In J Conem Mah Scences Vol 7 0 no 46 45-5 Inverse Jon Momens of Mulvarae Rom Varables M A Hussan Dearmen of Mahemacal Sascs Insue of Sascal Sudes Research ISSR Caro Unversy Egy Curren address: Kng Saud Unversy Faculy of scence De of sascs oeraons research PO Box 455 Ryadh 45 Saud Araba Absrac Le X be rom varables wh jon dsrbuon funcon X F XX jon momen generang funcon M X X In hs aer he auhor has roosed mehods for dervng nverse jon momens of mulvarae rom varables based on he jon momen generang funcon mgf of Two examles are gven n he bvarae case for X X llusraon Keywords: Inverse; momens; mulvarae; generang bvarae; jon; Gamma Inroducon Inverse momens of rom varables aear n many raccal alcaons [] They may be aled n Sen esmaon os-srafcaon [3] evaluang rsks of esmaors owers of ess [45] They also aear n ceran roblems of relably survval analyss [6] Ths ye of momens s also moran n lfe esng [7] nsurance fnancal mahemacs [8] comlex sysems [9] oher areas Garca Palacos [0] needed o evaluae E [ a X α n ] α > 0 wh X n havng he bnomal dsrbuon B n 0 5 n order o oban coulngs of rom walks on n- dmensonal cubes The calculaons of E [ a X n α ] α > 0 are almos dffcul o calculae her bounds aroxmaons were exlored nsead [37] Cresse ohers [3] showed ha nverse momens are hdden n he mgf of he robably dsrbuon rovded ha he mgf exss They roosed mehods based on mgf for comung nverse momens of unvarae dsrbuons Carryng ou he negraons may no be easy n many cases negraons may no yeld closedform exressons
46 M A Hussan Therefore many auhors looked for good aroxmaons or even numercal soluons o he gven negrals see [4-7] In hs aer he auhors derve new mehods for comung nverse jon momens of nonnegave rv's wh he knowledge of he jon mgf rovded ha he mgf exss These mehods are furher exended o rom varables defned on he dmensonal sace Resuls Jon Inverse Momens Suose ha X X s osve rom varables wh dsrbuon funcon F X X jon momen generang funcon M ex X E Snce X s nverse jon momen of X X s gven by E 0 0 ex - x x d d X 0 0 X df X X - 0 0 0 0 ex x x d d df X X ex- x x df X X d d 0 0 0 0 0 0 M X X d d X X hen he where he nerchange of he order of negraon s subjec o E ex X s negrable over R I s no easy o erform negeraon analycally secally n he mulvarae case however equaon s a mahemacal ool for evaluang nverse momen whch can be solved numercally Generalzaons of equaon may be he k h jon nverse momen of X X he k a smlar way as E E X Γ k k h jon nverse momen of X X ha can be gven n k k 0 0 M X X d d k k X M X X d d 3 0 0 Γ k The above fndngs can be furher exended accordng o he unvarae case roosed earler by Cresse ohers [3] f we le X X be rom varables wh dsrbuon funcon F X X Y sgn x Y sgn x be anoher se of rom varables where sgn z f z 0 f z < 0 hen he s k h k k h jon nverse momens can be gven as follows:
Inverse jon momens 47 E E E X k X 0 0 lm 0 0 k Γ k 0 0 M X X Y Y d d 4 lm X X Y Y d d 0 ṫ 0 k X k 0 0 Γ k lm d d X X Y Y 0 0 wh he resrcon F 0 F 0 j When X X are ndeenden x j x j rom varables we have ha X E[ X ] k E [ ] X k E X k k X E[ X ] E 5 6 E Jon Inverse Cenral Momens Suose ha X X s osve rom varables wh dsrbuon funcon F XX jon momen generang funcon M X X Cresse ohers [3] showed ha b E ax b 0 e M X a d where X s a osve rom varable havng dsrbuon funcon F X mgf M X for all real numbers 0 assumng ha E s fne Thus E a X b a X b df X X 0 0 0 0 0 0 ex b ex a x d d df X X 0 0 0 0 ex b ex a x df X X d d ex 0 0 b M X X a a d d 7 From 7 we can defne he s jon nverse cenral momen of E X X o be X e M X X d d 8 0 0
48 M A Hussan Equaon 8 can be generalzed for he k h jon cenral nverse momen of X o be gven as X k E X k e M X X d d 0 0 9 Γ k or even exended o he kk h jon nverse cenral momens of X X o be as k k E X e M X X d d 0 Γ k wh he resrcon Fx 0 F 0 j j x j As before f we le X X be rom varables wh dsrbuon funcon F X X Y sgn x 0 0 Y sgn x be anoher se of rom varables where sgn z f z 0 f z < 0 hen he s k h k k h jon nverse momens can be gven as follows: E X e E [ ] 0 0 lm d d X X Y Y 0 ṫ 0 X k [ ] k 0 0 e Γ k lm X X Y Y d d 0 0 E X k [ ] k 0 0 e Γ k lm X X Y Y d d 0 0 [ ] When X X are ndeenden we have ha k E X E X 3 Examles k [ ] In hs secon wo examles on he bvarae case wll be llusraed usng he roosed mehods 3
Inverse jon momens 49 Examle : Le X Y have a Kbble Moran bvarae gamma dsrbuon BG α β wh shae arameer α scale arameer β β β [8] hen he Lalace ransform of X Y s gven by β -α L β β β > 0 > 0 where α β β β > 0 β β β > 0 Snce for Y L M XY herefore by we have ha X nonnegave rv's α - β [ ] α β E X Y F α π α ββ csc πα β - ββ ββ β α where F abc z s he Hyergeomerc funcon ab z a a b b z a b z F abcz n n z < c! c c! n c n n! From 3 we have -k Γ α - k k k α k E[ X Y ] d k 0 β β β Γ α Γ -k r Γ α - k r k α k E[ X Y ] d r 0 β β β Γ α Γ k > 0 k > 0r > 0 When β β β The case X Y are ndeenden equaons 6 7 8 gve k - β E[ XY ] β α > -k β E[ XY ] β α > k k > 0 [ α ] [ α α α k ] k -k r β E[ X Y ] β α > max k r [ α α α k ][ α α α r ] Examle: Le X Y have a bvarae weghed exonenal dsrbuon roosed by Al-Muar oher [9] Y BWE λ λ The jon df of X Y s X λ3 λλ x y z f X Y x y λ λ e λ e λ e 3 λ3 z mn{ x y λ λ λ λ3 λ λ λ3 > 0 where } funcon of X Y s gven by M XY - - - λ λ λ Then by 3 we have ha - λλ [ ] λ X X π ln λ λ λ λ The momen generang λ E ln λ ln λ λ d log λ λ where d log z z ln u du u λ d log d log λ λ From 7 8 we have k k [ λ ] λ d k > 0 k -k k E[ XY ] csc k π π λλ λ 0 λ λ λ
50 M A Hussan k k [ λ ] λ d k > 0 r > 0 r -k r k E[ X Y ] csc k λλ λ π π 0 λ λ λ 4 Dscusson Momen generang funcons can descrbe dsrbuons ha are no easly defned Alhough some dsrbuons are naccessble momens nverse momens can be derved hrough mgf s In hs aer he use of mgf s for dervng nverse jon momens of mulvarae rom varables s shown In many cases erformng negraon on he jon mgf analycally s no easy raher comlex; neverheless he resuls of hs aer can be of grea neres for evaluang he negrals usng aroxmaons or even numercally Furher work s needed esecally for he nverse cenral momen case References [] X Sh Y Wu Y Lu Noe on asymoc aroxmaons of nverse momens of nonnegave rom varables Sas Probab Le 80 00 60-64 [] DA Wooff Bounds on recrocal momens wh alcaons develomens n Sen esmaon os-srafcaon J R Sa Soc Ser B 47 985 36-37 [3] AO Penger Shar mean-varance bounds for Jensen-ye nequales Sas Probab Le 0 990 9-94 [4] E Marcnak J Wesolowsk Asymoc Euleran exansons for bnomal negave bnomal recrocals Proc Amer Mah Soc 7 999 339-3338 [5] T Fujoka Asymoc aroxmaons of he nverse momen of he noncenral ch-squared varable J Jaan Sas Soc 3 00 99-09 [6] RC Gua O Akman Sascal nference based on he lengh-based daa for he nverse Gaussan dsrbuon Sascs 3 998 35-337 [7] W Mendenhall EH Lehman An aroxmaon o he negave momens of he osve bnomal useful n lfe-esng Technomercs 960 7-4
Inverse jon momens 5 [8] CM Ramsay A noe on rom survvorsh grou benefs ASTIN Bull 3 993 49-56 [9] A Jurlewcz K Weron Relaxaon of dynamcally correlaed clusers J Non-Crys Solds 305 00 - [0] NL Garca J L Palacos On mxng mes for srafed walks on he d- cube Rom Srucures Algorhms 0 00 540-55 [] MT Chao WESrawderman Negave momens of osve rom varables J Amer Sas Assoc 67 97 49-43 [] NL Garca J L Palacos On nverse momens of nonnegave rom varables Sas Probab Le 53 00 35-39 [3] N Cresse A S Davs FLeory G E Polcello II The momen generang funcon negave ower momens he Amercan Sascan 35 98 No 3 48-50 [4] M Kaluszka A Okolewsk On Faou-ye lemma for monoone momens of weakly convergen rom varables Sas Probab Le 66 004 45-50 [5] S-HHu G-J Chen X-J Wang E-B Chen On nverse momens of nonnegave weakly convergen rom varables Aca Mah Al Sn 30 00736-367 [6] T-J Wu X Sh B Mao Asymoc aroxmaon of nverse momens of nonnegave rom varables Sas Probab Le 79 009 366-37 [7] S H Sung On Inverse Momens for a Class of Nonnegave Rom Varables Journal of Inequales Alcaons Hndaw Publshng Cororaon Arcle ID 83767 00 3 ages [8] S Koz N Balakrshnan N L Johnson Connuous Mulvarae Dsrbuons Models Alcaons Wley seres n robably sascs Volume of Connuous Mulvarae Dsrbuons John Wley & Sons 004 [9] D K Al-Muar M E Ghany D Kundu A new bvarae dsrbuon wh weghed exonenal margnals s mulvarae generalzaon Sa Paers 5 0 9-936
5 M A Hussan Receved: Seember 0