A New Generalization of Lehmann Type - II Distribution: Theory, Simulation, and Applications to Survival and Failure Rate data

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Rinnakkaistallenteet Luonnontieteiden ja metsätieteiden tiedekunta

2021

A New Generalization of Lehmann Type - II Distribution: Theory,

Simulation, and Applications to Survival and Failure Rate data

Balogun, Oluwafemi Samson

Elsevier BV

Tieteelliset aikakauslehtiartikkelit

© 2021 The Author(s)

CC BY-NC-ND https://creativecommons.org/licenses/by-nc-nd/4.0/

http://dx.doi.org/10.1016/j.sciaf.2021.e00790

https://erepo.uef.fi/handle/123456789/25642

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Scientific African

journalhomepage:www.elsevier.com/locate/sciaf

A new generalization of Lehmann type-II distribution: Theory, simulation, and applications to survival and failure rate data

Oluwafemi Samson Balogun

^a

, Muhammad Zafar Iqbal

^b

,

Muhammad Zeshan Arshad

^b,∗

, Ahmed Z. Afify

^c

, Pelumi E. Oguntunde

^d

aSchool of Computing, University of Eastern Finland, Kuopio Campus, FI-70211, Finland

bDepartment of Mathematics and Statistics, University of Agriculture, Faisalabad, Pakistan

cDepartment of Statistics, Mathematics and Insurance, Benha University, Egypt

dDepartment of Mathematics, Covenant University, Ota, Ogun State, Nigeria

a rt i c l e i nf o

Article history:

Received 21 January 2021 Revised 5 May 2021 Accepted 8 May 2021

Editor: Dr B. Gyampoh

Mathematics subject classification 60E05

62P30 62P12 Keywords:

Lehmann type-II distribution Weighted distribution Power distribution Kumaraswamy distribution Hazard rate function Moments

Rényi entropy

Maximum likelihood estimation

a b s t ra c t

Inthispaper,apotentiatedlifetimemodelthatdemonstratesthebathtub-shapedhazard rate function isdeveloped. Thisproposedmodel is referred toas Kumaraswamy modi- fiedsize-biased LehmannType-II(Kum–MSBL–II)distribution.Variousmathematicaland reliabilitycharacteristics arederivedand discussed.The maximumlikelihoodestimation method isused toestimate themodel parameters and itsperformance is discussedby followingasimulationstudy.Three-lifetimesetsofdataareutilizedtoillustratetheflex- ibility oftheproposed modeland theresultsdemonstratethattheproposed modelcan adequatelyfitthereal-lifedatasetsthanthecompetitors.

© 2021TheAuthor(s).PublishedbyElsevierB.V.onbehalfofAfricanInstituteof MathematicalSciences/NextEinsteinInitiative.

ThisisanopenaccessarticleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/)

Introduction

Theoreticalandappliedresearchersmostoftensearchforalternativeskewedandskewedsymmetriclifetimeparametric modelsthatmayperformwellthanparentdistribution.Accordingly,tofillthisgap,theresearchers’excellentattentionto- wardsthedevelopmentofnewmodelshasbeenobservedtoexplorethehiddencharactersofbaselinemodelsoverthepast twodecades.Newmodelsopennewhorizonsforthetheoreticalandappliedresearcherstoaddressreal-worldproblems,so wellandadequatelyfittheasymmetricandcomplexrandomphenomena.Tothiseffect,severalmodelshavebeendeveloped anddiscussedintheliterature.Oneofthemoststraightforwardandhandylifetimemodelsinducedintheresearcherworld wastheLehmann[1]type-I(L-I)andtype-II(L-II)models.TheLehmanntype-I(L-I)modelismostoftendiscussedinfavor

∗Corresponding author.

E-mail address: profarshad@yahoo.com (M.Z. Arshad).

https://doi.org/10.1016/j.sciaf.2021.e00790

2468-2276/© 2021 The Author(s). Published by Elsevier B.V. on behalf of African Institute of Mathematical Sciences / Next Einstein Initiative. This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ )

ofPowerfunction(PF)distribution.AsfarasthePFdistributionisconcerned,itisdefinedasthesimpleexponentiationof anybaselinemodelwithassociatedCDFisgivenby;

F

(

^x

)

=G^α

(

^x

)

, x>0.

Inaddition,Guptaetal.[2]practicedtheL–Iclassonexponentialdistribution.ThesimplicityandusefulnessofPFdis- tributionhaveattractedresearcherstoexploreitsfurtherapplications,extensions,andgeneralizationsindifferentareasof science.Meanwhile,a dualtransformationyieldedtheLehmannType-II(L–II)GclassattributedtoCordeiroandDeCastro [3]withassociatedCDFisgivenby

F

(

^x

)

⁼¹⁻

(

^1−G

(

^x

) )

^{α, x>0,}

where

α

>0isashapeparameter.

The closed-formfeatures ofL-IIdistribution helpsone toderive andstudyitsnumerousproperties.In literature,both approaches(L-IandL-II)havebeenextensivelyutilizedtoexplorethenewcharacteristicsofclassicalandmodifiedmodels.

Formoredetails,readersareencouragedtoseethecredibleworkofsomenotablescientists includingDallas [4],whode- velopedanexcitingrelationshipbetweenPFandParetodistributions;MeniconiandBarry[5]foundthePFasabest-fitted model onelectronic componentsdata;independenceofrecord values-basedcharacterizationwasdiscussedby Chang [6]; orderstatistics,andlowerrecordvaluessupporteddescriptionwerestudiedby Tavangar[7];CordeiroandBrito[8]devel- oped thebeta PF distribution; Ahsanullahet al.[9]discussed the characterization ofthe PF distribution basedon lower record values; ZakaandFarooqi[10]developed variousestimation methods; Shahzadetal.[11] derived themoments by followingdifferenttechniques.

In the mostrecent ones, L-II G family-based development wasstudied by severalstatisticians including Arshadet al.

[12].Arshadetal.[12]developedabathtubshapedfailureratemodelandexploreditsapplicationusingengineeringdata;

Tomazellaetal.[13]discussednumerousmathematicalpropertiesofL-IIFréchetdistributionandexploreditsapplicationto aircraftmaintenancedata;Awodutireetal.[14]discussednumerousstatisticalmeasuresofL-IIgeneralizedhalflogisticand explored itsapplicationinsportsdata.Meanwhile,Badmusetal.[15]discussedweightedWeibullviaL-IIandexploredits applicationintextileengineeringdata,andOgundeetal.[16]extendedGumbeltype-IIviaexponentiated L-IIG classand provideditsapplicationinthebiologydata.

Besides, thiswork isbeingmotivatedbythecapabilitiesofweighteddistributionsinthe literature.Weighteddistribu- tions are considered asa milestone inmodelingandprediction whenstandard modelsare lacking; thiscontributionwas initiatedby Fisher [17] andRao [18]who developedthe concept ofweighteddistributions, toadjust thebias inunequal probabilitysamplingandlater,researchersexploredthesize-biasedandarea-biasedasexceptionalcases.Tremendouswork wasdonebynotablestatisticianssuchasCox[19],whoinvestigatedthelength-biasedcaseofweighteddistributions;Patil and Ord [20] developed an inter-relation between size-biased samplingand weighteddistributions; Gove[21] discussed some results inforestry ofsize-biaseddistributions;Mirand Ahmad[22] expressedthe Geeta distributionin size-biased perspective;Dara[23]developedsize-biasedversionofvariouslifetimemodels;Saghiretal.[24]comprehensivelyreviewed the severalweightedmodelsalong withthecharacterizations;Arshadetal.[25] developedthetransmutedversion ofex- ponentiated moment (size-biased)Pareto distribution; Perveen and Ahmad [26] explored the size-biased version of the Weibulldistribution;Ahsan-ul-Haqetal.[27]developedtheMarshall-Olkinlength-biasedexponentialdistributionishighly recommended.

LetXbeanon-negativerandomvariable(r.v.)havingtheprobabilitydensityfunction(pdf),f(x),thenp(x)issaidtobe size-biaseddistributionifitspdfhastheform;

p

(

^x

)

^{=x f}

(

^x

)

E

(

^x

)

^{, x>0.}

Toachievetheaimofthisresearch,aflexibleandsimplegeneralizedfamilyofdistributionproposedbyCordeiroandde Castro[3]wasusedtoextendthemodifiedsize-biasedpowerfunction.ItisworthytonotethatKumaraswamy’distribution [28] wasproposed asan alternative modelagainst the Betadistribution becauseofits mathematical complexities; hence our choice forKumaraswamy-G (Kum-G) family of distributions with cdf andpdf ofthe Kum-G family with two shape parameters,arerespectivelydefinedby;

F

(

^x

)

⁼¹⁻

1−G^β

(

^x

)

γ

, x>0 (1.1)

and

f

(

^x

)

=

βγ

^g

(

^x

)

^Gβ−1

(

^x

)

1−G^β

(

^x

)

γ−1

, x>0 (1.2)

where

β

,

γ

>0areshapeparameters.

The Kum-Gfamilyofdistributions hasbeenadaptedinextendingsome standard distributions,severalnotableauthors have also demonstrated its algebraicsimplicity andmodeling potentials. The KumaraswamyPareto distribution by Bour- guignonetal.[29],KumaraswamyMarshall-Olkin-GfamilybyAlizadehetal.[30],KumaraswamyMarshall-OlkinFréchetdis- tributionbyAfifyetal.[31],KumaraswamyPowerfunctiondistributionbyAbdul-Moniem[32],exponentiatedKumaraswamy powerfunctiondistributionbyBursaandOzel[33],KumaraswamyexponentiatedFréchetdistributionbyMansouretal.[34],

andtheexponentiatedKumaraswamy-GfamilyofdistributionsbySilvaetal.[35],Bayesianinferencebyreferencepriorfor FréchetdistributionstudiedbyRamosetal.[36],aresomenotableexamples.

Thispaperis assembledon thefollowingsteps:theKumaraswamymodifiedsize-biasedLehmann type-II(Kum-MSBL- II)distributionisdevelopedandstudiedindetailinSection 2,somemathematicalandreliabilitypropertiesare presented in Section 3,the parameters ofthe Kum-MSBL-II distribution are estimated by the method of maximum likelihood and simulation isperformed in Section 4, application to real data is discussed in Section 5,and finally, the conclusionsare reportedinSection6.

ThenewModel

In thissection,the newmodeldevelopedis presentedintwo stages.Inthefirst stage, amodifiedsize-biasedversion ofLehmannType-II(MSBL-II)distributionisdeveloped(presentedinEq.(2.1)),andthenKumaraswamyeditionofMSBL-II (Kum–MSBL-II)distribution(presentedinEq.(2.2))isdevelopedatthesecondstage.Formoreconvenience,themathemat- icalconstructionofKum-MSBL-IIdistributionthatisaquitestraightforwardapproachtounderstandisdemonstrated.

Forthis, thewell-known modelidentified asL-II distributionis consideredandits cdfisgivenasfollows;W(^x)=1− (¹− x)^a, 0<x<1.

Atthatpoint,thesize-biasedversionisdevelopedandmodifiedaccordingtotheneedandthenewversionreferredto asthemodifiedsize-biasedLehmanntype-II(MSBL-II)distributionisderived.Thecdfisgivenasfollows:

G

(

^x

)

⁼¹⁻

_1−x

1+

α

^x

α

, 0<x<1 (2.1)

where

α

>0isashapeparameter.

Tothebestofourknowledge,MSBL-IIdistribution(NEW)hasnotbeendevelopedanddiscussedinthepast.Now,the developmentof theKumaraswamyedition of theMSBL-II distributionis obtainedsimply by insertingEq. (2.1)into(1.1).

Hence,theassociatedcdfoftheKum-MSBL-IIdistributionwiththreeshapeparameterstakestheform;

F

(

^x

)

⁼¹⁻

1−

1−

_1−x

1+

α

^x

α

β γ

,0<x<1 (2.2)

thePDFcorrespondingtoEq.(2.2)reducesto:

f

(

^x

)

⁼

αβγ ⁽

¹⁺

α ) (

^{1− x}

)

^α−1

(

¹⁺

α

^x

)

^α+1

1−

_1−x

1+

α

^x

α

β−1

1−

1−

_1−x

1+

α

^x

α

β γ−1

(2.3)

where

α

,

β

,

γ

>0arethethreeshapeparametersthatcontroltheshapeandtailbehaviorofKum-MSBL-IIdistribution.

However, thepresentresearchisaimedataddressingsome modelingdeficienciesofsome oftheexistingmodels.Con- sequently,theKum-MSBL-IIdistributionisanappropriatechoiceasitisflexibleandexhibitsabathtub-shapedhazardrate function.Themodeloffersmorerealisticandrationalizedresultsspecificallyoncomplexskewedsymmetricdataspecifically boundedonaninterval(0,1).Itprovidesaconsistentlybetterfitoveritscompetingmodels,asshownintheapplicationsec- tion.Furthermore,itofferssimpleandstraightforwardcdf,pdf,andlikelihoodfunctions.

MathematicalProperties

Linear combinationprovides amuchinformalapproachto discussthecdfandpdfthantheconventionalintegral com- putationwhendeterminingthemathematicalproperties.Forthis,thefollowingbinomialexpansionsisconsidered:

(

^1−z

)

^β=∞

i=0

(

⁻¹

)

ⁱ

β

i

zⁱ,

|

^z

|

^<1

FromEq.(2.2),InfinitelinearcombinationsofCDFisgivenasfollows:

F

(

^x

)

= ∞ i,j,k,l=0

(

−1

)

^i+j+k

α

^l

γ

i

β

ⁱ

j

α

^j

k

−

α

^j

l

x^l+k (3.1)

F

(

^x

)

^{= ∞}

i,j,k,l=0

η

i jkl

(

⁻¹

)

^i+j+k

α

^{lxl+k (3.2)}

where

η

i jkl=(

γ

i)(

β

ⁱ

j)(

α

^j

k)(⁻

α

^j

l ),

α

,

β

,

γ

>0

FromEq.(2.3),mixturerepresentationofPDFisgivenasfollows:

f

(

^x

)

⁼

αβγ (

¹⁺

α )

^∞

i,j=0

(

−1

)

^i+j

γ

⁻¹

i

β

ⁱ⁺

β

⁻¹

j

(

^1−x

)

^αj+α−1

(

¹⁺

α

^x

)

^{αj+α+1 (3.3)}

PresentationofEq.(3.3)willbequitehelpfulintheforthcomingcomputationofr-thordinarymoment

f

(

^x

)

⁼

αβγ (

¹⁺

α )

^∞

i,j=0

ζ

^{i j}

⁽

^1−x

⁾

^αj+α−1

(

¹⁺

α

^x

)

^{αj+α+1 (3.4)}

where

ζ

i j=(−1)^i+j(

γ

−1

i )(

β

ⁱ+

β

−1

j )

f

(

^x

)

⁼

αβγ (

¹⁺

α )

^∞

i,j,k,l=0

⎛

⎜ ⎜

⎝

(

−1

)

^i+j+k

α

^l

γ

⁻¹

i

β

ⁱ⁺

β

⁻¹

j

×

α

^j+j−1 k

−

α

^j−j−1 l

⎞

⎟ ⎟

⎠

^xl+k

f

(

^x

)

⁼

αβγ (

¹⁺

α )

^∞

i,j,k,l=0

φ

i jkl

(

−1

)

^i+j+k

α

^lxl+k

(3.5)

where

φ

i jkl=(

γ

−1

i )(

β

ⁱ+

β

−1

j )(

α

^j+j−1

k )(⁻

α

^j−j−1

l )

ReliabilityCharacteristics

Oneofthecriticalrolesofprobabilitydistributioninreliabilityengineeringistoanalyzeandpredictacomponent’slife.

For this, numerous reliability measures of the Kum-SBL-II distribution are discussed here. One may explain the reliabil- ity/survivalfunctionastheprobabilityofacomponentthatsurvivestillthetimex.

AnalyticallyitiswrittenasS(^x)=1−F(^x)^. ThesurvivalfunctionofXisgivenby

S

(

^x

)

⁼

1−

1−

_1−x

1+

α

^x

α

β γ

(3.6)

In reliabilitytheory,one ofthe significantlycontributedfunctionsconsidered asa failure ratefunction, orhazard rate functionandsometimescalledtheforceofmortalityisusedtomeasurethefailurerateofacomponentinaparticulartime x.Thefailureratefunctionismathematicallyexpressedash(^x)= f(^x)/R(^x)^.

ThehazardratefunctionofXisgivenby:

h

(

^x

)

⁼

αβγ (

¹+

α ) (

¹− x

)

^α−1

1−

_1−x

1+α^x

α

β−1

(

¹⁺

α

^x

)

^α+1

1−

1−

_1−x

1+α^x

α

β

^(3.7)

Further, several notable reliability measures may be derived for X such as reverse hazard rate function by hr(^x)= f(^x)/F(^x),MillsratiobyM(^x)=R(^x)/f(^x),andOddfunctionbyO(^x)=F(^x)/R(^x).

WemayobtainthelinearexpressionsforreliabilitycharacteristicsbyfollowingEqs.(3.2)and(3.5).Thelinearexpressions ofsurvivalandhazardratefunctionsofXaregivenasfollows,respectively.

S^∗

(

^x

)

=1− ∞ i,j,k,l=0

η

i jkl

(

−1

)

^i+j+k

α

^lxl+k

and

h^∗

(

^x

)

⁼

αβγ (

¹⁺

α )

^∞i,j,k,l=0

φ

i jkl

(

−1

)

^i+j+k

α

^lxl+k

1−_∞

i,j,k,l=0

η

i jkl

(

⁻¹

)

^i+j+k

α

^lxl+k

where

η

i jkl=(

γ

i)(

β

ⁱ

j)(

α

^j

k)(⁻

α

^j

l ),

α

,

β

,

γ

>0,and

φ

i jkl=

γ

−1 i

β

ⁱ+

β

−1 j

α

^j+j−1 k

−

α

^j−j−1 l

Shape

Differentplotsofprobabilitydensity(pdf)andhazardratefunctions(HRF)ofXaredisplayedinFigs.1and2,fordiffer- entmodelparameters.Fig.1presentssome possibleshapesofPDFincludingupside-down,left-skewed,right-skewed,and symmetric,andFig.2demonstratesthepossibleshapesoftheHRFwhichincludeUshape,bathtubshape,andincreasing.

Fig. 1. Density Plots of Kum-MSBL-II distribution. Black( α= 1 . 5 , β= 7 . 2 , γ= 1 . 6 ), Blue( α= 1 . 1 , β= 5 . 6 , γ= 1 . 1 ), Red( α= 1 . 5 , β= 5 . 1 , γ= 1 . 3 ), Hot- pink ( α= 1 . 7 , β= 4 . 1 , γ= 1 . 2 ), Green ( α= 1 . 6 , β= 9 . 8 , γ= 0 . 8 ), and Yellow ( α= 1 . 9 , β= 1 . 9 , γ= 0 . 7 ) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) .

Fig. 2. Hazard rate function plot of Kum-MSBL-II distribution. Black( α^{= 2 . 3 ,} β^{= 0 . 1 ,} γ^{= 0 . 1 ), Blue(} α^{= 1 . 3 ,} β^{= 1 . 3 ,} γ^{= 0 . 3 ), Red(} α^{= 1 . 5 ,} β^{= 2 . 4 ,} γ⁼ 0 . 09 ), Hot-pink ( α= 1 . 7 , β= 3 . 5 , γ= 0 . 4 ), Green ( α= 1 . 9 , β= 4 . 6 , γ= 0 . 1 ), and Magenta ( α= 2 . 1 , β= 5 . 7 , γ= 0 . 4 ) (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.).

Quantiles,skewness,kurtosisandmeandeviation

Theq^thquantilefunction(QF)ofX∼Kum–MSBL–II(x;

α

,

β

,

γ

^{)isobtainedbyinvertingtheCDFinEq.(2.2)as;}

xq= 1−

1−

1−

(

^1−q

)

^1/γ

1/β

1/α 1+

α

1−

1−

(

¹−q

)

^1/γ

¹/β

1/α (3.8)

Henceforth, it is assumed that the cdf in Eq.(2.2) follows the uniform distribution u= U (0, 1) to generaterandom numbers.

Further,theskewnessandkurtosisofXcanbecalculatedusingthefollowingtwousefulmeasuresintroducedbyBowley andMoors,respectively.

SB=Q

(

³⁴

)

^+Q

(

¹⁴

)

^−2Q

(

¹²

)

Q

(

³⁴

)

^−Q

(

¹⁴

)

^{, and KM=}

Q

(

³⁸

)

^−Q

(

¹⁸

)

^−Q

(

⁵⁸

)

^+Q

(

⁷⁸

)

Q

(

⁶⁸

)

^−Q

(

²⁸

)

Thelastexpressionsofthedescriptivemeasuresarebasedonquartilesandoctiles.Moreover,thesemeasuresarealmost lessreactivetooutliersandworkmoreeffectivelyforthedistributionshavingadeficiencyinmoments.

Furthermore,thequartiledeviationofXisobtainedby;

QD_B=Q

(

³⁴

)

^−Q

(

¹⁴

)

2

Fig. 3. The fitted density plot for SAR data.

The Bowley[37]coefficientofMoors[38]coefficient ofkurtosisplotsispresentedintheFiguresinthesupplementary material,thisillustratesthepositivebehaviorforboththecases,aspervariouschoicesofthemodelparameters.Hereafter, thesharpdecreaseinthequartiledeviationofXisobservedintheFigureinthesupplementarymaterial.

MomentsanditsAssociatedMeasures

Moments havea remarkablerolein distributiontheory,itis frequentlyusedto discussthevariouscharacteristics and importantfeaturesofaprobabilitydistributioncomprisingmean,variance,skewness,andkurtosis.

Ther-thordinarymomentofXisgivenby;

μ

^/r=E ∞ i,j,l=0

Di jl

(

−1

)

^i+j

α

^lB

(

^r+l+1,

α

^j+

α )

, (3.9)

whereB(x;

α

,

β

⁾=∫^x

0

t^α(¹−t)^{β−1dtisthebetafunctionwithE}=

αβγ

(¹+

α

)^,and

Di jl=

γ

−1 i

α

+

α

^j−1 j

−

( α

+

α

^j+1

)

l

The derived expression in Eq.(3.9) may servea supportive andusefulrole indeveloping severalstatisticalmeasures.

For instance: to deducethe mean ofX,set r = 1in Eq. (3.9). Forthe higher-order ordinary moments ofX approximat- ing to 2^nd, 3^rd, and 4^th, and higher, these moments can be formulated by setting r = 2, 3, and 4 in Eq. (3.9) respec- tively. Further, to discussthe variability in X, the Fisher index(F. I = (Var(^X)/E(^X)^{)) may play a significant role in this} scenario.ToderivethenegativemomentsofXsimplysubstituterwith–winEq.(3.9).Onemayperhapsfurtherdetermine the well-establishedstatistics such asskewness (

τ

1=

μ

²3/

μ

³2), andkurtosis (

τ

2=

μ

4/

μ

²2), ofX by integrating Eq.(3.9). A well-established relationshipbetween thecentral moments (

μ

^s) ^{andcumulants(Ks) ofX mayeasily be defined byordi-} narymomentsby

μ

^s= ^s

k=0(_k^s)(−1)^k(

μ

^/1)^s

μ

^/_s−k^{.Hence,thefirstfourcumulantscanbecalculatedbyK}1=

μ

^/1,K₂=

μ

^/2−

μ

^/1², K₃=

μ

^/3−3

μ

^/2

μ

^/1+2

μ

^/1³,andK₄=

μ

^/4−4

μ

^/3

μ

^/1−3

μ

^/2²+12

μ

^/2

μ

^/1²−6

μ

^/1⁴.

ResidualandReversedResidualLifeFunctions

Theresiduallifefunction/conditionalsurvivorfunctionofrandomvariableXR(^t)=X−t/X>tistheprobabilitythata componentwhoselifesaysx,survivesinthedifferentintervalatt≥0.Analyticallyitcanbewrittenas;

S_R(t)

(

^x

)

^=S

(

^x+t

)

S

(

^t

)

ResiduallifefunctionofX

SR(^t)

(

^x

)

⁼

1−

1−

1−(^x+t) 1+α(^x+t)

α

β

γ

1−

1−

_1−t

1+α^t

_α

β

γ , x>0 (3.10)

withassociatedcdfaregivenasfollows;

FR(^t)

(

^x

)

⁼¹⁻

1−

1−

_1−(x+t)

1+α(^x+t)

α

β

γ

1−

1−

_1−t

1+α^t

α

β

γ (3.11)

MeanresiduallifefunctionofXisgivenby;

E

(

^SR

(

^x

) )

= 1 S

(

^t

)

μ

^/1−∫^t

0

x f

(

^x

)

^dx

−t, t≥0

Further,thereverseresiduallifecanbedefinedasR¯(^t)=t−X/X≤t S_R¯(t)

(

^x

)

=F

(

^t−x

)

F

(

^t

)

^{, t≥0}

ReverseresiduallifefunctionofX

SR¯(^t)

(

^x

)

⁼

1−

1−

1−(^t−x) 1+α(^t−x)

α

β

γ

1−

1−

_1−t

1+α^t

α

β

_γ ^(3.12)

withassociatedcdfaregivenasfollows;

F_R¯(t)

(

^x

)

⁼¹⁻

1−

1−

_1−(t−x)

1+α(t−x)

α

β

γ

1−

1−

_1−t

1+α^t

α

β

γ (3.13)

Meanreversedresiduallifefunction/Meanwaitingtimeisgivenby;

E

S_R¯

(

^x

)

=t− 1 F

(

^t

)

∫t 0

x f

(

^x

)

^{dx, t≥0}

Furthermore,thestrongmeaninactivitytimeofXmaybeobtainedbyfollowing M

(

^t

)

^=t2− _f

(

^1t

)

∫t 0

x²f

(

^x

)

^dxfort≥0

where r(^t)=∫^t

0

x^rf(^x)^dx, is the r–th incomplete moments and it is given by r(^t)=

E ^∞

i,j,l=0

D_{i jl}(−1)^i+j

α

^lBt(^r+l+1,

α

^j+

α

). One may perhaps determine the mean residual and reversed residual life functionsby placingr= 1,andforstrongmeaninactivitytime, placer= 2,in r(^t)respectively, withE=

αβγ

(¹+

α

) andD_{i jl}=(

γ

−1

i )(

α

+

α

^j−1

j )(⁻(

α

+

α

^j+1)

l )

Stress-strengthreliability

Let X₁ and X₂ be the strength and stress of a component respectively, followed by the same uni-variatedistribution family.TheinadequateandgoodworkingofacomponentdependonifX2>X1 andX2<X1,respectively.Stress– strength reliabilitycanbewrittenasR=P(^X2<X₁).

Theorem1. LetX₁∼Kum–MSBL–II(x;

α

,

β

,

γ

1) andX₂∼Kum–MSBL–II(x;

α

,

β

,

γ

2)betwoindependentrandomvariables followedbytheKum–MSBL–IIdistribution.ThenthereliabilityRofXtakestheform;

R= 2

γ

1+

γ

2

γ

¹⁺

γ

²

Proof. Reliability(R)isdefinedas R=∫f1

(

^x

)

^F2

(

^x

)

^dx.

FromEqs.(2.2)and(2.3),RofXcanbewrittenas

R=∫¹

0

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎝

αβγ

¹

⁽

¹⁺

α ) (

^{1− x}

)

^α−1

(

¹+

α

^x

)

^α+1

1−

_1−x

1+α^x

α

β−1

× 1−

1−

_1−x

1+α^x

α

β

γ1−1

× 1−

1−

1−

_1−x

1+α^x

α

β

γ2

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎠

dx

Hence,afterapplyingsomealgebraonthepreviousexpression,thestress-strengthreliabilityofXhastheform:

R= 2

γ

1+

γ

2

γ

1+

γ

2

asRispresumedafunctionof

γ

1.

Entropy

When a system isquantified by the disorderedness, randomness, diversity, or uncertainty, ingeneral, it is known as entropy.

Rényi[39]entropyofXisdescribedby;

I_δ

(

^X

)

= 1 1−

δ

^log

∫1 0

f^δ

(

^x

)

dx,

δ

>0and

δ

=1

f^δ

(

^x

)

⁼

⎛

⎜ ⎜

⎝

( αβγ (

¹+

α ) )

^δ

(¹−x)^α−1 (¹+α^x)^α+1

δ

1−

_1−x

1+α^x

α

β−1

_δ

×

1−

1−

_1−x

1+α^x

α

β

δ(γ−1)

⎞

⎟ ⎟

⎠

Thelastequationcanbewrittenbyfollowingthebinomialexpansionanditgivenasfollows;

f^δ

(

^x

)

⁼

⎛

⎜ ⎜

⎝

( αβγ (

¹⁺

α ) )

^{δ ∞}

i,j,k,l=0

(

−1

)

^i+j+k

α

^l

δ ( γ

⁻¹

)

i

β

ⁱ⁺

δ ( β

⁻¹

)

j

×

α

^j

k

δ ( α

⁺¹

)

⁻

α

^j

l

x^l+k

(

^1−x

)

^δ(α−1)

⎞

⎟ ⎟

⎠

^.

Byintegratingtheaboveequation,weefficaciouslydevelopamostsimplifiedversionoftheRényientropyofX,anditis givenasfollows:

I_δ

(

^X

)

=

1

1−δlog

( αβγ (

¹+

α ) )

^{δ ∞}

i,j,k,l=0

∇

i jkl

(

−1

)

^i+j+k

α

^l

B

(

^l+k+1,

δ ( α

⁻¹

)

⁺¹

)

^(3.14)

where

∇

i jkl=(

δ

(

γ

−1)

i )(

β

ⁱ+

δ

(

β

−1) j )(

α

^j

k)(

δ

(

α

+1)−

α

^j

l )

Orderstatistics

Inreliabilityanalysisandlifetestingofacomponentinqualitycontrol,orderstatistics(OS)anditsmomentsareconsid- eredworthymeasures.LetX₁, X₂, ...,XnbearandomsampleofsizendistributedaccordingtotheKum–MSBL–IIdistribu- tion andX₍₁₎<X₍₂₎< ...<X_(n)bethecorresponding orderstatistics.Therandomvariables X_(i), X₍₁₎,and X_(n)be thei-th, minimum,andmaximumorderstatisticsofX.

ThepdfofX_(i)isgivenby;

f_(i)

(

^x

)

⁼_B

(

^{i, n−1i+1}

)

^!

(

^F

(

^x

) )

ⁱ⁻¹

(

^1−F

(

^x

) )

^n−if

(

^x

)

^{, i=1,2,3,...,n.}

ByincorporatingEqs.(2.2)and(2.3),thepdfofX_(i)takestheform

f_(i:n)

(

^x

)

⁼

αβγ

B

(

^{i, n−i+1}

)

^!

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎝

(¹+α)(¹−x)^α−1 (¹+α^x)^α+1

1−

_1−x

1+α^x

α

β−1

× 1−

1−

_1−x

1+α^x

α

β

γ−1

×

1−

1−

1−

_1−x

1+α^x

α

β

_γ

ⁱ−1

×

1−

1−

_1−x

1+α^x

α

β

γ

ⁿ−i

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎠

Aftersomesimplifications,thelastequationreducesto

f₍i)

(

^x

)

= 1

B

(

i, n−i+1

)

^!

αβγ ( α

+1

)

^∞

j,k,l,m=0

τ

jklm

(

−1

)

^j+k+m

α

^lxl

(

¹− x

)

^αk+α−1

Theabove equationisquitehelpfulincomputingofw-thmomentorderstatisticsofX.Furthermore,theminimumand maximumorderstatisticsofX followsdirectlytheequationabovewithi=1andi=n,respectively.

Thew-thmomentorderstatistics,E(^X_OS^w)^,ofXis

E

(

^XOS^w

)

= 1

B

(

ⁱ, n−i+1

)

^!

αβγ ( α

+1

)

^∞

j,k,l,m=0

τ

jklm

(

−1

)

^j+k+m

α

^{l B}

(

^w+l+1,

α

^k+

α )

where

τ

jklm=(

γ

(¹+n−i+j)−1

j )(

β

ⁱ+

β

−1

k )(⁻

α

^k−

α

−1 l )(ⁱ⁻_m¹) ThecdfofX_(i)isgivenby

F₍i:n)

(

^x

)

⁼ n

r=i

n r

(

^F

(

^x

) )

^r

(

^1−F

(

^x

) )

^{n−r, i=1,2,3,...,n.}

F_(i:n)

(

^x

)

⁼

⎛

⎜ ⎜

⎝

n r=i

n r

1−

1−

1−

_1−x

1+α^x

α

β

γ

^r

×

1−

1−

_1−x

1+α^x

_α

β

γ

ⁿ−r

⎞

⎟ ⎟

⎠

Estimation

In this section, the X’s parameters were estimated by following the method ofmaximum likelihood, as this method providesfullinformationabouttheunknownmodelparameter.LetX₁,X₂,X₃,...,XnbearandomsampleofsizenfromX,then thelikelihoodfunctionL(

ζ

)= ⁿ

i=1

f(^xi;

α

,

β

,

γ

)^{ofXisgivenby:}

L

( ζ )

⁼

( αβγ (

¹⁺

α ) )

ⁿ n

i=1

(

^{1− xi}

)

^α−1

(

¹⁺

α

^xi

)

^α+1

⎛

⎜ ⎝

1−

_1−xi

1+α^xi

α

β−1

× 1−

1−

_1−xi

1+α^xi

α

β

γ−1

⎞

⎟ ⎠

Thelog-likelihoodfunction,l(

ϑ

),reducesto

l

( ζ )

⁼

⎛

⎜ ⎜

⎜ ⎜

⎜ ⎝

n

(

^log

α

^+log

β

^+log

γ

^+log

(

¹⁺

α ) )

⁺

( α

⁻¹

)

ⁿ

i=1

log

(

^1−xi

)

⁻

( α

⁺¹

)

ⁿ

i=1

log

(

¹⁺

α

^xi

)

⁺

( β

⁻¹

)

ⁿ

i=1

log

1−

_1−xi

1+α^xi

α

+

( γ

⁻¹

)

ⁿ

i=1

log

1−

1−

_1−xi

1+α^xi

α

β

⎞

⎟ ⎟

⎟ ⎟

⎟ ⎠

(3.15)

Partialderivativesof(3.15)yield