Exponentiated Lomax Geometric Distribution: Properties and Applications

Transcription

1 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos Amal Solma Hassa Mathematcal Statstcs Cao Uvesty Isttute of Statstcal Studes ad Reseach Egypt Mawa Abd-Allah Mathematcal Statstcs Cao Uvesty Isttute of Statstcal Studes ad Reseach Egypt Abstact I ths pape a ew fou-paamete lfetme dstbuto called the expoetated Lomax geometc (ELG) s toduced. The ew lfetme dstbuto cotas the Lomax geometc ad expoetated Paeto geometc as ew sub-models. Explct algebac fomulas of pobablty desty fucto suvval ad hazad fuctos ae deved. Vaous stuctual popetes of the ew model ae deved cludg; quatle fucto Re'y etopy momets pobablty weghted momets ode statstc Loez ad Bofeo cuves. The estmato of the model paametes s pefomed by maxmum lkelhood method ad feece fo a lage sample s dscussed. The flexblty ad potetalty of the ew model compaso wth some othe dstbutos ae show va a applcato to a eal data set. We hope that the ew model wll be a adequate model fo applcatos vaous studes. Keywods: Expoetated Lomax dstbuto Geometc dstbuto Maxmum lkelhood estmato.. Itoducto Lomax (954) toduced a mpotat ad wdely used lfetme model the so-called Lomax dstbuto ad t used fo stochastc modelg of deceasg falue ate.it has bee appled studes of come sze of ctes wealth equalty egeeg queug theoy ad bologcal aalyss. Studes about Lomax dstbuto have bee dscussed by seveal authos. Some popetes ad momets fo the Lomax dstbuto have bee dscussed by Balaksha ad Ahsaullah (994). The dscete Posso-Lomax dstbuto has bee povded by Al-Awadh ad Ghtay (). The Bayesa ad o-bayesa estmato of the elablty has bee studed by Abd-Elfattah et al. (7). Ghtay et al. (7) toduced Mashall-Olk exteded Lomax. Hassa ad Al-Ghamd (9) detemed the optmal tmes of chagg stess level fo smple stess plas ude a cumulatve exposue model fo the Lomax dstbuto. Hassa et al. (6) dscussed the optmal tmes of chagg stess level fo k-level step stess acceleated lfe tests based o adaptve type-ii pogessve hybd cesog wth poduct's lfe tme followg Lomax dstbuto. Some extesos of the Lomax dstbuto have bee costucted by seveal authos. Abdul-Moem ad Abdel-Hameed () toduced the expoetated Lomax (EL) by addg a ew shape paamete to the Lomax dstbuto. Lemote ad Codeo (3) vestgated beta Lomax Kumaaswamy Lomax ad McDoald Lomax dstbutos. The gamma-lomax dstbuto has bee suggested by Codeo et al. (3). Fve-

2 Amal Solma Hassa Mawa Abd-Allah paamete beta Lomax dstbuto has bee vestgated by Rajab et al. (3). The Webull Lomax ad Gumbel-Lomax dstbutos have bee toduced by Tah et al. (5a) ad (5 b). The cumulatve dstbuto fucto (cdf) of Lomax dstbuto wth shape paamete ad scale paamete s gve by G ( x ; ) ( x ) x. () The pobablty desty fucto (pdf) of Lomax dstbuto s as follows ( ) g ( x ; ) ( x ) x. () The expoetated Lomax has bee toduced by Abdul-Moem ad Abdel-Hameed () by addg shape paamete to the cdf (). The cdf of the EL takes the followg fom G ( x ; ) [ ( x ) ] x. (3) The coespodg pdf s as follows: ( ) g ( x ; ) ( x ) [ ( x ) ] x. (4) Also a dscete adom vaable N s a membe of a zeo-tucated geometc adom vaable depedet of X ' s wth pobablty mass fucto (pmf) gve by: P( N ; p) ( p) p 3... p (). (5) Recetly vaous compoudg pobablty dstbutos have bee poposed by seveal authos fo modelg lfetme data seveal aeas. Adamds ad Loukas (998) poposed a two-paamete expoetal-geometc dstbuto. I a smla mae some examples as the Webull geometc expoetated expoetal geometc ad Ldley geometc dstbutos have bee suggested by Baeto-Souza et al. () Rezae et al. () ad Zakezadeh ad Mahmoud () espectvely. Recetly expoetated Lomax Posso Lomax-logathm ad exteded Lomax Posso dstbutos have bee gve espectvely by Ramos et al. (3) Al-Zaha ad Sago (4) ad Al-Zaha (5). I ths atcle a ew compoud dstbuto s toduced by mxg EL ad geometc dstbutos. We hope that ths ew model wll seve as a sutable model seveal aeas. The desty cumulatve suvval ad hazad ate fuctos of the ew model ae obtaed Secto. Secto 3 devotes wth some mathematcal popetes such as; quatle pobablty weghted momets etopy ad ode statstcs. Secto 4 dscusses the estmato of the ukow paametes by maxmum lkelhood method ad feece fo lage sample s peseted. I Secto 5 applcatos to eal data sets ae gve. Fally cocludg emakes ae outled Secto

3 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos. Costucto of the Dstbuto I ths secto followg the same appoach of Adamds ad Loukas (998) we toduce ad study the expoetated Lomax geometc dstbuto. The pobablty desty fucto dstbuto fucto elablty ad hazad ate fucto ae obtaed. Let X X... X N be a adom sample of sze N fom the Expoetated Lomax dstbuto wth cdf(3) ad N be a zeo-tucated geometc adom vaable depedet of X ' s wth pmf (5). Defe X () m{ X X... X N } the the codtoal pobablty desty fucto of X () N s obtaed as follows: f x x x x X () ( ) ( ; ) ( ) ( ( ) ) [ ( ( ) ) ]. N The jot pobablty desty fucto of X adn s obtaed as follows: ( ) f ( x; ) ( p) p ( x ) ( ( x ) ) [ ( ( x ) ) ]. X () N The pobablty desty of expoetated Lomax geometc sdefed as the magal desty of X.e. ( ) ( p) ( x) ( ( x) ) f ( x; ) x (6) { p[ [ ( x) ] ]} whee p ad ( p) s the set of paametes. A adom vaable X wth desty fucto (6) shall be deoted by X ELG ( x ; ). Futhemoe the cumulatve dstbuto fucto of ELG coespodg to (6) s deved as follows: [ ( x ) ] F ( x ; ) x. (7) p[ [ ( x) ] ] Based o cdf (7) some specal dstbutos asefom the ELG dstbuto as follows:. As p the expoetated Lomax (EL) s a lmtg case of the ELG dstbuto.. Fo ad whe p the expoetated Lomax geometc educes to Lomax dstbuto. p ad 3. Whe the expoetated Lomax geometc educes to expoetated Paeto (see Gupta et al. (998)). 4. The expoetated Lomax geometc educes to Lomax geometc whe. 5. The expoetated Lomax geometc educes to expoetated Paeto geometc whe λ=. 547

4 desty fucto desty fucto Amal Solma Hassa Mawa Abd-Allah Fgue () gves some possble shapes of the desty (6) fo some selected paamete values..5 =.5=p=.5 = ==.5p=. = =3=.5p=.3 = =.5=.5p=.5 = ==3.5p=.8 = x =.=.p=.5 =. =6=5p=. =. =.8=p=.3 =3 =.6=4p=.9 =5 =.5=8p=.7 = Fgue : Plots of the ELG destes fucto fo some paamete values x I addto the elablty ad hazad ate fuctos of ELG dstbuto ae as follows: ( p) [ ( x) ] R( x; ) p[ [ ( x) ] ] ad ( ) ( x) ( ( x) ) ( x ; ) { p[ [ ( x ) ] ]}[ [ ( x ) ] ] 548

5 hazad fucto hazad fucto Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos whee ad p. Fgue () llustates the gaphcal behavo of hazad ate fucto fo ELG fo some selected values of paametes..5 =.5=p=.5 = ==.5p=. = =3=.5p=.3= =.5=.5p=.5 = ==3.5p=.8 = x.5 =.=.p=.5 =. =6=5p=. =. =.8=p=.3 =3 =.6=4p=.9 =5 =.5=8p=.7 = x Fgue. : Plots of the ELGhazad fucto fo some paamete values Fgue shows that the shapes of the hazad ate ae ceasg deceasg ad costat at some selected values of paametes.. Some Statstcal Popetes I ths secto some of statstcal popetes of the ELG dstbuto cludg expaso fo pdf (6) ad cdf (7) quatle fucto th momet ad pobablty weghted momets ae deved. Futhemoe Re'y etopy dstbuto of ode statstcs Bofeo ad Loez cuves ae povded. 549

6 Amal Solma Hassa Mawa Abd-Allah 3.. Usefuel expasos I ths subsecto two useful expasos fo the pdf (6) ad cdf (7) ae deved. We show that the pdf of ELG ca be expessed as lea combatos of EL dstbutos. Also the two expasos ae used to deteme some mathematcal popetes of the ELG dstbuto. Fstly the pdf (6) of ELG ca be expessed as lea combatos of EL dstbutos. Usg the followg sees expasos j k ( k j ) z ( z ) z k. (8) ( k) j! j The the pdf (6) ca be wtte as follows j ( ) j ( ; ) ( ) ( ) ( ) [ ( ) ] { [ ( ) ] }. f x j p p x x x j The by usg the bomal expaso fo the pevous pdf the t ca be wtte as k j j ( ) ( k ) f ( x ; ) ( ) ( j ) p ( p) ( x ) [ ( x ) ] (9) jk k whee pdf (9) leads to the followg fte lea combato f ( x ; ) W j k h ( k ) ( x ; ) () jk whee k j j W jk ( ) p ( p) k W jk jk ad h ( ; ) ( ) x deotes the pdf of EL wth paametes ( k ) ad. k Secodly; a expaso fo [ Fx ( ; )] s s deved fom cdf (7) though the expasos defed (8) as follows s h s ( s ) [ F ( x ; )] p [ ( x ) ]. () h h s 3. Quatle measues The quatle fucto of ELG dstbuto deoted by Q( u) F ( u) of X s deved as follows: u( p) Qu ( ) () ( up) 55

7 Skewess Skewess Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos whee u s a ufom adom vaable o the ut teval (). I patcula the meda of the ELG dstbuto deoted by m s obtaed by substtutgu.5 () as follows.5( p) m. (.5 p) The Bowleyskewess (see Keey ad Keepg (96)) based o quatles ca be calculated by 3 Q Q Q 4 4 B. 3 Q Q 4 4 Futhe the Moos kutoss (see Moos (988)) s defed as Q Q Q Q M 6 Q Q 8 8 whee Q(.) deotes the quatle fucto. Plots of the skewess ad kutoss fo some choces of the paamete as fucto of p ad fo some choces of the paamete p as fucto of ae show Fgues 3 ad 4. We ca detect fom these fgues that the skewess ad kutoss fo p deceases as α ceases fom.5 to.5. Also the skewess ad kutoss fo α deceases as p ceases fom. to =.5 =.5 = = (a) p (b) p=. p=.3 p=.5 p=.7 Fgue 3: Skewess of the ELG dstbuto. (a) As fucto of p fo some values of wth.5 ad (b) As fucto of fo some values of p wth.5 ad 55

8 Kutoss Kutoss Amal Solma Hassa Mawa Abd-Allah 3 =.5 =.5 = = p=. p=.3 p=.5 p= p (a) (b) Fgue 4: Kutoss of the ELG dstbuto.(a) As fucto of p fo some values of wth.5 ad (b) As fucto of fo some values of p wth.5 ad 3.3 Momets The momets of ay pobablty dstbuto ae ecessay ad mpotat ay statstcal aalyss especally appled wok. Some of the most mpotat featues ad chaactestcs of a dstbuto ca be studed though momets such as; dspeso skewess ad kutoss. A explct expesso fo the th momet of ELG dstbuto about the og s obtaed by usg pdf () as follows: ' j k ( k ) jk W x h ( x ; ) dx ' ( ) ( k) j k jk W ( k ) x ( x ) [ ( x ) ] dx. Let y ( x) ad usg bomal expaso the the above tegal s educed to W y y dy ( ) ' ( k ) m m ( k) j k ( ) [ ] j k m m m ( ) ( k ) ( m ) ( k ) m ' W j k j k m 55

9 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos whee (..) stads fo beta fucto. Hece the th momet fo ELG dstbuto takes the followg fom: whee D ' j k m D j k m ( ) W. m m j k m k j ( k ) ( ( m )) ( ( k )) (3) ( ( m ) ( k )) Futhemoe t s easy to show that the momet geeatg M X () t fucto ca be wtte as follows: t M X ( t) '! whee ' s the th momet. The by usg (3) the momet geeatg fucto of ELG dstbuto ca be wtte as follows: t ( k ) ( ( m )) ( ( k )) M X ( t ) D j k m. j k m! ( ( m ) ( k )) 3.4 The pobablty weghted momets Pobablty weghted momets (PWMs) wee devsed by Geewood et al. (979) pmaly as a ad to estmate the paametes of dstbutos that ae aalytcally expessble oly vese fom. PWMs ae the expectatos of ceta fuctos of a adom vaable defed whe the oday momets of the adom vaable exst. The PWMs of a adom vaable X ae fomally defed by s s s E [ X F( x ) ] x f( x )(F( x )) dx. (4) The PWMs of ELG dstbuto s obtaed by substutg pdf () ad cdf () (4) as follows: x k x x dx s j k m j k h ( ) ( k s h) ( ) ( ) [ ( ) ] whee s. h s h j k h pwj k 553

10 Amal Solma Hassa Mawa Abd-Allah Let y ( x) ad usg bomal expaso the s takes the followg fom: m m ( ) ( k ) ( k s h) s j k h [ ] j k h m z z dz The m ( ) ( k ) ( m ) s j k h ( k s h ) j k h m Hece the PWMs of ELG dstbuto ca be expessed as: ( m) ( k ) ( ) ( ( k s h )) s j k m h ( ) j k h m m ( ( m s h ) ) j k m h k m h j s j ( ) p ( p). k h s 3.5 Re'y etopy The etopy of a adom vaable Xs a measue of ucetaty vaato. If X s a adom vaable whch dstbuted as ELG the the Re'y etopy fo ad s defed by: I ( x ) log f ( x ; ) dx. ( ) b The by usg pdf (6) the Re'y etopy of ELG ca be wtte as follows: Let ( ) ( p) ( x ) ( ( x ) ) b ( ) { p[ [ ( x) ] ]} I ( x ) log dx. ( ) ( ) ( x) ( ( x) ) IP ( p) dx. { p[ [ ( x) ] ]} By usg the sees expaso (8) the the above tegato IP ca be wtte as follows j j j ( ) ( ) IP ( ) p ( p) [ ( x ) ] ( x ) dx j Let y ( x) the the IP s educed to ) j j j p p IP ( ) ( ) ( ) ( ). j ( ) 554

11 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos Theefoe the Re'y etopy of ELG dstbuto s as follows ( ) ( ) I R ( x ) ( ) log b L j j ( ) ( ) whee j j j ( ) p ( p) L j. ( ) 3.6 Ode statstcs I ths subsecto a closed fom expesso fo the pdf of the th ode statstcs of the ELG dstbuto wll be deved. Let X X X be a smple adom sample fom ELG dstbuto ad let X: X:... X: deote the ode statstcs obtaed fom ths sample. Accodg to Davd (98) the pdf of th ode statstcs s as follows whee B (..) f : ( x ; ) F ( x ; F ( x ; f ( x ; ) B ( ) stads fo beta fucto. Usg the bomal sees expaso of F( x; the f : ( x; ) ca be wtte as: f : ( x ; ) ( ) F ( x ; ) f ( x ; ). B ( ) Substtutg cdf (7) the the pobablty desty fucto of th ode statstcs X: fom ELG dstbuto s deved as follows ( ( x) ) f : ( x; ) ( ) f ( x; ). (5) B( ) p[ [ ( x) ] ] Usg the powe sees (8) ad bomal expaso the (5) takes the followg fom j j f x p x f x k j ( k ) : ( ; ) ( ) [ ( ) ] ( ; ). B ( ) j k k The by substtutg pdf (6) usg the sees expaso (8) ad bomal expaso the pevous equatowe have j j f : ( x ; ) ( ) p [ ( x ) ] B ( ) j k k h m h ( ) ( m ) ( ) ( h ) p ( p) ( x ) [ ( x ) ] h m m k j ( k ) 555

12 Amal Solma Hassa Mawa Abd-Allah Theefoe the pdf of th ode statstcs of ELG ca be expessed as follows f ( x ; ) h ( x ; ) x (6) : h j k m ( m k ) j k m h whee k m ( ) ( h ) j j h h j h j k m ( p) p ( )( m k ) k m ad h ( ; ) ( ) x deotes the pdf of EL wth paametes ( m k ) ad. mk I patcula the pdf of the smallest ode statstc X: s obtaed by substtutg = (6) as follows f ( x ; ) h x : h j k m ( m k ) j k m h k m ( ) ( h ) j j h h j j k m h ( p) p ( m k ) k m ad h ( ; ) ( ) x deotes the pdf of EL wth paametes ( m k ) ad. mk Also the pdf of the lagest ode statstc X: s obtaed by substtutg = (6) as follows f ( x ; ) h ( x ; ) x : j k m h ( k m ) j k m h km ( ) h j j h h j j k m h ( p) p. ( k m ) k m ad aga h ( ; ) ( ) x deotes the pdf of EL wth paametes mk ( m k ) ad. 3.7 Bofeo ad Loez cuves Bofeo ad Loez cuves ae come equalty measues that ae also useful ad applcable to othe aeas cludg elablty demogaphy medce ad suace. The Bofeo cuve s calculated by the followg fom: x B F [ F ( x )] uf ( u) du F( x) The by usg pdf () the Bofeo cuve ca be expessed as follows: x ( ) ( k ) F [ ( )] j k ( ) ( ) [ ( ) ].(7) F( x) jk B F x W u k u u du 556

13 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos Let z ( u) the the tegated pat (7) ca be wtte as follows ( x ) ( k ) ( ) ( )[ ] k jk. jk I W z z dz Usg the sees expaso the the pevous tegeal s as follows j k h ( x ) h h ( k ) ( k ) h I W jk ( z z ) h h h ( ) h ( k ) k ( x ) ( x ) W jk. j k h h h h Fo smplcty put Q ( k ) ( k ) h the the Bofeo h j k h W j k cuve ca be wtte as follows h h ( x) ( x) Qj k h [ ( ) ] x j k h h h B F [ F ( x )]. [ p[ [ ( x ) ] ] W k j( k ( ( k )) ( ( k )) k j Also the Loez cuve s calculated by the followg fom z L[ Z ] xf ( x ) dx. The Loez cuve of ELG dstbuto takes the followg fom h h ( x) ( x) Q j k h j k h h h LZ [ ]. W k j( k ( ( k )) ( ( k )) k j 4. Paamete Estmato I ths secto estmato of the ELG model paametes; s obtaed by usg maxmum lkelhood method of estmato. 557

14 Amal Solma Hassa Mawa Abd-Allah Let X X... X be a smple adom sample fom the ELG dstbuto wth set of paametes ( p). The log-lkelhood fucto deoted by ll based o the obseved adom sample of sze fom desty (6) s gve by: ll ( x ; ) l p l l l l x whee S x S p ps. l l espect to the ukow paametes ae gve by: ll ps l( S ) ls p ps The patal devatves of the log-lkelhood fucto wth ( S ) lx ps S lx l x ll S p ps x p S x x ll x x x S p ps ll S. p p p ps The maxmum lkelhood estmatos of the model paametes ae detemed by solvg ll ll ll ll umecally the o-lea equatos ad p smultaeously. Fo teval estmato ad hypothess tests o the model paametes the obseved Fshe s fomato matx must be obtaed. The 4 4 ut obseved fomato matx I ( ) s gve as follows whee I I I I I p I I I I p I( ) I I I I p I p I p I p I pp ll. j j The etes of Fshe s fomato matx fo ELG ae gve the Appedx. 558

15 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos The appoxmate ( )% two sded cofdece tevals fo p ae espectvely gve by: Hee ˆ Z va( ˆ ) ˆ Z va( ˆ ) ˆ Z va( ˆ ) pˆ Z va( pˆ) th Z s the uppe deote the dagoal elemets of 5. Applcatos pecetle of the stadad omal dstbuto ad va (.) s I ( ) coespodg to the model paametes. I ths secto the flexblty of ELG model s examed by compag t wth some othe dstbutos. Two eal data sets ae used to show that ELG dstbuto ca be appled pactce ad ca be a bette model tha some othes. Fo the two sets of data; the ELG s compaed to Lomax (L) expoetated Lomax kumaaswamy Lomax (KL) Webull Lomax (WL) ad expoetated Paeto (EP) dstbutos. The desty fuctos fo kumaaswamy Lomax Webull Lomax ad expoetated Paeto dstbutos ae as follows; f x a b ab x x a x b b b KL ( ; ) ( ) [ ( ) ] exp[ [( ) ] ] ( ) a b WL( ; ) ( ) [ ( ) ] [ [ ( ) ] ( ) EP ( ; ) ( ) [ ( ) ]. f x a b ab x x x f x x x The fst data set epesets 84 obsevatos of falue tmes ( hous) fo a patcula wd sheld model epoted by (Muthy et al. (4)): The maxmum lkelhood method s employed to obta the pot estmates of the model paametes. To compae the ftted models some selected measues ae appled. The selected measues clude; -log-lkelhoodfucto evaluated at the paamete estmates Akake fomato cteo (AIC) Bayesa fomato cteo (BIC) cosstet 559

16 Amal Solma Hassa Mawa Abd-Allah Akake fomato cteo (CAIC) Haa-Qu fomato cteo (HQIC) ad the Kolmogoov-Smov (k-s) statstc. The mathematcal fom of these measues s as follows p( p ) AIC p l L CAIC AIC p BIC k l( ) l L HQIC p l l( ) l L k s sup F ( y ) F ( y ) y whee k s the umbe of models paamete s the sample sze ad l L s the maxmzed value of the log- lkelhood fucto ude the ftted models. The bette model s coespodg to the lowe values of AIC CAIC BIC ad k-s statstcs. The esults fo the pevous measues to the metoed models ae lsted Table. Table: Measuemets fo all models based o the fst data set Models Statstcs -logl AIC CAIC BIC HQIC k-s ELG EL EP KL WL L The values Table dcate that the most ftted dstbuto to the data s ELG dstbuto compaed to othe dstbutos cosdeed hee (EL EP KL WL L). Plots of the estmated cumulatve ad estmated destes of the ftted models fo the fst set of data ae descbed below Fgue 5. Estmated destesof models fo the fst data set 56

17 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos Fgue 6. Estmated cumulatve destes fo the fst data set Aga fom Fgues 5 ad 6 we ca otce that the most ftted dstbuto compaed wth the othe models to the fst set of data s ELG. The secod data set cotas obsevatos o beakg stess of cabo fbes ( Gba) studed by Nchols ad Padgett (6). The secod set of data ae as follows: The same models (ELG EL EP KL WL L) ae ftted fo the secod set of data ad the values of the measuemets ae lsted Table. It s cleafom Table that the ELG s the most ftted dstbuto compaed wth the othe dstbutos fo fttg the secod set of data. 56

18 Destes Amal Solma Hassa Mawa Abd-Allah Table: Measuemets fo all models based o the secod data set Statstcs Models -logl AIC CAIC BIC HQIC k-s ELG EL EP KL WL L Futhemoe the gaphcal compaso coespodg to these fttedmodelsto cofom ou clam s llustated Fgues 7 ad Data ELG L EL KL WL EP. 5 data Fgue 7. Estmated destes of models fo the secod data Fgue 8. Estmated cumulatve destes fo the secod data set 56

19 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos Aga Fgues 7 ad 8show that ELG model s the best ftted model fo the secod data set. 6. Cocluso I the peset study we popose a ew dstbuto called expoetated Lomax geometc dstbuto. The subject dstbuto s deved by compoudg expoetated Lomax ad geometc dstbutos. The desty fucto of ELG ca be expessed as a mxtue of EL desty fuctos. The ELG dstbuto cludes the Lomax geometc ad expoetated Paeto geometc as ew dstbutos. Explct expessos fo momets pobablty weghted momets Bofeo ad Loez cuves ode statstcs ad R ey s etopy ae deved. The estmato of paametes alog wth the fomato matx s deved. Applcatos of the expoetated Lomax geometc dstbuto to ealdata show that the ew dstbuto ca be used qute effectvely to povde bette fts as compaed to Lomax expoetated Lomax Kumaaswamy Lomax Webull Lomax ad expoetated Paeto dstbutos. Refeeces. Abd-Elfattah A. M. Alaboud F. M. ad Alhaby A. H. (7). O sample sze estmato fo Lomax dstbuto Austala Joual of Basc ad Appled Sceces Abdul-Moem I. B. ad Abdel-Hameed H. F. (). O expoetated Lomax dstbuto Iteatoal Joual of Mathematcal Educato 33 (5) Adamds K. ad Loukas S. (998). A lfetme dstbuto wth deceasg falue ate Statstcs ad Pobablty Lettes Al-Awadh S. A. ad Ghtay M. E. (). Statstcal popetes of Posso- Lomax dstbuto ad ts applcato to epeated accdets data Joual of Appled Statstcal Scece (4) Al-Zaha B. (5). A exteded Posso-Lomax dstbuto Advaces Mathematcs: Scetfc Joual 4() Al-Zaha B. ad Sago H. (4). Statstcal aalyss of the Lomax- Logathmc dstbuto. Joual of Statstcal Computato ad Smulato Balaksha N. ad Ahsaullah M. (994). Relatos fo sgle ad poduct momets of ecod values fom Lomax dstbuto Sakhya B Baeto-Souza W. Moas A.L. ad Codeo G.M. (). The Webullgeometc dstbuto Joual of Statstcal Computato ad Smulato Codeo G. M. Otega E. M. M. ad Popovć B. V. (3). The gamma-lomax dstbuto. Joual of Statstcal Computato ad Smulato 85() Davd H. A. (98). Ode Statstcs d ed. Wley New Yok. 563

20 Amal Solma Hassa Mawa Abd-Allah. Ghtay M. E. Al-Awadh F. A. ad Alkhalfa L. A. (7). Mashall-Olk exteded Lomax dstbuto ad ts applcato to cesoed data Commucatos Statstcs Theoy ad Methods Geewood J.A. Ladweh J.M. Matalas N.C. ad Walls J.R. (979). Pobablty weghted momets: defto ad elato to paametes of seveal dstbutos expessble vese fom Wate Resouces Reseach 5(5) Gupta R. C. Gupta R. D. ad Gupta P. L. (998). Modelg falue tme data by Lehma alteatves Commucatos Statstcs Theoy ad Methods 7(4) Hassa A. S. ad Al-Ghamd A. S. (9). Optmum step stess acceleated lfe testg fo Lomax dstbuto Joual of Appled Sceces Reseach 5(): Hassa A. S. Assa S. M. ad Shelbaa A. (6). Optmum step stess acceleated lfe test pla fo Lomax dstbuto wth a adaptve type-ii pogessve hybd cesog Btsh Joual of Mathematcs & Compute Scece. 3() Keey J. F. ad Keepg E. (96). Mathematcs of Statstcs. D. Va Nostad Compay. 7. Lemote A. J. ad Codeo G. M. (3). A exteded Lomax dstbuto Statstcs: A Joual of Theoetcal ad Appled Statstcs Lomax K. S. (954). Busess falues: aothe example of the aalyss of falue data Joual of Ameca Statstcs Assocato Muthy D. N. P. Xe M. ad Jag R. (4). Webull Models Joh Wley ad Sos New Jesey.. Moos J. J. A. (988). A quatle alteatve fo kutoss Joual of the Royal Statstcal Socety. Sees D (The Statstca) 37() Nchols M. D. ad Padgett W.J. (6). A bootstap cotol chat fo Webull pecetles Qualty ad Relablty Egeeg Iteatoal Rajab M. Aleem M. Nawaz T. ad Dayal M. (3). O fve paamete beta Lomax dstbuto Joual of Statstcs Ramos M. W. A. Maho P. R. D. da Slva R. V. ad Codeo G. M. (3). The expoetated Lomax Posso dstbuto wth a applcato to lfetme data Advaces ad Applcatos Statstcs Rezae S. Nadaajah S. ad Tahghgha N. (). A ew thee-paamete lfetme dstbuto Statstcs: Joual of Theoetcal ad Appled Statstcs.DOI..8/ Tah M. H. Codeo G. M. Masoo M. ad Zuba M. (5 a). The Webull-Lomax dstbuto: popetes ad applcatos. Hacettepe Joual of Mathematcs ad Statstcs 44 ()

21 Expoetated Lomax Geometc Dstbuto: Popetes ad Applcatos 6. Tah M. H. Hussa M. A. Codeo G. M Hameda G. G. Masoo M. ad Zuba M. (5 b). The Gumbel-Lomax dstbuto: popetes ad applcatos Joual of Statstcal Theoy ad 5 () Zakezadeh H. ad Mahmoud E. (). A ew two paamete lfetme dstbuto: model ad popetes axv:4.448 [stat.co] Appedx: Etes of Obseved Ifomato Matx fo ELG. Dstbuto ls S p ps ll p p ll S l( x ) ( S) l( ) ( ) x S p S S ps p p ps ll x S S ( x ) S S p p ps S S S ll S S S S p ps p p p ps S p S S ls ps ls ll S p S p S ( l( ) ( ( l( ps S ) x p) l S ps S p ps ll ( S) l( S) p p ps ll S) x ) 565

22 Amal Solma Hassa Mawa Abd-Allah ll x x p x x x S S S S p ps. S. x. l.. x p S x S p ps ll x S x p p ps ll S S )l( x p p ps ( ) S x l( x ( S )l( x S x x. 566