A new generalized Lindley distribution

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Gwendoline Oliver
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1 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso Asc A w lzd dly dsuo Ih Ell o Movc * M Elhy 3 Isu o Sscl Suds d Rsch Dpo Mhcl Sscs Co Uvsy Dp o Mhcs Uvsy o Psh "Hs Psh" Rpulc o Kosovo 3 Isu o Sscl Suds d Rsch Dpo Mhcl Sscs Co Uvsy El: _ll@scudu * El o h cospod uho: ovc@yhooco 3 El: _lhy85@yhooco I hs pp w ps w clss o dsuos clld Nw lzd dly DsuoND Ths clss o dsuos cos svl dsuos such s pol d dly s spcl css Th hzd uco vs hzd uco os d o uco d uly sus od Moov w dscuss h u lklhood so o hs dsuo Th usulss o h w odl s llusd y s o wo l d ss W hop h h w dsuo poposd h wll sv s lv odl o oh odls vll h lu o odll posv l d y s Kywods: lzd dly Dsuo; dsuo Mu lklhood so; Mo uco Ioduco d Movo I y ppld sccs such s dc d c os ohs odl d lyz l d cucl Svl l dsuos hv usd o odl such kds o d o sc h pol Wull Rylh dsuos d h lzos s up d Kudu [] Ech dsuo hs s ow chcscs du spcclly o h shp o h lu uco whch y oly oooclly dcs o cs o cos s hvo s wll s oooo hu shpd o v uodl H w cosd h dly dsuo whch ws oducd y dly [3] h l do vl X hs dly dsuo wh p h poly dsy uco pd o X s v y ; > > I c s h hs dsuo s u o pol cospod cuulv dsuo uco cd o D s od s d > > dsuos Th wh s scl p Th dly dsuo s po o sudy sss sh lly odl Bsds so schs hv poposd w clsss o dsuos sd o odcos o h dly dsuo clud lso h pops Th d s lwys dcd y dd o dsuos o o ll sucus Sk [6] oducd h dsc Posso dly dsuo y co h Posso d dly dsuos hy l [5] hv dscussd vous pops o hs dsuo d showd h y wys Euo povds odl o so pplcos h h pol dsuo A dsc vso o hs dsuo hs susd y Dz d Od [3] hv s pplcos cou d ld o suc hy l [7 8] od szsd d zoucd vso o Posso dly dsuo d dscussd h vous pops d pplcos hy d Al Mu [6] dscussd s vous so hods o h dsc Posso dly dsuo Bkouch l [] od dd dly dsuo d dscussd s vous pops d pplcos Mzuchl d Achc [4] dscussd h pplcos o dly dsuo o cop sks l d R d 3

2 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 3 Msh [5] sudd us dly dsuo hy l [9] dvlopd wop whd dly dsuo d dscussd s pplcos o suvvl d Zkzdh d Dol [8] od lzd dly dsuo d dscussd s vous pops d pplcos Ths pp os w dsuo wh h p clld lzs h dly dsuo hs dsuo cluds s spcl css h ody pol d dsuos Th pocdu usd h s sd o c us o h dsuos Th sudy s vous pops o h w odl Th s o h pp s ozd s ollows: ous sscl pops cluds o uco d uly sus o h N dsuo plod Sco Th dsuo o h od sscs s pssd Sco 3 W povd h sso sd hod o ls sus d whd ls sus sos Sco 4 Mu lklhood ss o h ps d o h dsuo dscussd Sco 5 Sco 6 povds pplcos o l d ss Sco 7 ds wh so coclusos Sscl Pops d Rlly Msus I hs sco w vs h sc sscl pops pcul h o o uco d uly sus o h N dsuo Dsy suvvl d lu ucos Th w lzd dly dsuo s dod s ND Ths lzd odl s od o u o h d dsuos s ollows: p p > > ; 3 wh d p Th cospod cuulv dsuo uco cd o lzd dly s v y 4 wh d s s s clld low copl Also h upp copl s v y d o o dls ou h do o copl s Wll [] us d lluss so o h possl shps o h pd d cd o h N dsuo o slcd vlus o h ps d spcvly Th suvvl uco ssocd wh 4 s v y 5 o 3 4 d 5 h lu o hzd uco h d vs hzd ucos h o lzd dly dsuo v y

3 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 3 h 6 d u : Th pd s o vous N dsuos o vlus o ps: ; ; wh colo shps pupl lu plu d lck d dkcy spcvly

4 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No u : Th cd s o vous N dsuos o vlus o ps: ; wh colo shps d pludcy lck d pupl spcvly 7 spcvly u 3 lluss so o h possl shps o h hzd uco o h N dsuo o slcd vlus o h ps d spcvly

5 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso u 3: Th hzd s o vous N dsuos o vlus o ps: ; ; wh colo shps pupl lu plu dlckd dkcy spcvly Th ollow spcl css o h lzd dly dsuo D I d w h dly dsuo o l w h dsuo wh p l 3 I w h pol dsuo wh p Mos My o h s chcscs d us o dsuo c sudd houh s os Mos cssy d po y sscl lyss spclly pplcos I c usd o sudy h os po us d chcscs o dsuo dcy dspso skwss d kuoss Tho I X hs h h h o o X s v y h ollow ' 8 34

6 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No Poo: X do vl ollow h dsuo wh ps d Epssos o hcl pco vc d h h o o h o o X c od us h wllkow oul d X E ' d d 9 Whch copls h poo Bsd o h s ou os o h dsuo h sus o skwss A d kuoss k o h dsuo c od s [ ] A d [ ] k 3 Mo uco I hs susco w dvd h o uco o dsuo Tho : I X hs dsuo h h o uco M X hs h ollow o M X Poo W s wh h wll kow do o h o uco v y d E M X X d d Whch copls h poo I h s wy h chcsc uco o h dsuo cos M X X wh s h u y u

7 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No Iuly Msus oz d Boo cuvs h os wdly usd uly sus co d wlh dsuo Kl 4 Z cuv ws psd y Z [9] I hs sco w wll dv oz Boo d Z cuvs o h dsuo Th oz Boo d Z cuvs dd y X E d X E d B 3 d A 4 wh X E d d d spcvly Th sdul l l uco copus h pcd suvvl o suc v suvvl up o W hv ldy dd h l s h pco o h suvvl v suvvl up o s k uss d k Posch [4] 3 Dsuo o h od sscs I hs sco w dv closd o pssos o h pds o h h od ssc o h dsuo lso h sus o skwss d kuoss o h dsuo o h h od ssc spl

8 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso o sz o d chocs o ; psd hs sco X X X spl do spl o dsuo wh pd d cd v y?? d 4 spcvly dsy uco o X X X do h od sscs od o hs spl W ow v h poly X : sy : d h os o X : Tho h sus o X : psd Th poly dsy uco o X : s skwss d kuoss o h dsuo o h v y : [ ] [ ] B 5 wh d h pd d cd o h dsuo v y 3 d 4 spcvly d B s h uco sc < < o > y us h ol ss pso o [ ] v y w hv [ ] [ ] æö è ø æö : è ø susu o 3 d 4 o 7 w c pss h sy E k s l coo o h h X : [ ] 6 7 Tho h sus o skwss d kuoss o h dsuo o k h ody o o h h od sscs X : k os o h dsuo wh d shp ps X : c clculd 4 s Sus d Whd s Sus Esos I hs sco w povd h sso sd hod sos o h ukow ps o h dsuo whch ws olly susd y Sw k d Wlso [7] o s h ps o dsuos I c usd so oh css lso Suppos Y Y s do spl o sz o dsuo uco d suppos Y ; dos h odd spl Th poposd hod uss h dsuo o o spl o sz w hv Y Y Y E k dcovy Y k ;o < k s Johso Koz d Blksh [] Us h pcos d h vcs wo vs o h ls sus hods c usd Mhod s Sus Esos O h sos y z æ Y è ö ø wh spc o h ukow ps Tho cs o dsuo h ls sus sos o d sy SE SE d SE spcvly c od y z 37

9 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No wh spc o d Mhod Whd s Sus Esos Th whd ls sus sos c od y z ø ö è æ Y w wh spc o h ukow ps wh Y w Tho cs o dsuo h whd ls sus sos o d sy WSE WSE d WSE spcvly c od y z w wh spc o h ukow ps oly 5 Mu klhood Esos I hs sco w cosd h u lklhood sos ME s o dsuo T od o s h ps d o dsuo l do spl o sz o h h lklhood uco c w s Õ ø ö è æ Õ 8 By ccuulo k loh o uo 8 d h lo lklhood uco c w s lo lo lo lo lo 9 D lo wh spc o ch p d d s h sul uls o zo w o u lklhood ss Th pl dvvs o lo wh spc o ch p o h sco uco s v y ø ö è æ U lo lo lo wh lo

10 Mhcl Thoy d Modl wwwso ISSN 4584 Pp ISSN 55 Ol ol3 No lo lo y ' d lo lo y ' wh y s h d uco By solv hs ol sys o uos hs soluos wll yld h M sos o d o h h ps lzd dly dsuo pd ll h scod od dvvs s Thus w hv h vs dspso s v y ø ö è æ ø ö è æ ø ö è æ N : E 3 wh ' B A y l l l l A ' l l Y Y y B D C

11 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso C l D y l l l l l Y Y y l y l l Y Y l y l y l ' ' y y y l l l l Y l By solv hs vs dspso hs soluos wll yld sypoc vc d covcs o hs M sos o d Us w ppo % codc vls o d dd spcvly s ± z ± z d ± z wh z s h upp h pcl o h sdd ol dsuo Us R w c sly copu h Hss d s vs d hc h sdd os d sypoc codc vls W c copu h zd uscd d scd lolklhood ucos o cosuc h lklhood o R s ssc o s o so h w lzd dly suodls o pl w c us h R s ssc o chck whh h w lzd dly dsuo o v d s s ssclly supo o h dly dsuo I y cs hypohss ss o h yp H : vsus H : ¹ c pod us R s I hs cs h R s ssc o s H vsus H s 4

12 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso w l ; l ; wh ĵ d s sypoclly s dsud s Th R s cs H w ck wh > ; ĵ h MEs ud H d H spcvly Th ssc w c k wh k s h lh o h p vco o s c k ; dos h upp % ul o h c k dsuo 6 Applcos I hs sco w us wo l d ss o show h h dly dsuo c odl h o sd o h dly dsuo D s : Th d s v Tl pss ucsod d s cospod o sso s ohs o do spl o 8 ldd cc ps pod d W []: Tl : Th sso s ohs o ldd cc ps D s : Th ollow d ps h suvvl s dys o 7 u ps cd wh vul ucl cll osvd d pod y Bkdl [] Th d s ollows: Tl : Th M ss sdd o d olklhood o d s Modl M Ess Sdd Eo dly 4959 ND â Th vc covc o d s s copud s l I o h MEs ud h w lzd dly dsuo æ 3 5ö è5 4 9ø 4

13 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso Thus h vcs o h ME o d s v v 7 d v 95 Tho 95% codc vls o d [3 5][5743] d [9876 6] spcvly I od o cop h wo dsuo odls w cosd c lk l AIC Akk oo co AICC cocd Akk oo co BICBys oo co d KSKolooovSov s o h d s Th dsuo cospods o sll l AIC d AICC vlus: k k AIC k l AICC AIC k wh I BIC k * lo l dk S sup s pcl dsuo uco s coulv dsuo uco k s h u o ps h sscl odl h spl sz d l s h zd vlu o h lolklhood uco ud h cosdd odl s Tl 3: Th AIC AICC BIC d KS o h odls sd o d s Modl AIC AICC BIC KS dly ND Th R s ssc o s h hypohss : w 3539 > 599 c ;5 so w c h ull hypohss H vsus : / Ú H o d s / u 4: Esd dss o h odls o d s 4

14 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso u 5: Esd cuulv dss o h odls o d s u 6: PP plos o d dly d h ND o d s 43

15 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso Tl 4: Th M ss sdd o d olklhood o d s Modl M Ess S Eo l dly ND â Th vc covc copud s l I o h MEs ud h dly dsuo o d s s Thus h vcs o h ME o 37 [98 9][586] d [6538 3] spcvly æ ö è ø d s 39 v 39 v d v Tho 95% codc vls o d Modl Tl 5: Th AIC AICC BIC d KS o h odls sd o d s l AIC AICC BIC KS dly ND Th R s ssc o s h hypohss : w 5493 > 599 c ;5 H vsus : / Ú H o d s s / so w c h ull hypohss Tls d 4 shows p MEs o ch o o h wo d dsuos o d s d Tls 3 d 5 shows h vlus o lo AIC AICC BIC d KS vlus Th vlus Tls 3 d 5 dc h h w lzd dly dsuo s so copo o oh dsuo usd h o d s d d s A dsy plo cops h d dss o h odls wh h pcl hso o h osvd d 3 d 5 Th d dsy o h w lzd dly odl s clos o h pcl hso h h s o h dly odl 44

16 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso u 7: Esd dss o h odls o d s u 8: Esd cuulv dss o h odls o d s 45

17 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso u 9: PP plos o d dly d h ND o d s 7 Cocluso H w popos w odl h soclld h w lzd dly dsuo whch ds h dly dsuo h lyss o d wh l suppo A ovous so o lz sdd dsuo s cus h lzd o povds l lly odll l d W dv psos o h os d o h o uco Th so o ps s ppochd y h hod o u lklhood lso h oo s dvd W cosd h lklhood o ssc o cop h odl wh s sl odl Two pplcos o h w lzd dly dsuo o l d show h h w dsuo c usd u cvly o povd s h h dly dsuo Rcs Bkouch HS AlZh BM AlSho AA Mch AA ouzd A dd dly dsuo J Ko S Soc Bkdl T 96 Acuso o Rssc u Ps cd wh D Doss o ul Tucl Bcll Ac Joul o Hy Dz E Od EC Th dsc dly dsuo Pops d Applcos J S Copu Sul k uss d k Posch M sdul l: hoy d pplcos 985 Tchcl Rpo 8578 Noh Col S Uvsy d lod S Uvsy Tllhsslod Ju 985 hy ME Ah B d Ndh S 8 dly dsuo d s Applcos Mh Copu Sul 784: hy ME d AlMu DK 9 Eso hods o h dsc Posso dly dsuo J S Copu Sul 79:9 hy ME d AlMu DK 8 Szsd Possodly dsuo d s Applcos Mo I J S XI3: 99 3 hy ME AlMu DK d Ndh S 8c Zoucd Possodly dsuo d s Applcos Mh Copu Sul 793:7987 hy ME Alll AlMu DK d Huss H A A wo p whd dly dsuo d s pplcos o suvvl d Mh Copu Sul

18 Mhcl Thoy d Modl ISSN 4584 Pp ISSN 55 Ol ol3 No3 3 wwwso up RD d Kudu D 999 lzd pol dsuos Ausl d Nw Zld Joul o Sscs Johso N Koz S d BlkshN 995 Couous Uv Dsuo d do Nw Yok Wly ET d W JW3Sscl Mhods o Suvvl D Alyss 3d dwly NwYok dly D 958 ducl dsuos d Bys ho J Royl S Soc Ss B :7 Mzuchl J d Achc J A Th dly dsuo ppld o cop sks l d Copu Mhods Pos Bod 4:889 R S d Msh A 3 A us dly dsuo Ac Joul o Mhcs d Copu Scc Rsch Sk M 97 Th dsc Possodly dsuo Bocs pp 4549 Sw J ks d Wlso J 988 s sus so o dsuo uco Johso s slo sys Joul o Sscl Copuo d Sulo Zkzdh H Dol A lzd dly dsuo J Mh E 3:35 Z M7 Iuly cuv d uly d sd o h os w low d upp hc s Ssc Applczo 4: 3 7 Wll H S 948 Alyc Thoy o Coud cos Nw Yok: Chls 47

Convergence tests for the cluster DFT calculations

Convergence tests for the cluster DFT calculations Covgc ss o h clus DF clculos. Covgc wh spc o bss s. s clculos o bss s covgc hv b po usg h PBE ucol o 7 os gg h-b. A s o h Guss bss ss wh csg s usss hs b us clug h -G -G** - ++G(p). A l sc o. Å h c bw h