On Five-Parameter Lomax Distribution: Properties and Applications

Transcription

1 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos M. E. Mead Depatmet of Statstcs ad Isuace Faculty of Commece, Zagazg Uvesty, Egypt Abstact A fve-paamete cotuous model, called the beta expoetated Lomax dstbuto, s defed ad studed. The model has as specal sub-models some mpotat lfetme dstbutos dscussed the lteatue, such as the logstc, Lomax, expoetated Lomax, beta Lomax dstbutos, amog seveal othes. We deve the oday ad complete momets, geeatg ad quatle fuctos, mea devatos, Bofeo, Loez ad Zega cuves, mea esdual lfe, mea watg tme ad Réy of etopy. The method of maxmum lkelhood s poposed fo estmatg the model paametes. We obta the obseved fomato matx. Thee eal data sets demostate that the ew dstbuto ca povde a bette ft tha othe classcal lfetme models. Keywods: Beta dstbuto; Lomax dstbutos; Maxmum lkelhood estmato.. Itoducto The Lomax o Paeto II dstbuto has wde applcatos may felds such as come ad wealth equalty, medcal ad bologcal sceces, egeeg, sze of ctes actuaal scece, lfetme ad elablty modelg. I the lfetme cotext, the Lomax model belogs to the famly of deceasg falue ate see Chahkad ad Gajal, 29 ad ases as a lmtg dstbuto of esdual lfetmes at geat age see Balkema ad de Ha, 974. Fo moe fomato about the Lomax dstbuto ad Paeto famly ae gve Aold 983 ad Johso et al Vaous geealzatos of Lomax dstbuto have bee studed, the expoetated Lomax, dscussed by Abdul-Moem ad Abdel-Hameed 22, Mashall-Olk exteded Lomax defed by Ghtay et al. 27, McDoald Lomax vestgated by Lemote ad Codeo 23, gamma Lomax toduced by Codeo et al. 25, the Webull Lomax dstbuto studed by Tah et al. 25 ad ecetly the tasmuted Webull Lomax dstbuto gve by Affy et al. 25. The adom vaable X wth expoetated Lomax EL dstbuto has the cumulatve dstbuto fucto cdf gve by G x;,, x, fo,, ad x. The pobablty desty fucto pdf coespodg to takes the fom ;,,, g x x x 2 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

2 M. E. Mead 2. The Beta Expoetated Lomax Dstbuto Let Gx be the cdf of ay adom vaable X. The cdf of a geealzed class of dstbutos defed by Eugee et al. 22 s gve by G x a b G x,, B a, b F x I a b v v dv 3 whee a, b ae the shape paametes, I a, b B a, b B a, b s the complete a b beta fucto ato, B a, b v v dv y B a, b a b a b s the beta fucto ad. coespodg pdf fo 3 s gve by a f x g x G x G x, B a, b y y y s the complete beta fucto, s the gamma fucto. The b 4 whee g x G x x s the basele desty fucto. Replacg 3, we obta a ew dstbuto, called beta expoetated Lomax BEL, wth cdf gve by w a b F x, Iw a, b v v dv, B a, b 5 fo a ad b. Hee w x ad ab,,,, model paametes. Equato 5 ca be expessed as follows whee a x F x; 2F a, b; a ; x, a B a, b 2 b cb t t F a, b; c; z dt, a B b, c b t z s the well kow hypegeometc fucto Gadshtey ad Ryzhk, 27. s the vecto of the The pdf coespodg to 5 s gve by f x x x x B a, b a b ;. 6 I Fg., we Plot of the BEL desty fucto fo dffeet values of ab,,,,. Fo a lfetme adom vaable t, the suvval fucto St, hazad ate fucto ht, evesed hazad ate fucto t ad the cumulatve hazad ate fucto Ht of BEL dstbuto ae gve by S t F t I a, b, t 86 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

3 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos.8.6 a=4, b=5.8, =7.5,,=.3, =.5 a=2, b=2.8, =5.5,,=.4, =.5 a=2, b=2.8, =.2,,=.9, =.6 a=2, b=2.8, =2.5,,=.2, = Fg.. The pdf of the BEL fo dffeet values of the paametes ad a f t t t t ht, St B a, b B a, b t a f t t t t t 8 F t B a, b t H t S t I a, b t. b b 7 Plots of the HRF fo dffeet values of ab,,,, ae gve Fg a=.2, b= 6, =4, =.95, =.5 a=2.5,b=5.5,=4.8, =.2, =.5 a=2.8, b=5.4, =4.8, =., =.4 a=.5,b=.8,=2.65,=.6,= Fg. 2. The HRF of the BEL fo dffeet values of the paametes Pak.j.stat.ope.es. Vol.XII No. 26 pp

4 M. E. Mead 2. Sub-models The followg dstbutos ae specal of the BEL dstbuto:. Whe, equato 6 educes to the beta Lomax BL dstbuto. 2. Settg, we obta the beta expoetated stadad Lomax dstbuto BESL dstbuto. 3. If b, the desty 6 coespods to the expoetated Lomax dstbuto EL dstbuto. 4. Whe ab, we obta the expoetated Lomax dstbuto EL dstbuto. 5. If a b, the BEL gves the expoetated stadad Lomax ESL dstbuto. 6. Equato 6 becomes the Lomax dstbuto fo the choce a b 7. Settg a b, the desty 6 yelds the stadad Lomax SL dstbuto. 3. Some Statstcal Popetes We gve a mathematcal teatmet of the ew dstbuto cludg expasos fo the desty fucto, momets, complete momets, quatle fucto, mea devatos, Bofeo, Loez ad Zega cuves, mea esdual lfe, mea watg tme ad Réy etopy. 3. Expasos fo the Dstbuto ad Desty Fuctos Equatos 5 ad 6 ae staghtfowad to compute usg ay statstcal softwae. Howeve, we obta expasos fo Fx ad f x tems of a fte o fte weghted sums of cdf 's ad pdf 's of adom vaables havg Lomax dstbutos, espectvely. Fo ay postve eal umbe b ad fo z <, a geealzed bomal expaso holds j b j z. j! b j 9 b z j Theefoe, the cdf of BEL ca be expaded to obta a b F x, Iw a, b v v dv, B a, b w w j a b a j b F x, v dv a b j! b j j j p G x;,, a j, j 88 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

5 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos whee j a b p j, a j! b j a j ad G x;,, a j deotes the cdf of EL wth paametes, ad a j. Smlaly, we ca wte the pdf 6 as a b f x x x a b j j ; j j p H x;,, a j, j aj whee H x;,, a j deotes the EL desty fucto wth paametes, ad a j. Aga, by usg bomal expaso equato, we obta whee j a b a j f x, x! a j! b j a j q j j q h x;,, j a b a j,! a j! b j a j ad h x;, deotes the Lomax desty wth paametes ad. If b s a tege, the the summato equatos, ad 2 stops at b.thus, the BEL desty fucto ca be expessed as a fte lea combato of Lomax destes. Thus, some of ts mathematcal popetes ca be obtaed dectly fom those popetes of the Lomax dstbuto. 3.2 Momets ad Momet Geeatg Fucto As wth ay othe dstbuto, may of the teestg chaactestcs ad featues of the BEL dstbuto ca be studed though the momets. If we assume that Y s a Lomax dstbuted adom vaable, wth paametes ad, the the th momet of Y s gve E Y B,,. Let X be a adom vaable havg the BEL dstbuto 6. Usg equato 2, t s easy to obta the th momet of X as 2 3 E X q B,,. The mea, vaace, Skewess ad Kutoss ca be obtaed fom 3. If b s tege ad, the sum stops at b. Pak.j.stat.ope.es. Vol.XII No. 26 pp

6 M. E. Mead The momet geeatg fucto mgf, say M t Eexp tx k t k M t E X k! k k whee EX follows fom equato 3. of BEL s gve by 3.3 Quatle Fucto Let Q, u be the beta quatle fucto wth paametes a ad b. The quatle fucto ab of the BEL dstbuto, say x Q u, ca be easly obtaed as ab, x Q u Q u, u,. 4 Ths scheme s useful to geeate BEL adom vaates because of the exstece of fast geeatos fo beta adom vaables most statstcal packages,.e. f V s a beta adom vaable wth paametes a ad b, the X V, follows the BEL dstbuto. Fom 4 we coclude that the meda m of X s m Q 2. The Bowley skewess SK measue ad Moos kutoss KR based o octles of the BEL dstbuto ca be calculated usg the fomulae gve below ad Q3 4 Q 4 2 Q 2 SK Q3 4 Q 4 KR 3.4 Icomplete Momets Q Q Q Q Q6 8 Q If Y s a adom vaable wth BXII dstbuto wth paametes ad, the th complete momet ofy, fo, s gve by z m z y g y;, dy B,,. z s Fom ths equato, we ote that Ms z EY whe z, wheeve. Let X be a adom vaable havg the BEL dstbuto 6. The th complete momet of X s the equal to z m z q B,,. 5 9 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

7 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos 3.5 Mea Devatos The mea devatos about the mea ad the meda ca be used as measues of spead a populato. Let EX ad be the mea ad the meda of the BEBXII dstbuto espectvely. The mea devatos about the mea ad about the meda of X ca be calculated as ad D E X x f x dx 2 F 2 m d D E X x f x dx 2 m, d espectvely, whee m deotes the fst complete momet ad F follows fom Mea Resdual Lfe ad Mea Watg Tme The mea esdual lfe fucto MRL at a gve tme t measues the expected emag lfetme of a dvdual of age t. It s deoted by mt. The MRL o lfe expectacy of BEL s defed as t m t E t t f t dt t, St whee t q B 2, Bt 2, t, I a, b t t f t dt = q Bt 2,., The mea watg tme MWT of a tem faled a teval [, t ] fo BEL s defed as t, t t f t dt Ft t t q Bt 2,, I a, b t. 3.7 Loez, Bofeo ad Zega Cuves Loez ad Bofeo cuves have bee appled may felds such as ecoomcs, elablty, demogaphy, suace ad medce, see Klebe ad Kotz, 23 fo addtoal detals. Zega cuve was peseted by Zega 27. The Loez L x, F Pak.j.stat.ope.es. Vol.XII No. 26 pp

8 M. E. Mead Bofeo B F x ad Zega Ax 23 as the followg cuves ae defed by Oluyede ad Rajasooya x LF x t f t dt E X, B F x t f t dt F x E X LF x F x ad A x M x M x espectvely, whee x x M x t f t dt F x ad M x t f t dt F x ae the lowe ad uppe meas espectvely. Fo the BEL dstbuto, these quattes ae deved below. Loez cuve: L x FG x;. 2. Bofeo cuve: q B 2, q B 2,, x G x;. B F j j 3. Zega cuve: x F x t f t dt Ax q B 2, p G x;,, a j q B 2, ;, F x t f t dt x x p jg x;,, a j q B x 2, j. p jg x;,, a j q B 2, Bt 2, j 3.8 Réy Etopy The etopy of a adom vaable X s a measue of ucetaty vaato. The Réy etopy s defed as IR log I, whee I f x dx, ad. Usg equato 5 we obta a b I x x x dx. ab, 92 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

9 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos Based o the bomal expaso to the last facto the above tegad yelds b I x x dx a. ab, Aga, usg the bomal expaso to the last facto, we obta a b j j. ab, j j I x dx Usg tegato above expesso ad smplfyg, a b j I j. ab, j j Hece, the Réy etopy educes to b a j IR log log j log. B a, b j j 4. Estmato of Paametes The maxmum lkelhood estmato MLE s oe of the most wdely used estmato method fo fdg the ukow paametes. Let X, X2,..., X be a depedet adom sample fom BEBXII. The total log-lkelhood s gve by B a, b u a z b z, 6 whee u x ad z. u The scoe vecto,,,, has compoets a b a b a z, a a b b z, b a z u x b z z u x u x, u a z u u b z z u u Pak.j.stat.ope.es. Vol.XII No. 26 pp

10 M. E. Mead a z b z z z whee p s the dgamma fucto whch s the devatve of log.. The maxmum lkelhood estmates MLEs of the ukow fve paametes ca be obtaed by solvg the system of olea equatos, teatvely. Fo teval estmato of ab,,,, ad hypothess tests o these paametes, we obta the obseved fomato matx sce ts expectato eques umecal tegato. The 5 5 obseved fomato matx J s Jaa Jab Ja Ja Ja Jba Jbb Jb Jb J b J J a J b J J J, Ja Jb J J J J a J b J J J whose elemets ae gve Appedx. 5. Applcatos I ths secto we povde thee applcatos of the BEL dstbuto to thee eal data sets. The fst data set, stegth data, whch wee ogally epoted by Bada ad Pest 982 ad t epesets the stegth measued GPA fo sgle cabo fbes ad mpegated -cabo fbe tows. Sgle fbes wee tested ude teso at gauge legths of mm wth sample sze = 63. Ths data set cossts of obsevatos:.9, 2.32, 2.23, 2.228, 2.257, 2.35, 2.36, 2.396, 2.397, 2.445, 2.454, 2.474, 2.58, 2.522, 2.525, 2.532, 2.575, 2.64, 2.66, 2.68, 2.624, 2.659, 2.675, 2.738, 2.74, 2.856, 2.97, 2.928, 2.937, 2.937, 2.977, 2.996, 3.3, 3.25, 3.39, 3.45, 3.22, 3.223, 3.235, 3.243, 3.264, 3.272, 3.294, 3.332, 3.346, 3.377, 3.48, 3.435, 3.493, 3.5, 3.537, 3.554, 3.562, 3.628, 3.852, 3.87, 3.886, 3.97, 4.24, 4.27, 4.225, 4.395, 5.2. Ths data set s pevously studed by Affy et al. 25 to ft the expoetated tasmuted geealzed Raylegh dstbuto. As a secod applcato, we aalyze a eal data set o the actve epa tmes hous fo a aboe commucato tasceve. Ths data set was aalyzed by Jogese 982. The data ae as follows:.5,.6,.6,.7,.7,.7,.8,.8,.,.,.,.,.,.3,.5,.5,.5,.5, 2., 2., 2.2, 2.5, 2.7, 3., 3., 3.3, 4., 4., 4.5, 4.7, 5., 5.4, 5.4, 7., 7.5, 8.8, 9.,.2, 22., Recetly, Lemot et al. 23 studed these data usg the expoetated Kumaaswamy dstbuto. Based o the thd applcato, we use the lfetme data set gve by Goss ad Clak 975. The data set epesets the elef tmes of twety patets ecevg a aalgesc. The data ae as follows:.,.4,.3,.7,.9,.8,.6, 2.2,.7, 2.7, 4.,.8,.5,.2,.4, 3.,.7, 2.3,.6, 2.. Recetly, ths data set s pevously studed by Rodgues et al. 24 to ft the beta expoetated Ldley dstbuto. 94 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

11 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos We use these thee data sets to compae the ft of the BEL dstbuto ad the submodels, BL ad EL wth fou models: McDoald Lomax McL Lemote ad Codeo, 23, gamma Lomax GL Codeo et al., 25, Webull Lomax WL Tah et al., 25 ad the tasmuted Webull Lomax TWL Affy et al., 25. Tables, 2 ad 3 lst the maxmum lkelhood estmates MLEs the coespodg stadad eos paetheses of the paametes of all the models fo the thee data sets espectvely. Table : MLEs stadad eos paetheses fo BEL, BL, EL, McL, TWL, WL ad GL models ad the statstcs 2, W ad A ; fst data set Model BEL ab,,,, BL ab,,, Estmates Statstcs W A EL,, McL ab,,,, TWL ab,,,, ab WL,,, GL a,, Table 2: MLEs stadad eos paetheses fo BEL, BL, EL, McL, TWL, WL ad GL models ad the statstcs 2, W ad A ; secod data set. Model BEL ab,,,, BL ab,,, EL,, McL ab,,,, TWL ab,,,, ab WL,,, GL a,, Estmates Statstcs W A Pak.j.stat.ope.es. Vol.XII No. 26 pp

12 M. E. Mead Table 3: MLEs stadad eos paetheses fo BEL, BL, EL, McL, TWL, WL ad GL models ad the statstcs 2, W ad A ; thd data set Model BEL ab,,,, BL ab,,, EL,, McL ab,,,, TWL ab,,,, ab WL,,, GL a,, Estmates Statstcs W The statstcs: 2 whee deotes the log-lkelhood fucto evaluated at the maxmum lkelhood estmates, the Adeso Dalg A ad Camé-vo Mses W ae epoted Tables, 2 ad 3. I geeal, the dstbuto whch has the smalle values of these statstcs s the bette the ft to the data. The esults show that the BEL dstbuto povdes a sgfcatly bette ft tha the othe sx models. All the computatos wee doe usg the MATH- CAD PROGRAM. 6. Cocludg emaks I ths pape, we poposed a ew dstbuto, amed the beta expoetated Lomax dstbuto whch exteds the Lomax dstbuto. Seveal popetes of the ew dstbuto wee vestgated, cludg oday ad complete momets, mea devatos, Réy etopy, ad elablty. The model paametes ae estmated by maxmum lkelhood ad the fomato matx s deved. Thee applcatos of the beta expoetated Lomax dstbuto to eal data show that the ew dstbuto ca be used qute effectvely to povde bette fts tha the expoetated Lomax Abdul-Moem ad Abdel-Hameed, 22, beta Lomax ad McDoald Lomax Lemote ad Codeo, 23, Webull Lomax Tah et al., 25, gamma Lomax Codeo et al., 25 ad ecetly, the tasmuted Webull Lomax Affy et al., 25. We hope that the poposed model may attact wde applcatos may aeas such as egeeg, suvval aalyss, hydology, ecoomcs, ad so o. Ackowledgmets The autho thaks the Edto ad the Refeees fo the helpful emaks that mpoved the ogal mauscpt. A 96 Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

13 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos Refeeces. Abdul-Moem, I. B. & Abdel-Hameed, H. F. 22. O expoetated Lomax dstbuto, Iteatoal Joual of Mathematcal Achve 3, Affy, A. Z., Nofal, Z. M. & Ebahem, A. N. 25. Expoetated tasmuted geealzed Raylegh dstbuto: a ew fou paamete Raylegh dstbuto. Paksta Joual of Statstcs ad Opeatos Reseach,, Affy, A. Z., Nofal, Z. M., Yousof, H.M., El Gebaly,Y.M. & Butt, N.S.25. Tasmuted Webull Lomax dstbuto. Paksta Joual of Statstcs ad Opeatos Reseach,, Aold, B.C Paeto Dstbutos. Iteatoal Coopeatve Publshg House, Maylad. 5. Bada, M.G. & Pest, A.M Statstcal aspects of fbe ad budle stegth hybd compostes. I: Hayash, T., Kawata, K., Umekawa, S. Eds., Pogess Scece ad Egeeg Compostes. ICCM-IV, Tokyo, Balkema, A.A & de Ha, L Resdual lfe at geat age, Aals of Pobablty 2, Chahkad, M. & Gajal, M. 29. O some lfetme dstbutos wth deceasg falue ate, Computatoal Statstcs ad Data Aalyss 53, Codeo, G. M., Otega, E. M. M. & Popovc, B. V. 25. The gamma-lomax dstbuto, Joual of Statstcal computato ad Smulato, 85, Eugee, N., Lee, C. & Famoye, F. 22. The beta-omal dstbuto ad ts applcatos. Commucato Statstcs - Theoy ad Methods, 3, Ghtay, M. E., AL-Awadh, F. A & Alkhalfa, L. A. 27. Mashall-Olk exteded Lomax dstbuto ad ts applcatos to cesoed data, Commucatos Statstcs-Theoy ad Methods 36, Gadshtey, I. S. & Ryzhk, I. M. 27. Table of tegals, sees ad poducts, Seveth. 2. Goss, A. J. & Clak, V. A Suvval dstbutos: Relablty applcatos the bomedcal sceces, Joh Wley ad Sos, New Yok. 3. Johso, N. L., Kotz, S. & Balaksha, N Cotuous Uvaate Dstbutos: Vol., 2d edto Wley, New Yok. 4. Jogese, B Statstcal popetes of the geealzed vese Gaussa dstbuto. Spge-Velag, New Yok. 5. Klebe, C. & Kotz, S. 23. Statstcal sze dstbutos ecoomcs ad actuaal sceces. Wley Sees Pobablty ad Statstcs. Joh Wley & Sos. 6. Lemot, A.J., Baeto-Souza, W. & Codeo, G.M. 23. The expoetated Kumaaswamy dstbuto ad ts log-tasfom. Bazla Joual of Pobablty ad Statstcs, 27, Pak.j.stat.ope.es. Vol.XII No. 26 pp

14 M. E. Mead 7. Lemote, A. J. & Codeo, G. M. 23. A exteded Lomax dstbuto, Statstcs 47, Oluyede, B. O. & Rajasooya, S. 23. The Mc-Dagum dstbuto ad ts statstcal popetes wth applcatos. Asa Joual of Mathematcs ad Applcatos, 44, Rodgues, J.A., Slva, A.P.C.M. & Hameda, G.G. 24. The beta expoetated Ldley dstbuto. Joual of Statstcal Theoy ad Applcatos, 4, Tah, M. H., Codeo, G. M., Masoo, M. & Zuba, M. 25. The Webull- Lomax dstbuto: popetes ad applcatos. To appea the Hacettepe Joual of Mathematcs ad Statstcs. 2. Zega, M. 27. Iequalty cuve ad equalty dex based o the atos betwee lowe ad uppe athmetc meas. Statstca & Applcazo,, Pak.j.stat.ope.es. Vol.XII No. 26 pp85-99

15 O Fve-Paamete Lomax Dstbuto: Popetes ad Applcatos Appedx The elemets of the obseved fomato matx J fo the paametes ab,,,, ae J a b a, aa J a b, ab a J z u x a J z u u, J a z, J a b b, bb b, J z z u x b, J z z u u b, J z z z, J u x a z u x z u b z z u x, J b z z u x u z z u u z u u u x a z u x u u u, 2 J b z z u x z z z z a z u x, J b z z u u z z u z u 2 2 a z u u u z,, J b z z u u z z z z a z u u 2 2 J b z z z z z. Pak.j.stat.ope.es. Vol.XII No. 26 pp