EXPECTATION IDENTITIES OF GENERALIZED RECORD VALUES FROM NEW WEIBULL-PARETO DISTRIBUTION AND ITS CHARACTERIZATION

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1 Joral of Statstcs: Advaces Theor ad Applcatos Vole 8, Nber 2, 27, Pages 87-2 Avalable at DOI: EXPECTATION IDENTITIES O GENERALIZED RECORD VALUES ROM NEW WEIBULL-PARETO DISTRIBUTION AND ITS CHARACTERIZATION M A R KHAN, R U KHAN ad M A KHAN 2 Departet of Statstcs ad Operatos Research Algarh Msl Uverst Algarh-222 Ida e-al: are@redffalco 2 Japra Isttte of Maageet Lcow-226 Ida Abstract I ths paper, we cosder New Webll-Pareto dstrbto, was recetl trodced b Nasr ad Lgterah [6] Eact epressos as well as soe recrrece relatos for sgle ad prodct oets of geeralzed pper record vales -th pper record vales are derved rther, varos dedctos ad partclar cases are dscssed At the ed, the characterzato reslts based o codtoal epectato ad recrrece relatos are preseted ad soe coptatoal wor also carred ot 2 Matheatcs Sbect Classfcato: 62G3, 62E, 6E5 Kewords ad phrases: order statstcs, geeralzed record vales, -th record vales, New Webll-Pareto dstrbto, sgle oets, prodct oets, recrrece relatos, codtoal epectato ad characterzato Receved October 25, Scetfc Advaces Pblshers

2 88 M A R KHAN et al Itrodcto The statstcal std of record vales a seqece of depedet ad detcall dstrbted d cotos rado varables was frst carred ot b Chadler [2] Dzbdzela ad Kopocńs [3] have geeralzed the cocept of record vales of Chadler [2] b rado varables of a ore geeralzed atre ad called the the -th record vales Later, Mol ad Thoas [4] call the record vales defed b Dzbdzela ad Kopocńs [3] also as the geeralzed record vales, sce the r-th eber of the seqece of the ordar record vales s also ow as the record statstcs Applcatos of r-th record vale Settg, we obta ordar -th record vales ca be fod the lteratre, for stace, see the eaples cted Kaps [9] or Daela ad Raqab [4] relablt theor Sppose that a techcal sste or pece of eqpet s sbect to shocs, eg, peas of voltages If the shocs are vewed as realzatos of a d seqece, the the odel of ordar records s adeqate If t s ot the records theselves, bt secod or thrd vales are of specal terest, the the odel of -th record vales s adeqate Whe record vales theselves are vewed as otlers, the the secod or thrd largest vales are of specal terest Record statstcs are appled estatg stregth of aterals, predctg atral dsasters, sport acheveets, etc or statstcal ferece based o ordar records, seros dffcltes arse f epected vales of ter arrval te of records s fte ad occrreces of records are ver rare practce Ths proble s avoded oce we cosder the odel of -th record statstcs Let { X, } be a seqece of d rado varables wth dstrbto fcto df ad probablt dest fcto pdf f The -th order statstc of a saple X, X2,, X s

3 EXPECTATION IDENTITIES O GENERALIZED 89 deoted b X : or a fed postve teger, we defe the seqece { U, } of -th pper record tes of { X, } as follows: U, { U > U : X : > X } : U U The seqece {, }, where X s called the seqece of U -th pper record vales or geeralzed pper record vales of { X, } Note that for, we wrte XU,, whch are the pper record vales of { X, } as defed Ahsallah [] Moreover, we see that ad X, X2,, X X: The pdf of ad the ot pdf of Dzbdzela ad Kopocńs [3]; Grdzeń [6] ad are gve b, l f f!, ad f,, f l!! [ l l ] [ ] f, <, <, 2, 2 where or soe recet developets o geeralzed pper record vales wth specal referece to those arsg fro epoetal, Gble, Pareto, geeralzed Pareto, Brr, Webll, Gopertz, Maeha, epoetal-

4 9 M A R KHAN et al Webll, odfed Webll, ad addtve Webll dstrbtos, see Grdzeń ad Szal [7], Pawlas ad Szal [7, 8, 9], Mol ad Thoas [4, 5], Kha et al [3], Kha ad Kha [], ad Kha et al [2], respectvel I ths wor, we al focs o the std of geeralzed pper record vales arsg fro the New Webll-Pareto dstrbto A rado varable X s sad to have a New Webll-Pareto dstrbto Nasr ad Lgterah [6], f ts pdf s of the for f e, < <, >,, >, 3 wth the correspodg df e, < <, >,, > It s eas to see that 4 f 5 Nasr ad Lgterah [6] poted ot the New Webll-Pareto dstrbto provdes greater fleblt the odellg of lfete data The reslts, o real lfe data copared wth other ow dstrbtos, revealed that ths dstrbto provdes a better ft for odellg real lfe data The a propertes of the New Webll-Pareto dstrbto are as follows: If, the falre rate s costat Ths aes New Webll- Pareto dstrbto stable for odellg sste or copoets wth costat falre rate If >, the hazard rate s a creasg fcto of, whch aes the New Webll-Pareto dstrbto stable for odellg copoets that wears faster wth te

5 EXPECTATION IDENTITIES O GENERALIZED 9 If <, the falre rate s a decreasg fcto of, whch aes the New Webll-Pareto dstrbto stable for odellg copoets that wears slower wth te The relato 5 wll be eploted ths std to derve eplct epressos ad soe recrrece relatos for the oets of geeralzed pper record vales fro the New Webll-Pareto dstrbto 2 Relatos for Sgle Moets Here we wll derve the eact epressos ad recrrece relatos for sgle oets of geeralzed pper record vales fro the New Webll-Pareto dstrbto rst we descrbe the eact epresso for sgle oets of geeralzed pper record vales the followg theore: Theore 2 or the dstrbto gve 4 a postve teger, for, ad,, E Γ[ ]! 2 Proof ro ad 5, we have E [ l ] [ ] d 22! Settg t l 22, we fd that E t t e dt, 23! ad hece the reslt gve 2 Corollar 2 The eact epresso for sgle oets of pper record vales fro the New Webll-Pareto dstrbto has the for

6 92 M A R KHAN et al U E E X Γ[ ]! Nercal coptatos for the frst for oets of pper record vales fro New Webll-Pareto dstrbto for arbtrar chose vales of,, ad varos saple sze, 2,, 5 are gve Table 2 Table 2 rst for oets of pper record vales, 2, 2, 2, E X 2 E X 3 4 E X E X E X 2 E X 3 4 E X E X , 2, 2 2, 2, 2 E X 2 E X 3 4 E X E X E X 2 E X 3 4 E X E X Now, we obta the recrrece relatos for sgle oets of geeralzed pper record vales fro the New Webll-Pareto dstrbto the followg theore: Theore 22 or the dstrbto gve 4 a postve teger, for, ad,, E E E 24

7 EXPECTATION IDENTITIES O GENERALIZED 93 Proof ro ad 5, for ad,,, we have E [ l ] [ ] d 25! Now 24 ca be see b otg that vew of Kha et al [2] E E [ l ] [ ] d! Rear 2 Settg 24, we get the relato for sgle oets of geeralzed pper record vales fro the Webll dstrbto, whch verf the reslts obtaed b Pawlas ad Szal [7] Pttg,, 24, we dedce the recrrece relato for sgle oets of -th pper record vales fro the epoetal dstrbto, establshed b Pawlas ad Szal [8] Pttg 2, 24, the recrrece relato for sgle oets of -th pper record vales s dedced for the Ralegh dstrbto as obtaed b Kha et al [3] Corollar 22 The recrrece relato for sgle oets of pper record vales fro the New Webll-Pareto dstrbto has the for EX U EX U EX U 3 Relatos for Prodct Moets Ths secto cotas the eplct epressos ad recrrece relatos for prodct oets of geeralzed pper record vales fro the New Webll-Pareto dstrbto We shall frst establsh the eplct epresso for the prodct oets of geeralzed pper record vales the followg theore: Theore 3 or the dstrbto gve 2 a postve teger, for 2 ad,,,

8 M A R KHAN et al 94 [ ]!! E Γ 3 Proof ro 2, we have [ ] f E l!! [ ] l l dd f 32 O epadg [ ] l l boall 32, we get [ ] E!!, l d I f 33 where l 2 d f I 34 B settg t l 34, we obta dt t I 2 l l O sbstttg the above epresso of I 33, we fd that

9 EXPECTATION IDENTITIES O GENERALIZED 95 [ ] E!! l d f 35 Aga b settg z l 35 ad splfg the resltg epresso, we get [ ], dz e z A E z where,!! A ad hece the reslt gve 3 Idett 3 or, <!!! 36 Proof Pttg 3, we get the reqred reslt Rear 3 At 3, we have E!! Γ 37 Mag se of 36 37, we fd that,! Γ E

10 96 M A R KHAN et al whch s the eact epresso for sgle oets fro the New Webll- Pareto dstrbto, as obtaed 2 Corollar 3 The eact epresso for prodct oets of pper record vales fro the New Webll-Pareto dstrbto has the for U [ E ] E X X U!! Γ[ ] [ ] The followg theore gves the recrrece relatos for prodct oets of geeralzed pper record vales ad Theore 32 or the dstrbto gve 4 ad,,,, [ ] [ E E E ], 38 ad for 2,,,, [ ] [ E E ] [ E ] 39 Proof ro 2 ad 5, we have [ E ] [ l ]!! [ l l ] [ ] dd f 3 3 ca be proved vew of Kha et al [2] b otg that [ ] [ E E ]

11 EXPECTATION IDENTITIES O GENERALIZED 97!! [ l ] [ l l ] [ ] dd f Proceedg a slar aer for the case, the recrrece relato gve 38 ca easl be establshed Oe ca also ote that Theore 22 ca be dedced fro Theore 32 b pttg Rear 32 Settg 39, we get the recrrece relato for prodct oets of geeralzed pper record vales fro the Webll dstrbto, whch verf the reslts obtaed b Pawlas ad Szal [7] Pttg,, 39, we dedce the recrrece relato for prodct oets of geeralzed pper record vales fro the epoetal dstrbto, establshed b Pawlas ad Szal [8] v Pttg 2, 39, the recrrece relato for prodct oets of geeralzed pper record vales s dedced for the Ralegh dstrbto as obtaed b Kha et al [3] Corollar 32 The recrrece relatos for prodct oets of pper record vales fro New Webll-Pareto dstrbto has the for U U U U E X X E X X E X X U U 4 Characterzatos Theore 4 a postve teger ad let be a o-egatve teger, a ecessar ad sffcet codto for a rado varable X to be dstrbted wth f gve b 3 s that E E E, 4 for, 2, ad

12 98 M A R KHAN et al Proof The ecessar part follows edatel fro 25 O the other had f the recrrece relato 4 s satsfed, the o sg, we have! 2! [ l ] [ ] f d 2 [ l ] [ ] f d! [ l ] [ ] f d 42 Itegratg the left had sde 42 b parts ad splfg the resltg epresso, we fd that! [ l ] [ ] f d 43 Now applg a geeralzato of the Mütz-Szász Theore see, for eaple, Hwag ad L [8] to 43, we obta f, whch proves that f has the for as 5 Corollar 4 Uder the assptos of Theore 4 wth, the followg eqato: E E,, 2, characterze the New Webll-Pareto dstrbto Rear 4 If, we obta the followg characterzato of the New Webll-Pareto dstrbto:

13 EXPECTATION IDENTITIES O GENERALIZED 99 EX EX U U,, 2, Theore 42 Let X be a o-egatve rado varable havg a absoltel cotos df wth ad for all >, the [ ] E ξ e l l, l,, 44 f ad ol f e, < <, >,, >, where ξ e Proof ro 2 ad, we have [ E ξ ]! e [ ] l l f d 45 B settg e fro 4 45, we have e [ E ξ ] e! l d 46 We have Gradshte ad Rzh [5], p-55 µ v Γµ l d, µ >, v > µ v 47 O sg 47 46, we have the reslt gve 44

14 M A R KHAN et al To prove sffcet part, we have! e [ ] l l [ ] f d [ ] g, 48 where g e Dfferetatg 48 both sdes wth respect to, we get f 2! e 2 [ ] l l [ ] f d g [ ] g [ ] f, or g Therefore, [ ] f g [ ] g [ ] f f g [ g g ] Kha et al, 49 where g e, g g e Itegratg both sdes of 49 wth respect to betwee,, the sffcec part s proved

15 EXPECTATION IDENTITIES O GENERALIZED Refereces [] M Ahsallah, Record Statstcs, Nova Scece Pblshers, New or, 995 [2] K N Chadler, The dstrbto ad freqec of record vales, J Ro Statst Soc Ser B 4 952, [3] W Dzbdzela ad B Kopocńs, Ltg propertes of the -th record vale, Appl Math Warsaw 5 976, 87-9 [4] K Daela ad M Z Raqab, Sharp bods for epectatos of -th record creets, Ast N Z J Stat 46 24, [5] I S Gradshte ad I M Rzh, Table of Itegrals Seres ad Prodcts, Acadec Press, New or, 27 [6] Z Grdzeń, Characterzato of dstrbto of te lts record statstcs as well as dstrbtos ad oets of lear record statstcs fro the saples of rado bers, Praca Dotorsa UMCS Lbl 982 [7] Z Grdzeń ad D Szal, Characterzato of cotos dstrbtos va oets of -th record vales wth rado dces, J Appl Statst Sc 5 997, [8] J S Hwag ad G D L, O a geeralzed oets proble II, Proc Aer Math Soc 9 984, [9] U Kaps, A Cocept of Geeralzed Order Statstcs, B G Teber Stttgart, Gera, 995 [] A H Kha, R U Kha ad M aqb, Characterzato of cotos dstrbtos throgh codtoal epectato of fcto of geeralzed order statstcs, J Appl Probab Stat 26, 5-3 [] M A Kha ad R U Kha, -th pper record vales fro odfed Webll dstrbto ad characterzato, It J Cop Theo Stat 3 26, 75-8 [2] R U Kha, M A Kha ad M A R Kha, Relatos for oets of geeralzed record vales fro addtve Webll dstrbto ad assocated ferece, Stat Opt If Copt 5 27, [3] R U Kha, A Klshrestha ad M A Kha, Relatos for oets of -th record vales fro epoetal-webll lfete dstrbto ad a characterzato, J Egpta Math Soc 23 25, [4] S Mol ad P agee Thoos, O soe propertes of Maeha dstrbto sg geeralzed record vales ad ts characterzato, Braz J Probab Stat 27 23, [5] S Mol ad P agee Thoos, O characterzato of Gopertz dstrbto b propertes of geeralzed record vales, J Stat Theor Appl 3 24, [6] S Nasr ad A Lgterah, The ew Webll-Pareto dstrbto, Pa J Stat Oper Res 25, 3-4

16 2 M A R KHAN et al [7] P Pawlas ad D Szal, Recrrece relatos for sgle ad prodct oets of -th record vales fro Webll dstrbtos ad a characterzato, J Appl Statst Sc 2, 7-26 [8] P Pawlas ad D Szal, Relatos for sgle ad prodct oets of -th record vales fro epoetal ad Gble dstrbtos, J Appl Statst Sc 7 998, [9] P Pawlas ad D Szal, Recrrece relatos for sgle ad prodct oets of -th record vales fro Pareto geeralzed Pareto ad Brr dstrbtos, Co Statst Theor Methods , g