be calculated as

fT(1)(t) nα2tminus 3

eminus αt 1 minus 1 +

αt

1113874 1113875eminus αt

1113874 1113875nminus 1

(17)

fT(n)(t) nα2tminus 3 1 +

αt

1113874 1113875nminus 1

eminus nαt

(18)

respectively

24 Mean Residual Life Function It has an important ap-plication of theMOs of residual lifetime function)eMRLSof ILBE distribution is

υ(t) E(T minus t|Tgt t) 1

F(t)1113946

infin

t

xf(x)dt minus t

αΓ 1αt

1113874 1113875 1 minus 1 +αt

1113874 1113875eminus αt

1113876 1113877minus 1

minus t

(19)

)eMINTrepresents the amount of time that has passedafter an item has failed assuming that this failure has oc-curred )e MINT of ILBE distribution is

ϖ(t) E(T minus t|Tle t) t minus1

F(t)1113946

t

0

xf(x)dx

t minus αc 1αt

1113874 1113875 1 +αt

1113874 1113875minus 1

eαt

(20)

25 ProbabilityWeightedMoments )e PRWMOs are oftenused to investigate additional aspects of the probabilitydistribution )e PRWMOs of the random variable Tdenoted by Srp are defined as

Srp 1113946infin

minus infintrf(t)[F(t)]

pdt (21)

where r and p are positive integers Substituting (3) and (4)into (21) yields the PRWMOs of the ILBE distribution asfollows

15

10

05

00

f (t)

h (t)

00 05 10 15t t

20 25 30

20

15

10

05

00

0 1 2 3 4

α = 09α = 12α = 15

α = 18α = 20α = 25

α = 09α = 12α = 15

α = 18α = 20α = 25

Figure 1 Plots of PDF and HZRF for the ILBE distribution

Journal of Mathematics 3

Srp α2 1113946infin

0trminus 3 1 +

αt

1113874 1113875p

eminus (p+1)αt

1113944

p

j0

p

j

⎛⎜⎜⎝ ⎞⎟⎟⎠αj+21113946

infin

0

trminus 3minus j

eminus (p+1)αtdt

(22)

As a result of the simplification the PRWMOs of theILBE distribution assume the following structure

Srp 1113944

p

j0

p

j

⎛⎝ ⎞⎠αrΓ(j minus r + 2)

(p + 1)jminus r+2 (23)

3 Statistical Inference

31 MLL Estimator Based on TIIC AssumeT(1) T(2) T(n) are the recorded TIICS of size r whoselifetimes have the ILBE distribution with PDF (4) and theexperiment is completed once the r-th object fails for justsome fixed values of r )e log-likelihood function (LLF)according to TIIC is provided by

ln l2 ln C + 2r ln α minus 31113944r

i1ln ti minus 1113944

r

i1

αti

+(n minus r)ln 1 minus 1 +αtr

1113888 1113889eminus αtr1113890 1113891

(24)

and for the sake of simplification we abbreviate ti ratherthan t(i) As a result the partial derivatives of the LLF withregard to the component of the score U(α) z ln l2zαmay be computed as follows

U(α) 2r

αminus 1113944

r

i1

1ti

+(n minus r)αe

minus αtr

t2r 1 minus 1 + αtr( 1113857( 1113857e

minus αtr1113872 11138731113960 1113961 (25)

)e model parametersrsquo MLL estimator is produced bynumerically solving equation (18) after assigning it to zeroIn the case of a complete sample we acquire the MLL es-timators of the model parameters for r n

32 Simulation Results A simulation is used to evaluate theestimatorsrsquo behavior considering a set of parameter choicesMean square error (weierp) bias (I) lower limit ([) of the COIsupper bound (H) of the COIs and average length (ℶ) of 90and 95 are among the metrics computed All numericalcalculations are made using the R programming (R 411))e following algorithm are used

(i) On aggregate the ILBE distribution produces 1000random samples with sizes of n 100 200 and 300

(ii) Values for a few parameters are α 12 and α 15(iii) )ere are three degrees of censorship r 60 80

(TIIC) and 100 (complete sample)(iv) weierp I [ H and ℶ of estimates are computed

Tables 1 and 2 include the numerical findings for thecomplete and TIIC measurements respectively

Table 1 MLE weierp I [ H and ℶ of the ILBE distribution for α 12 under TIIC

n tr () MLE weierp I90 95

[ H ℶ [ H ℶ

10060 16091 04091 02111 13068 19113 06045 12489 19692 0720380 13548 01548 00546 10803 16294 05490 10278 16819 06542100 12316 00316 00263 09700 14931 05231 09199 15432 06233

20060 15648 03648 01575 13372 17924 04552 12937 18360 0542480 13195 01195 00319 11125 15266 04142 10728 15663 04935100 11993 00007 00146 10020 13965 03946 09642 14343 04701

30060 15777 03777 01551 14154 17399 03245 13844 17710 0386680 13307 01307 00256 11831 14784 02953 11548 15067 03519100 12097 00097 00071 10690 13504 02814 10420 13773 03353



[ H ℶ [ H ℶ

10060 19815 04815 02846 16091 23538 07447 15378 24251 0887380 16720 01720 00666 13332 20107 06775 12683 20756 08073100 15196 00196 00313 11969 18423 06454 11351 19041 07690

20060 19785 04785 02638 16907 22664 05757 16356 23215 0685980 16695 01695 00544 14075 19315 05240 13573 19817 06243100 15170 00170 00215 12674 17665 04991 12196 18143 05946

30060 19562 04562 02273 17550 21573 04023 17165 21958 0479380 16486 01486 00359 14656 18315 03659 14306 18666 04359100 14988 00012 00114 13245 16731 03487 12911 17065 04154


From these tables we conclude the following

(i) As the sample size grows weierpI and ℶ of all esti-mates decrease

(ii) weierpI 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 modelrsquos adaptabilityby examining three real-world datasets Comparing the fit of

the ILBE model with known distributions such as theHLOIR [19] TIITOLIR [20] and TRIR [21] distributionsthe ILBE model performs better )e PDFs of competitivemodels are

fHLOIR(t) 4λα2tminus 3 exp minus (αt)21113872 1113873 1 minus exp minus (αt)21113872 11138731113872 1113873

λminus 1

1 + 1 minus exp minus (αt)21113872 11138731113872 1113873λ

1113874 11138752

fTIITOLIR(t) 4θα2tminus 3 exp minus 2(αt)21113872 1113873 1 minus exp minus 2(αt)21113872 11138731113872 1113873θminus 1

(26)

10

08

06

04

02

00

Den

sity

0 1 2 3t

4 5 6 0 1 2 3t

4 5 6 7

ILBEHLOIR

TIITOLIRTRIR ILBE

Empirical

HLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)Figure 2 Fitted PDFs and CDFs of comparison models for the first dataset

020

015

010

005

000

Den

sity

0 5 10 15t

20 25 30

ILBEHLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)

0 10 20 30 40t

ILBEEmpirical

HLOIR

TIITOLIRTRIR

Figure 3 Fitted PDFs and CDFs of comparison models for the second dataset


fTRIR(t) 2θtminus 3 exp minus

θt21113888 1113889 1 + λ minus 2λ exp minus

θt21113888 11138891113888 1113889

(27)

In order to make a comparison between various modelssome information criteria (INC) like maximized likelihood(1) Akaike INC (2) consistent Akaike INC (3)Bayesian INC (4) and HannanndashQuinn INC (5) are usedAccording to the given data the optimal model is one withthe lowest value of 1 2 3 4 and 5

e first dataset [22] it describes 72 guinea pigs in-fected with highly pathogenic tubercle bacilli and theirsurvival periods (in days)e second dataset acquired and documented in [23]the dataset comprises the waiting times (in minutes) of100 bank clientse third dataset [24] it offers 32 observations on thefailure time for vertical boring machines

Figures 2ndash4 indicate the fitted PDFs fitted CDFs of theILBE distribution and those of the comparison models(HLOIR TIITOLIR and TRIR) for the three datasets

Table 3 Numerical results of MLE SE and1 and2 and3 and4 and and5 for the first dataset

Model MLEs and SE and1 and2 and4 and5 and3ILBE (α) 2272 (0189) 224111 226111 225969 227018 226168HLIR (α λ) 0436 (005) 0579 (007) 260586 264586 264301 266399 26476TIITLIR (α λ) 0325 (0036) 0404 (0058) 280492 284492 284207 286305 284666TIR (α λ) 0352 (0426) minus 0942 (0351) 280538 284538 284253 286351 284712

ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR

Figure 4 Fitted PDFs and CDFs of comparison models for the third dataset

Table 4 Numerical results of MLE SE and1 and2 and3 and4 and and5 for the second dataset


Table 5 Numerical results of MLE SE and1 and2 and3 and4 and and5 for the third dataset



It can be observed from Figures 2ndash4 that the ILBEdistribution exhibits good matches attesting its applicabilityfor the three datasets

Tables 3ndash5 show the ML estimates (MLEs) and standarderrors (SEs) for the ILBE model when compared to variousknown distributions such like HLOIR TIITOLIR and TRIR)ey also include the relevant measures of fit statistic

Furthermore Tables 3ndash5 show that the ILBE distributionis the best match among the other models for the threedatasets since the ILBE distribution has the lowest values ofthe suggested metrics

5 Conclusions

)is paper developed a new one-parameter lifetime distri-bution named as inverse length-biased exponential distri-bution )e new model is quite flexible in nature and canacquire a variety of shapes of density and hazard ratefunctions MOs PRWMOs inverse MOs incomplete MOsMRLS and MINT are all explored as key characteristics ofthe new distribution In both complete and censoredsamples the maximum likelihoodmethodology is developedto calculate the parameters of the new distribution To in-vestigate the conduct of estimations a simulation analysis isdiscussed )ree real-world examples show that the inverselength exponential distribution gives a pretty good fit andmay be used as a competitive model to fit real-world data Itis hoped that this distribution would be helpful to scholars ina variety of disciplines In the future we plan to use the newproposed model to study the statistical inference of it underdifferent censored schemes using various methods of esti-mation to assess the performance of its parameters Alsoresearchers can extend and generalized it because this modelis very simple and has more flexibility to fitting moredatasets

Data Availability

Interested parties can reach out to the author in order toreceive the numerical dataset used to perform the researchdescribed in the paper

Conflicts of Interest

)e author declares that there are no conflicts of interest

Acknowledgments

)e author would like to thank the Deanship of ScientificResearch Qassim University for funding the publication ofthis project

References

[1] S T Dara and M Ahmad Recent Advances in MomentDistribution and their Hazard Rates Lap Lambert AcademicPublishing GmbH KG Sunnyvale CA USA 2012

[2] S A Hasnain Exponentiated Moment Exponential Distri-butions National College of Business Administration ampEconomics Lahore Pakistan 2013

[3] Z Iqbal S A Hasnain M Salman M Ahmad andG Hamedani ldquoGeneralized exponentiated moment expo-nential distributionrdquo Pakistan Journal of Statistics vol 30no 4 pp 537ndash554 2014

[4] M A u Haq R M Usman S Hashmi and A I Al-Omerildquo)e Marshall-Olkin length-biased exponential distributionand its applicationsrdquo Journal of King Saud University Sciencevol 31 no 2 pp 246ndash251 2019

[5] A Ali H Ahmadini A S Hassan R E MohamedS S Alshqaq and H F Nagy ldquoA New four-parametermoment exponential model with applications to lifetimedatardquo Intelligent Automation amp Soft Computing vol 29 no 1pp 131ndash146 2021

[6] A Keller and A R R Kamath ldquoAlternative reliability modelsfor mechanical systemsrdquo in Proceedings of the gtird Inter-national Conference on Reliability and MaintainabilityToulse France October 1982

[7] R I Abd EL-Kader G R AL-Dayian and S A AL-GendyldquoInverted Pareto type I distribution properties and estima-tionrdquo Journal of Faculty of Commerce AL-Azhar Universityvol 21 pp 19ndash40 2013

[8] G R AL-Dayian ldquoInverted Pareto type II distributionproperties and estimationrdquo Journal of Faculty of CommerceAL-Azhar University Girlsrsquo Branch vol 22 pp 1ndash18 2004

[9] A Aljuaid ldquoEstimating the parameters of an exponentiatedinverted Weibull distribution under type-II censoringrdquo Ap-plied Mathematical Sciences vol 7 no 35 pp 1721ndash17362013

[10] V K Sharma S K Singh U Singh and V Agiwal ldquo)einverse Lindley distribution a stress-strength reliability modelwith application to head and neck cancer datardquo Journal ofIndustrial and Production Engineering vol 32 no 3pp 162ndash173 2015

[11] A M Abd AL-Fattah A A EL-Helbawy and G R AL-Dayian ldquoInverted Kumaraswamy distribution properties andestimationrdquo Pakistan Journal of Statistics vol 33 no 1pp 37ndash61 2017

[12] M H Tahir G M Cordeiro S Ali S Dey and A Manzoorldquo)e inverted Nadarajah-Haghighi distribution estimationmethods and applicationsrdquo Journal of Statistical Computationand Simulation vol 88 no 14 pp 2775ndash2798 2018

[13] A S Hassan andM Abd-Allah ldquoOn the inverse power Lomaxdistributionrdquo Annals of Data Science vol 6 no 2pp 259ndash278 2019

[14] A S Hassan and R E Mohamed ldquoParameter estimation forinverted exponentiated Lomax distribution with right cen-sored datardquo Gazi University Journal of Science vol 32 no 4pp 1370ndash1386 2019

[15] A S Hassan and R E Mohamed ldquoWeibull inverse Lomaxdistributionrdquo Pakistan Journal of Statistics and OperationResearch vol 15 no 3 pp 587ndash603 2019

[16] A S Hassan S M Assar and A M Abd Elghaffar ldquoStatisticalproperties and estimation of power-transmuted inverseRayleigh distributionrdquo Statistics in Transition new seriesvol 21 no 3 pp 1ndash20 2020

[17] A S Hassan M Elgarhy and R Ragab ldquoStatistical propertiesand estimation of inverted Topp-Leone distributionrdquo Journalof Statistics Applications amp Probability vol 9 no 2pp 319ndash331 2020

[18] R A Bantan M Elsehetry A S Hassan et al ldquoA two-pa-rameter model properties and estimation under rankedsamplingrdquo Mathematics vol 9 pp 1ndash16 2021


[19] A M Almarashi M M Badr M Elgarhy F Jamal andC Chesneau ldquoStatistical inference of the half-logistic inverseRayleigh distributionrdquo Entropy vol 22 no 4 p 449 2020

[20] H F Mohammed and N Yahia ldquoOn type II Topp Leoneinverse Rayleigh distributionrdquo Applied Mathematical Sci-ences vol 13 no 13 pp 607ndash615 2019

[21] A Ahmad S P Ahmad and A Ahmed ldquoTransmuted inverseRayleigh distribution a generalization of the inverse Rayleighdistributionrdquo Mathematical gteory and Modeling vol 4no 7 pp 90ndash98 2014

[22] T Bjerkedal ldquoAcquisition of resistance in Guinea pigs in-fected with different doses of virulent tubercle Bacilli1rdquoAmerican Journal of Epidemiology vol 72 no 1 pp 130ndash1481960

[23] M E Ghitany B Atieh and S Nadarajah ldquoLindley distri-bution and its applicationrdquo Mathematics and Computers inSimulation vol 78 no 4 pp 493ndash506 2008

[24] M Pear Hossain M Hafidz Omar and M Riaz ldquoEstimationof mixture Maxwell parameters and its possible industrialapplicationrdquo Computers amp Industrial Engineering vol 107pp 264ndash275 2017


inverted distributions have been handled by several re-searchers for instance Reference [6] introduced the IWeibull distribution Reference [7] studied the I Pareto type1 distribution Reference [8] investigated the I Pareto type 2distribution Reference [9] handled exponentiated I Weibulldistribution Reference [10] provided the I Lindley distri-bution Reference [11] suggested the I KumaraswamymodelReference [12] presented the I Nadarajah-Haghighi modelReference [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 Refer-ence [17] investigated the I ToppndashLeone distribution andhalf logistic I ToppndashLeone distribution was studied in [18]

Our motivation here is (i) introducing a new distribu-tion 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 pa-rameter from complete and censored samples and (iv)examining its applicability using three real datasets

)e inverted LBE (ILBE) distribution is constructed byusing the random variable T1X where X follows (2) )eILBE distributionrsquos CDF is described as

F(t α) 1 +αt

1113874 1113875eminus αt

tge 0 αgt 0 (3)

)e ILBE distributionrsquos PDF is specified as

f(t α) α2

t3 e

minus αt tge 0 αgt 0 (4)

)e survival function (SRF) and HZRF of the ILBEdistribution are provided by

F(t α) 1 minus 1 +αt

1113874 1113875eminus αt

h(t α) α2eminus αt

t3 1 minus (1 + αt)eminus αt1113960 1113961

(5)

Figure 1 depicts PDF and HZRF plots for the ILBEdistribution According to Figure 1 the density of thesuggested distribution is highly flexible in nature and cantake on a number of forms including positively skewed andunimodal )rough the parameter space the HZRF can takeon many forms such as decreasing rising or upside down

)is paper is organized as follows In Section 2 the basiccharacteristics of the ILBE distribution are obtained )eMLL estimators for the ILBE model are described in Section3 and are established on complete and censored samplesaccompanied by a simulation analysis )e application toactual data collection is covered in Section 4 Section 5concludes the paper with some remarks

2 Fundamental Mathematical Properties ofILBE Distribution

Here we give some essential properties of the ILBE dis-tribution like QuF MOs PRWMOs incomplete MOs andinverse MOs

21 Quantile Function A generated random number fromthe ILBE distribution is obtained by solving the followingequation numerically

Q(u) α

minus 1 minus Wminus 1 minus eminus 1

u1113872 1113873 0lt ult 1 (6)

where Wminus 1 denotes the negative branch of the Lambert Wfunction (ie the solution of the equation W(Z)eW(Z) z

)e median say Q2 is achieved by adjusting u 05 in (6)and the first quartile and third quartile denoted by Q1 andQ3 are obtained by setting u 025 and 075 respectively in(6) Note that equation (6) is solved numerically by usingMathematica 9

Q1 α


1113872 1113873

Q2 α


1113872 1113873

Q3 α


1113872 1113873

(7)

22 Moments Due to its relevance in any statistical studywe shall give the n-th MO of the ILBE distribution here Forthe ILBE model the n-th MO of T about the origin iscomputed as follows

μnprime E T

n( 1113857 1113946

infin

0tn α

2

t3 e

minus αtdt αnΓ(2 minus n) nlt 2 (8)

)e following formula may be used to determine theMOGF of the ILBE distribution

Mx(t) 1113944

infin

n0

tn

nμnprime 1113944

infin

n0

tn

nαnΓ(2 minus n) nlt 2 (9)

)e incomplete (IN) MO say ςn(x) is

ςn(t) α2t 1113946 0tnminus 3

eminus αtdt αn

c 2 minus nαt

1113874 1113875 nlt 2 (10)

where c( t) is the upper IN gamma functionFurther the conditional MO say ϖs(x) is

ϖs(t) α2 1113946infin

ttαminus 3

eminus αtdt αnΓ 2 minus n

αt

1113874 1113875 nlt 2 (11)

where Γ( t) is the lower IN gamma functionFor the ILBE distribution the n-th inverse MO is cal-

culated on the basis

τk(x) α2 1113946infin

0tminus nminus 3

eminus αtdt αminus kΓ(n + 2) (12)

For n 1 we get the harmonic mean of the ILBEdistribution

)e Lorenz and Bonferroni curves are obtained asfollows


LF(t) ς1(t)

E(T) c 1

αt

1113874 1113875 (13)

BF(t) LF(t)

F(t)



fT(k)(t)

n



nminus k

(15)


fT(k)(t)

nα2


αt

1113874 1113875kminus 1


αt

1113874 1113875eminus αt

1113874 1113875nminus k

(16)




αt

1113874 1113875eminus αt

1113874 1113875nminus 1

(17)


αt

1113874 1113875nminus 1

eminus nαt

(18)

respectively



F(t)1113946

infin

t

xf(x)dt minus t

αΓ 1αt

1113874 1113875 1 minus 1 +αt

1113874 1113875eminus αt

1113876 1113877minus 1

minus t

(19)



F(t)1113946

t

0

xf(x)dx

t minus αc 1αt

1113874 1113875 1 +αt

1113874 1113875minus 1

eαt

(20)


Srp 1113946infin


pdt (21)


15

10

05

00

f (t)

h (t)

00 05 10 15t t

20 25 30

20

15

10

05

00

0 1 2 3 4

α = 09α = 12α = 15

α = 18α = 20α = 25

α = 09α = 12α = 15

α = 18α = 20α = 25




0trminus 3 1 +

αt

1113874 1113875p

eminus (p+1)αt

1113944

p

j0

p

j

⎛⎜⎜⎝ ⎞⎟⎟⎠αj+21113946

infin

0

trminus 3minus j

eminus (p+1)αtdt

(22)


Srp 1113944

p

j0

p

j


(p + 1)jminus r+2 (23)





r

i1

αti


1113888 1113889eminus αtr1113890 1113891

(24)


U(α) 2r

αminus 1113944

r

i1

1ti

+(n minus r)αe

minus αtr

t2r 1 minus 1 + αtr( 1113857( 1113857e

minus αtr1113872 11138731113960 1113961 (25)









[ H ℶ [ H ℶ

10060 16091 04091 02111 13068 19113 06045 12489 19692 0720380 13548 01548 00546 10803 16294 05490 10278 16819 06542100 12316 00316 00263 09700 14931 05231 09199 15432 06233

20060 15648 03648 01575 13372 17924 04552 12937 18360 0542480 13195 01195 00319 11125 15266 04142 10728 15663 04935100 11993 00007 00146 10020 13965 03946 09642 14343 04701

30060 15777 03777 01551 14154 17399 03245 13844 17710 0386680 13307 01307 00256 11831 14784 02953 11548 15067 03519100 12097 00097 00071 10690 13504 02814 10420 13773 03353



[ H ℶ [ H ℶ

10060 19815 04815 02846 16091 23538 07447 15378 24251 0887380 16720 01720 00666 13332 20107 06775 12683 20756 08073100 15196 00196 00313 11969 18423 06454 11351 19041 07690

20060 19785 04785 02638 16907 22664 05757 16356 23215 0685980 16695 01695 00544 14075 19315 05240 13573 19817 06243100 15170 00170 00215 12674 17665 04991 12196 18143 05946

30060 19562 04562 02273 17550 21573 04023 17165 21958 0479380 16486 01486 00359 14656 18315 03659 14306 18666 04359100 14988 00012 00114 13245 16731 03487 12911 17065 04154










λminus 1


1113874 11138752


(26)

10

08

06

04

02

00

Den

sity

0 1 2 3t

4 5 6 0 1 2 3t

4 5 6 7

ILBEHLOIR

TIITOLIRTRIR ILBE

Empirical

HLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t


020

015

010

005

000

Den

sity

0 5 10 15t

20 25 30

ILBEHLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)

0 10 20 30 40t

ILBEEmpirical

HLOIR

TIITOLIRTRIR





θt21113888 11138891113888 1113889

(27)






ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR










5 Conclusions


Data Availability




Acknowledgments


References



























LF(t) ς1(t)

E(T) c 1

αt

1113874 1113875 (13)

BF(t) LF(t)

F(t)



fT(k)(t)

n



nminus k

(15)


fT(k)(t)

nα2


αt

1113874 1113875kminus 1


αt

1113874 1113875eminus αt

1113874 1113875nminus k

(16)




αt

1113874 1113875eminus αt

1113874 1113875nminus 1

(17)


αt

1113874 1113875nminus 1

eminus nαt

(18)

respectively



F(t)1113946

infin

t

xf(x)dt minus t

αΓ 1αt

1113874 1113875 1 minus 1 +αt

1113874 1113875eminus αt

1113876 1113877minus 1

minus t

(19)



F(t)1113946

t

0

xf(x)dx

t minus αc 1αt

1113874 1113875 1 +αt

1113874 1113875minus 1

eαt

(20)


Srp 1113946infin


pdt (21)


15

10

05

00

f (t)

h (t)

00 05 10 15t t

20 25 30

20

15

10

05

00

0 1 2 3 4

α = 09α = 12α = 15

α = 18α = 20α = 25

α = 09α = 12α = 15

α = 18α = 20α = 25




0trminus 3 1 +

αt

1113874 1113875p

eminus (p+1)αt

1113944

p

j0

p

j

⎛⎜⎜⎝ ⎞⎟⎟⎠αj+21113946

infin

0

trminus 3minus j

eminus (p+1)αtdt

(22)


Srp 1113944

p

j0

p

j


(p + 1)jminus r+2 (23)





r

i1

αti


1113888 1113889eminus αtr1113890 1113891

(24)


U(α) 2r

αminus 1113944

r

i1

1ti

+(n minus r)αe

minus αtr

t2r 1 minus 1 + αtr( 1113857( 1113857e

minus αtr1113872 11138731113960 1113961 (25)









[ H ℶ [ H ℶ

10060 16091 04091 02111 13068 19113 06045 12489 19692 0720380 13548 01548 00546 10803 16294 05490 10278 16819 06542100 12316 00316 00263 09700 14931 05231 09199 15432 06233

20060 15648 03648 01575 13372 17924 04552 12937 18360 0542480 13195 01195 00319 11125 15266 04142 10728 15663 04935100 11993 00007 00146 10020 13965 03946 09642 14343 04701

30060 15777 03777 01551 14154 17399 03245 13844 17710 0386680 13307 01307 00256 11831 14784 02953 11548 15067 03519100 12097 00097 00071 10690 13504 02814 10420 13773 03353



[ H ℶ [ H ℶ

10060 19815 04815 02846 16091 23538 07447 15378 24251 0887380 16720 01720 00666 13332 20107 06775 12683 20756 08073100 15196 00196 00313 11969 18423 06454 11351 19041 07690

20060 19785 04785 02638 16907 22664 05757 16356 23215 0685980 16695 01695 00544 14075 19315 05240 13573 19817 06243100 15170 00170 00215 12674 17665 04991 12196 18143 05946

30060 19562 04562 02273 17550 21573 04023 17165 21958 0479380 16486 01486 00359 14656 18315 03659 14306 18666 04359100 14988 00012 00114 13245 16731 03487 12911 17065 04154










λminus 1


1113874 11138752


(26)

10

08

06

04

02

00

Den

sity

0 1 2 3t

4 5 6 0 1 2 3t

4 5 6 7

ILBEHLOIR

TIITOLIRTRIR ILBE

Empirical

HLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t


020

015

010

005

000

Den

sity

0 5 10 15t

20 25 30

ILBEHLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)

0 10 20 30 40t

ILBEEmpirical

HLOIR

TIITOLIRTRIR





θt21113888 11138891113888 1113889

(27)






ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR










5 Conclusions


Data Availability




Acknowledgments


References




























0trminus 3 1 +

αt

1113874 1113875p

eminus (p+1)αt

1113944

p

j0

p

j

⎛⎜⎜⎝ ⎞⎟⎟⎠αj+21113946

infin

0

trminus 3minus j

eminus (p+1)αtdt

(22)


Srp 1113944

p

j0

p

j


(p + 1)jminus r+2 (23)





r

i1

αti


1113888 1113889eminus αtr1113890 1113891

(24)


U(α) 2r

αminus 1113944

r

i1

1ti

+(n minus r)αe

minus αtr

t2r 1 minus 1 + αtr( 1113857( 1113857e

minus αtr1113872 11138731113960 1113961 (25)









[ H ℶ [ H ℶ

10060 16091 04091 02111 13068 19113 06045 12489 19692 0720380 13548 01548 00546 10803 16294 05490 10278 16819 06542100 12316 00316 00263 09700 14931 05231 09199 15432 06233

20060 15648 03648 01575 13372 17924 04552 12937 18360 0542480 13195 01195 00319 11125 15266 04142 10728 15663 04935100 11993 00007 00146 10020 13965 03946 09642 14343 04701

30060 15777 03777 01551 14154 17399 03245 13844 17710 0386680 13307 01307 00256 11831 14784 02953 11548 15067 03519100 12097 00097 00071 10690 13504 02814 10420 13773 03353



[ H ℶ [ H ℶ

10060 19815 04815 02846 16091 23538 07447 15378 24251 0887380 16720 01720 00666 13332 20107 06775 12683 20756 08073100 15196 00196 00313 11969 18423 06454 11351 19041 07690

20060 19785 04785 02638 16907 22664 05757 16356 23215 0685980 16695 01695 00544 14075 19315 05240 13573 19817 06243100 15170 00170 00215 12674 17665 04991 12196 18143 05946

30060 19562 04562 02273 17550 21573 04023 17165 21958 0479380 16486 01486 00359 14656 18315 03659 14306 18666 04359100 14988 00012 00114 13245 16731 03487 12911 17065 04154










λminus 1


1113874 11138752


(26)

10

08

06

04

02

00

Den

sity

0 1 2 3t

4 5 6 0 1 2 3t

4 5 6 7

ILBEHLOIR

TIITOLIRTRIR ILBE

Empirical

HLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t


020

015

010

005

000

Den

sity

0 5 10 15t

20 25 30

ILBEHLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)

0 10 20 30 40t

ILBEEmpirical

HLOIR

TIITOLIRTRIR





θt21113888 11138891113888 1113889

(27)






ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR










5 Conclusions


Data Availability




Acknowledgments


References



































λminus 1


1113874 11138752


(26)

10

08

06

04

02

00

Den

sity

0 1 2 3t

4 5 6 0 1 2 3t

4 5 6 7

ILBEHLOIR

TIITOLIRTRIR ILBE

Empirical

HLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t


020

015

010

005

000

Den

sity

0 5 10 15t

20 25 30

ILBEHLOIR

TIITOLIRTRIR

10

08

06

04

02

00

Fn (t

)

0 10 20 30 40t

ILBEEmpirical

HLOIR

TIITOLIRTRIR





θt21113888 11138891113888 1113889

(27)






ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR










5 Conclusions


Data Availability




Acknowledgments


References





























θt21113888 11138891113888 1113889

(27)






ILBEHLOIR

TIITOLIRTRIR

Fn (y

)

5e-04

4e-04

3e-04

2e-04

1e-04

0e+00

Den

sity

0 1000 2000 3000y

4000 5000

10

08

06

04

02

00

1000 2000 3000y

4000

ILBEEmpirical

HLOIR

TIITOLIRTRIR










5 Conclusions


Data Availability




Acknowledgments


References






























5 Conclusions


Data Availability




Acknowledgments


References


































Documents

Inverted Length-Biased Exponential Model: Statistical