Exp-Kumaraswamy Distributions: Some Properties and Applications

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1 Joul of Sccs, Islmc Rpulc of I 26: Uvsy of Th, ISSN 6-4 hp://sccs.u.c. Ep-Kumswmy Dsuos: Som Pops d Applcos Z. Jvsh, A. H Rd *, d N.R. Aghm Dpm of Sscs, Fculy of Mhmcl Sccs, Fdows Uvsy of Mshhd, P. O. Bo , Mshhd, Islmc Rpulc of I Rcvd: 28 My 24 / Rvsd: 6 Dcm 24 / Accpd: Juy 25 Asc I hs pp, w popos d sudy p-umswmy dsuo. Som of s pops dvd, cludg h dsy fuco, hzd fuco, qul fuco, moms, swss d uoss. A d s s usd o llus pplco of h poposd dsuo. Also, w o w dsuo y sfomo o p-umswmy dsuo. Nw dsuo s lv o sw-oml dsuo. Bsc pops of hs w dsuo, such s mom gg fuco, moms, swss, uoss d mmum llhood smo sudd. Is pplcly s llusd y ms of wo l d ss. Kywods: Log-p-umswmy dsuo; Moms; Mom gg fuco; Kuoss; Swss. * Cospodg uho: Tl: ; F: ; Eml: h@um.c. 57

3 Ep-Kumswmy Dsuos Ioduco B dsuo s mly vsl o modl d scd o y f vl. Ths dsuo hs wdspd pplcos dff s. O h oh hd, Kumswmy [] plds h dsuo dos o f hydologcl dom vl wll d poposd w doul oudd dsuo md f hm, sy Kw dsuo, s lv o dsuo. Th poly dsy fuco pdf d cumulv dsuo fuco cdf of Kw dsuo s f=,<<, d F=, <<, 2 spcvly, fo,>. Kw dsuo hs wdly usd hydology d ld s [2-6]. Accodg o Ndh [7], my pps h hydologcl lu hv usd hs dsuo cus s dmd s lv o h dsuo [8]. Jos [9] od som smls d dffcs w d Kw dsuos. H mphszd svl dvgs of Kw dsuo ov h dsuo: h omlzg cos s vy smpl; smpl plc fomul fo h dsuo d qul fucos whch do o volv y spcl fucos; smpl fomul fo dom v go; d plc fomul fo moms of od sscs d L-moms. Fuh, ccodg o Jos [9], h dsuo hs h followg dvgs ov h Kw dsuo: smpl fomul fo moms d mom-gg fuco; o-pm su-fmly of symmc dsuos; smpl mom smo d mo wys of gg h dsuo v physcl pocsss. Glzd Kw dsuos hv wdly sudd sscs d umous uhos hv dvlopd vous clsss of hs dsuos. Codo d Cso [] dfd w fmly of Kw glzd Kw-G dsuos o d svl wdly-ow dsuos such s h oml, Wull, gmm d Guml dsuos. Bougugo l. [] Ioducd Kumswmy Po Kw-P dsuo d povdd som sucul pops of h poposd dsuo cludg plc pssos fo h moms d gg fuco. Psco l. [2] sudd fou pm lfm dsuo, so-clld Kumglzd gmm KumGG dsuo whch s smpl so of h glzd gmm dsuo. Codo l. [3] poposd w fou-pm dsuo clld h Kumswmy glzd hlfoml Kw-GHN dsuo o d hhlfoml HN d glzd hlf-oml GHN dsuos. Pscm l. [4] dfd w fmly of Kumm glzd KBG dsuos o d svl wdly ow dsuos such s h oml, Wull, gmm d Guml dsuos. Ndh d El [5] poposd, smpl glzo of GP dsuo, Kumswmy GP KumGP dsuo. I hs pp, w popos glzo of Kw dsuo. Th glzo s movd y h followg gl clss, λg F = { λ, 3 G = wh λεr. Th cdf G could qu y d F s md p-g dsuo. O mo dvg of p-g dsuo s s flly fo fg wd spcum of l d ss. Bo-Souz d Sms [6] od svl mhmcl pops of hs clss of dsuos d dscussd h wo spcl css: p-wull d p- dsuos. Jvsh l. [7] oducd wh s ow s h p-ufom dsuo d povdd closd fom pssos fo hzd fuco, dsy fuco d moms of od sscs d dscussd h mmum llhood smo pocdu, h sympoc pops of smos d vous chczos. W oduc h pms dsuo, p- Kumswmy p-kw dsuo, s lv o d Kw dsuos. Th pdf, cdf d hzd fucohf of p-kw dsuo s follows, f = λ λ λ, < <, 4 F = λ λ λ, < <, 5 d h = λ λ, < <, 6 fo, >, λ. W povd h possl movos fo oducg p-kw dsuo. Fs, p-kw dsuo cos, s su-modls, ucd pol dsuo d ucd Wull dsuo. Fo ==, Equo 4 gvs ucd pol dsuo ov,. Fo 57

4 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I =, ylds ucd Wull dsuo ov,. Scod, lly d lf sg pms, my ms d modld y f g dsuos [8]. Th p-kw dsuo, du o h flly of s hf could mpo modl vy of polms suvvl lyss. Is hf c hu shpd, moooclly csg d upsd-dow hu dpdg sclly o h vlus of h pms. Ou fl d mo movo s sd o sfomo. L X, sfomo fom h u Y log X vl o h whol l l. Th dsuo of Y s sg hvy-l lv o swoml dsuo. Ths dsuo hs h pms d s pdf s umodl l sw-oml dsuo. I hs som dvgs ov swoml dsuo. Is cdf, hf d qul fuco hv closd fom. Also, h gs of swss d uoss fo hs w dsuo lg h sw-oml dsuo. So, h w dsuo could mo ppop fo fg sw d hvy ld d. Shp of p-kw dsuo No fom 4 h f~ s, f ~ s d, f >, > d λ R, h f s umodl. f <, < d λ R, h f s umodl. f >, d λ R, h f s csg o umodl. f, > d λ R, h f s dcsg o umodl. f == d λ>, h f s dcsg. f == d λ<, h f s csg. No fom 6 h h ~ s, h~ λ s. Also fom 6 w s h h c hu shpd, moooclly csg d upsd-dow hu. Plos of h pdf 4 d hf 6 fo som vlus, d λ gv Fgus d 2. Qul fucos wdspd us gl sscs, h qul fuco of p-kw dsuo s Qu = + log u λ. 7 λ So, p-kw dsuo s sly smuld y X=QU, wh U s ufom dom vl o h u,. Assum h X s p-kw dom vl wh pms λ, d. W sumz h loshp w p-kw dsuo d som ld dsuos s follows, f ==, h X hs ucd pol dsuo o, wh pm λ. If =, h X hs ucd dsuo o, wh pms d λ. X hs p-kw dsuo wh pms, d λ. X hs ucd Wull dsuo o, wh pms d λ. If =, h hs ucd Fch X dsuo o,+ wh pms d λ. hs ucd Féch dsuo X o,+ wh pms, d λ. If λ=, h log X hs ucd Guml dsuo o,+ wh pms d. Msus sd o quls Th ffc of h pms o swss d uoss c cosdd sd o qul msus. Bowly swss [9] d h Moos uoss [2] dfd s 58

5 Ep-Kumswmy Dsuos 3 Q Q 2Q B 4 4 2, 3 Q Q 4 4 d Q Q Q Q M Q Q 8 8 spcvly. Th ov msus s v fo dsuos whou moms. W plo h msus B d M fo p-kw dsuo fo som d, s fucos of λ Fgu 3. Ths plos show h h Bowly swss s csg wh λ fo y vlus of d, s dcsg wh fo y vlus of λ d d s csg wh foy vlus of λ d. I cos, h Moos uoss fs dcss o h mmum vlu d h css, wh λ fo som vlus of d. Th Bowly swss c posv d gv 8 9 vlus. Tl gvs vls fo h pm λ fo som vlus of d, wh h Bowly swss s posv d gv. Tl 2 gvs h pm λ fo som vlus of d h gv h mmum d mmum Moos uoss. Fgu. Pdf plos fo som vlus of, d λ. 59

6 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I Fgu 2. Hf plos fo som vlus of,d λ. Fgu 3. Th Bowly swss d Moos uoss of p-kw dsuo s fucos of λ fo som vlus of d. 6

7 Ep-Kumswmy Dsuos Tl. Ivls fo h pm fo fd d fo posv d gv B. Som vlus of d B< B> =.5,=.6 -, ,+ =.5,= -,.87.87,+ =.5,=7 Ø -,+ =.6,=.5 -, ,+ =,=.5 -, ,+ =7,=.5 -,+ Ø Tl 2. Vlus of h pm λ fo fd d fo whch M hs mmum o mmum vlu. Som vlus of d Mmum vlu fo M Mmum vlu fo M =.5,= =.5,= =.5,= =.6,= =,= =7,= Moms Cosd Y d X dom vls wh G d p- G dsuos, spcvly. Th -h mom of X c pssd ms of Y GY fo =,, s dfd y Bo-Souz d Sms [6]. X Y G Y.! Fom h -h mom of p-kw dsuo s X.! u u du Now w gv sml lv psso o X Epdg X Y,! d Mclus ss ylds wh Y hs Kw dsuo wh pms d +. Hc w hv X B,! Th swss d uoss msus c clculd usg h followg loshps, spcvly S X X X X X 4 X X K X 2 X X , 5 Fgu 4 shows swss d uoss fo som chocs of d s fuco of. Esmo pms of p-kw dsuo Suppos,, cosu dom smpl fom p-kw dsuo wh dsy 4. Th logllhood fuco s gv y,, log log log log log. I follows h h mmum llhood smos MLEs h soluo of h quos: log log,,, Th ov quos c solvd umclly y usg R sofw [22]. 6

8 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I Applcos of p-kw dsuo Ths sco cos pplco of p-kw dsuo o l d. Th d shp msums of 48 oc smpls fom polum svo. D w cd fom BP Rsch, mg lyss y Ro Kz, U. Ofod. Th d s s:.93296,.48622,.8332,.763,.2247,.6745,.8965,.6427,.23654,.62394,.5944,.484, ,.23623,.72567,.5348,.2434, ,.27,.448,.3852,.2929,.2477,.6865,.28887,.79455,.982,.3383,.22524,.34273,.3646,.2766,.97653, ,.5492,.2766,.76969,.43872,.63586, ,.32864,.238,.46425,.42477,.2744,.26265,.82453, Fgu 4. Th swss d uoss of p-kw dsuo s fucos of λ fo somvlus of d. ss Fgu 5. Poly plos fo h f of h p-kw dsuo, Kw dsuo d dsuo. 62

9 Ep-Kumswmy Dsuos Tl 3. MLEs of pms, ˆ, p-vlus of Kolmogooov-Smov s, AICc d BIC. Modl Esmd pms ˆ p-vlu AICc BIC Ep-Kw,, ˆ 32.7, ˆ.67, ˆ Kw, ˆ 2.7, ˆ B, ˆ 5.94, ˆ Fgu 6. Hsogm d smd pdfs fo h shp msums of 48 oc smplsfom polum svo. Now, w f p-kw dsuo, Kw dsuo d dsuo o h d. Th MLEs of h pms d h mmzd log-llhood fo hs dsuos compud. Th suls of goodss of f ss sd o Kolmogoov-Smov s, h vluo of h cocd A fomo co AICc [23,24] d Bys fomo co BIC [25] show Tl 3. Fom Tl 3, w s h h p- Kw dsuo s f, udgg o h ss of p- vlus, AICcs d BICs. Th poly plos gv Fgu 5 d h hsogm of h d s show Fgu 6 log wh h smd dss of h p- Kw, Kw d dsuos. Apply, h p-kw dsuo gvs h s f. Log-p-Kw dsuo Tsfomo X Y log ylds log-p-kw X dsuo s lv o sw-oml dsuo wh pdf, cdf d hf spcvly s g y G y h y y y y y y y y, y y c y y fo,>, λ. Plos of h pdf 6 d hf 8 fo som spcl vlus of, d λ gv Fgus 7 d 8. Th qul fuco cospodg o 6 s 63

10 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I Fgu 7. Plos of log-p-kw pdf fo som vlus of, d Fgu 8. Plos of log-p-kw hf fo som vlus of, d 64

11 Ep-Kumswmy Dsuos 65 9 log log log u u Q u Smulo of log-p-kw dsuo s sy y Y = QU, wh U s ufom ov,. Epsos fo mom gg fuco d moms Th mgf of log-p-kw dsuo s gv y 2. d M Epdg Mclus ss ylds 2! d M Now, w pd Mclus ss 22.! d M Th ov psso shows h h mgf of log-p- Kw dsuo s f <. So, fo < w hv 23,,! B M

12 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I 66 Th -h mom of h log-p-kw dsuo c od foms Y d M d Y E. So, w hv 24.,! B Y E I pcul, h fs fou moms of Y s follows Fgu 9. Swss d uoss of log-p-kw dso fo som chocs of d sfuco of λ Fgu. Th gs of swss d uoss of h log-p-kw dsuo d sw-oml dsuo.

13 Ep-Kumswmy Dsuos 67, 3 2!, 2!, 2! Y E Y E Y E d, ! Y E Wh ψ s. log d d Th swss d uoss of Y c od fom 5. Plos of h swss d uoss fo som chocs of d s fucos of λ, gv Fgu 9. I Fgu, w comp h gs of uoss d swss of log-p-kw dsuo wh hos of sw-oml dsuo, spcvly. Ths plos dc h h gs of swss d spclly uoss log-p-kw dsuo lg h hos of sw-oml dsuo. Esmo of pms of log-p-kw dsuo H, w cosd smo y mhod of mmum llhood. Suppos y, y cosu dom smpl fom 6, h log-llhood fuco s. log log log log log,, y y y y I follows h h MLEs h soluo of h quos:, log, log, Wh y fo =,...,. h ov quos c solvd umclly y usg R sofw [22]. Applcos of log-p-kw dsuo I hs sco w f log-p-kw dsuo o l d ss. W povd wo mpls. Empl : Th d s od fom Smh d Nylo [26] ps h sghs of.5 cm glss fs, msud h Nol Physcl Looy, Fgu. Poly plos fo h f of h log-p-kw dsuo d sw-oml dsuo.

14 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I Egld. Th d s s:.55,.93,.25,.36,.49,.52,.58,.6,.64,.68,.73,.8, 2,.74,.4,.27,.39,.49,.53,.59,.6,.66,.68,.76,.82, 2.,.77,.,.28,.42,.5,.54,.6,.62,.66,.69,.76,.84, 2.24,.8,.3,.29,.48,.5,.55,.6,.62,.66,.7,.77,.84,.84,.24,.3,.48,.5,.55,.6,.63,.67,.7,.78,.89. Now, w f log-p-kw dsuo d swoml dsuo o h d. Th MLEs of h pms d h fo hs dsuos compud.th suls of goodss of f ss sd o poly plos gv Fgu, lso show h logp-kw dsuo gvs f h sw-oml dsuo. Tl 4. MLEs of pms,, P-vlus of Kolmogoov-Smov s, AIC c d BIC. Modl Esmd pms p-vlu AIC c BIC Log-p-Kw λ,, 2.28, 6.84, 34.9 Sw-oml, 2,.5, 2.32, Fgu 2. Poly plos fo h f of h w d sw-oml dsuo o h d Kolmogoov-Smov s, h vluoof h AIC c d BIC show Tl 4. Fom Tl 4, w s h h log-p-kw dsuo s f, udgg o h ss of p-vlus, AIC c s d BICs. Th Empl 2: Th d s od fom Nchols d Pdg [2] d pss h g sss of co fs G, =. Log-p-Kw dsuo d sw-oml dsuo fd o h d s. Th suls of Kolmogoov-Smov s, AIC c d BIC pod Tl 5. Fom Tl 5, w s h oh css, w co c h ull hypohss h d comg fom logp-kw dsuo o sw-oml dsuo. O h oh hd, h vlus of AICc, BIC d h poly plos gv Fgu 2 lmos h sm fo oh modls. So, w comp hm sd o hfs. Tl 5. MLFs of pms,, P-vlus of Kolmogoov-Smov s, AIC c s d BIC s Modl Esmd pms p-vlu AIC c BIC Log-p-Kw λ,,.28, 9.77,.6 Sw-oml, 2,.68, 2.38,

15 Ep-Kumswmy Dsuos Fgu 3. Esmd hfs of h w d sw-oml dsuos fo h d s. Th plos of mpcl Hf of d d fd hfs show Fgu 3. Ths Fgu dcs h h hf of log-p-kw dsuo hs f h h of sw-oml dsuo. As fuh chc, w clcul h m squ dffc of mpcl hf d smd hf fo oh dsuos. Ths o s.9 fo log-p-kw dsuo d s.7 fo swoml dsuo. Hc, h log-p-kw dsuo hs f o h d, ms of hf. Rsuls d Dscusso W sudd p-kw dsuo. W dvd vous pops of hs dsuo, cludg h shp of pdf d hf, ld dsuos, moms, swss, uoss d smo of pms. Th p-kw dsuo, du o h flly of s hf could mpo modl vy of polms suvvl lyss. Also, w dsuo y sfomo o p-kw dsuo, s lv o sw-oml dsuo, s od. Ths dsuo hs h pms d s pdf s umodl l sw-oml dsuo. I hs som dvgs ov sw-oml dsuo. Is Cdf, hf d qul fuco hv closd fom. Also, h gs of swss d uoss fo hs w dsuo lg h sw-oml dsuo. So, h w dsuo could mo ppop fo fg sw d hvy ld d. Applco of dsuos o l d ss ws show y h mpls, showg h hs dsuos, c usd ffcvly lysg d. Rfcs. Kumswmy P. A glzd poly dsy fuco fo doul-oudd dom pocsss. J. Hydol. 46-2: Sud V. d Suh K. Applco of doul oudd poly dsy-fuco fo lyss of oc wvs. Oc Eg.6: Flch S. C. d Pomlm K. Esmo of svo yld d sog dsuo usg moms lyss. J. Hydol. 82: Sf A., Pomlm K., d Vlch J. Mmzo of mufcug yld of sysms wh y dsuos of compo vlus. A. Op. Rs.99: Pomlm K., Sf A., d Vlch J. Polsc dsg of sysms wh gl dsuos of pms. I. J. Ccu Thoy Appl.29: G A., Pomlm K., d Khll D. G yld lly lyss wh copw dmd ucy. Soch. Evo. Rs. Rs Assss.2: Ndh S. O h dsuo of Kumswmy. J. Hydol.348: Kousoys D. d Xhopoulos T.O h pmc ppoch o u hydogph dfco. W Rs. Mg. 3: Jos M. C. Kumswmys dsuo: A -yp dsuo wh som cly dvgs. S. Mh.6: Codo G. M. d d CsoM. A w fmly of glzd dsuos. J.S. Compu. Sm.8: Bougugo M., Slv R. B., Z L. M. d Codo G. M. Th Kumswmy Po dsuo. J. S. T. A.2, No. 2: 29-44July Psco M. A. R., Og E. M. M., d Codo G. M. 69

16 Vol. 26 No. W 25 Z. Jvsh, l. J. Sc. I. R. I Th Kumswmy glzd gmm dsuo wh pplco suvvl lyss. S. Mh.8: Codo G. M., Pscm R. R. d Og E. M. M. Th Kumswmy Glzd Hlf-Noml Dsuo fo Swd Posv D. J. D. S.: Pscm R. R., Codo G. M., Dméo C. G. B., Og E. M. M. d Ndh S. Th w clss of Kumm glzd dsuos. So 36 2: Ndh S. d El S. Th Kumswmy GP Dsuo. J. D. S.: Bo-Souz W., Sms A.B. Th p-g fmly of poly dsuos. Bz.J. Po. S. Volum 27: Jvsh Z., H Rd A. d Hmd G. G. Ep- Ufom Dsuo: Pops d Chczos. J. Ss. Rs. I: Blow R. E. d Posch F. Sscl hoy of lly d lf sg: poly modls. Hol, Rh d Wso, w Yo, Lodo Ky J. F. d Kpg E. S. Mhmcs of Sscs, P, 3d do. V Nosd, Nw Jsy Moos J. J. A. A qul lv fo uoss. J. Roy. S. Soc. D-S.37: Nchols M. D. d Pdg W. J. A oosp cool ch fo Wull pcls.qul. Rl. Eg. I. 22: R Co Tm, R: A lgug d vom fo sscl compug, R Foudo fo Sscl Compug, V, Aus. ISBN: , URL:hp:// Sugu N. Fuh lyss of h d y As fomo co d h f cocos. Comm. Ss.A 7: Huvch C. F. d Ts C.L. Rgsso d m ss modl slco smll smpls. Bom.76 2: Schwz E. Gd o Esmg h dmso of modl. A. S.6 2: Smh R. L. d Nylo J. C. A compso of mmum llhood d Bys smos fo h h-pm Wull dsuo. Appl. S.36:

Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues

Boyce/DiPrima 9 th ed, Ch 7.6: Complex Eigenvalues BocDPm 9 h d Ch 7.6: Compl Egvlus Elm Dffl Equos d Boud Vlu Poblms 9 h do b Wllm E. Boc d Rchd C. DPm 9 b Joh Wl & Sos Ic. W cosd g homogous ssm of fs od l quos wh cos l coffcs d hus h ssm c b w s ' A