Order statistics from non-identical Standard type II Generalized logistic variables and applications at moments

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1 Amerca Joural of Theoretcal ad Appled Statstcs 05; 4(: -5 Pulshed ole Jauar 3, 05 ( do: 0.648/.atas ISSN: (Prt; ISSN: (Ole Order statstcs from o-detcal Stadard tpe II Geeralzed logstc varales ad applcatos at momets Z. AL-Saar Departmet of Statstcs, Kg Adulazz Uverst, Sceces acult for Grls, Sauda Araa Emal address: To cte ths artcle: Z. AL-Saar. Order Statstcs from No-Idetcal Stadard Tpe II Geeralzed Logstc Varales ad Applcatos at Momets. Amerca Joural of Theoretcal ad Appled Statstcs. Vol. 4, No., 05, pp. -5. do: 0.648/.atas Astract: I ths paper the momet geeratg fucto of order statstcs arsg from depedet o-detcall dstruted (INID Stadard tpe II Geeralzed logstc (SGLII varales s estalshed. A recurrece relato for all momets of all order statstcs arsg from INID SGLII s computed. Specal cases for momets are deduced usg polgamma fucto. Some umercal examples are gve. Kewords: Momets, Nodetcall Order Statstcs, Stadard Tpe II Geeralzed Logstc dstruto, Polgamma ucto. Itroducto The momets of order statstcs (o.s of depedet odetcall dstruted (INID radom varales (r. v., s have ee estalshed lterature three drectos. The frst drecto s tated (Balakrsha, 994, requres a asc relato etwee the proalt dest fucto (p.d.f ad the cumulatve dstruto fucto (c.d.f. Ths techque s referred to ths as dfferetal equato techque (DET. It eales oe to compute all the sgle ad product momets of all order statstcs a smple recursve maer ad the dervato of the momets depeds mal o tegrato parts. The secod techque s estalshed Adelkader ad Barakat wll e referred to ths as (BAT. It s a easer maer to evaluate the momets of (INID. o.s ut ca ol e appled to dstrutos wth (c.d.f the form: ( x λ( x ad we caot evaluate the product momets. The thrd drecto was developed Jamoom ad Alsaar (0 whch s the momet geeratg fucto techque. I ths paper we compute the momets of order statstcs arsg from depedet o-detcall dstruted Stadard tpe II geeralzed logstc radom varales usg the momet geeratg fucto techque. The suect o odetcall order statstcs s dscussed wdel the lterature (Davd ad Nagaraa, 003. (Vaugha ad Veales, 97 deoted the ot p.d.f ad margal p.d.f of order statstcs of (INID radom varales meas of permaet. Applcatos of the prevous two methods are also foud the lterature for several cotuous dstrutos. The (DET was used (Balakrsha, 994 to derve recurrece relatos satsfed e sgle ad product momets of (INID order statstcs for the Expoetal ad rght trucated dstrutos. (Chlds ad Balakrsha, 006 appled (DET to derve the momets of (INID order statstcs for Logstc radom varales. (Mohe Eld et al., 007 appled ths method to derve the momets of (INID order statstcs for several dstrutos. The frst appled of (BAT was (Barakat ad Adelkader, 000 to weull dstruto ad the the method was geeralzed (Barakat ad Adelkader, 003 ad appled to Erlag, Postve Expoetal, Pareto, ad Laplace dstrutos. (Adelkader, 004 ad (Adelkader, 008 used a closed expresso for the survval fucto of Gamma ad Beta dstrutos to compute the momets of (INID (o.s usg ths techque. (Jamoom, 006 appled t to Burr tpe XII radom varales.. Nodetcal Order Statstcs from Stadard Tpe II Geeralzed Logstc Dstruto Let,,, e depedet radom varales havg cumulatve dstruto fuctos,,, ad proalt dest fuctos,,,, respectvel. Let : : : deote the order

2 Z. AL-Saar: Order Statstcs from No-Idetcal Stadard Tpe II Geeralzed Logstc Varales ad Applcatos at Momets statstcs otaed arragg the, creasg order of magtude. The the p.d.f ad the c.d.f of the rth order statstc : ( ca e wrtte as: { } r f ( x ( x f ( x ( x ( r-!( - r! p a a r c r + c ( 0.8 Where deotes the summato over all! p permutatos,,,,,. (Bapat ad Beg, 989 put t the form of permaet as: f ( x per x ( f( x ( x ( r!( r! r r { } ( ( r ( x ( x ( x r p a a a a + ( gure. Graph of c.d.f. of frst (d o.s. from SGL II dstruto for selected, ;, 3. The Momet Geeratg ucto of the r th Nodetcall (o.s from Stadard Tpe II Geeralzed Logstc Dstruto p s all permutatos of (,,, for (,, whch We kow that the dstruto fucto of Stadard tpe II Geeralzed Logstc dstruto ca e wrtte as: satsf < < < ad + + permaet as < < <. Ad usg ( x ( x ( r ( x per, < x < r!(! ( x ( x The p.d.f. ad the c.d.f. of SGL II dstruto are gve as e x f( x, x, > 0 + ( + e x e x ( x, x, 0 ( e x > + Now we ca ota the p.d.f. ad the c.d.f. of the rth INID o.s. of SGLII dstruto from eq ( ad (4 usg (Permaet ad Mathematca Program as ( ( 3 3 ( ( x e f : 3 x x + e x ( ( :3 x ( + e gure. Graph of p.d.f. of frst(d o.s. from SGL II dstruto for selected,, (5 (6 (4 x e ( x, x, 0,,,, ( e x > + See (Balakrsha & Hossa 007. To derve the momets of INID o.s. arsg from ( 6, we eed the followg theorem. 3.. Theorem Let X, X,, X e depedet o detcall dstruted r.v,.s. The momet geeratg fucto of the r th order statstcs, M ( t, for r ad k,, s gve : Where: ( r+ I (7 r r+ ( ( M t t < < < 0 tx I e [ ( x] dx,,,, (8 where (,,, are a permutatos of (,,, for whch < <. Proof: See (Jamoom ad Al-Saar Theorem or r ad k,, I t β( t, t (9 < < < Proof: O a pplg theorem. ad usg ( 6, we get:

3 Amerca Joural of Theoretcal ad Appled Statstcs 05; 4(: -5 3 x tx e I t e dx, x < < < t ( e + x tx e t e dx, < < < x ( e + x( t e t dx, < < < x ( + e Susttutg : I e x x l dx d < < < 0 B usg the kow relato: 0 [ + ] ( t d, p x β( p, q dx; where p ( t, q t p + q ( x + we wll get relato ( 9. ( k ( k ( r+ µ Mr : (0 ( r r+ < < < [ ] [ k] Γ ( t + Γ ( t k k l l 0 t 0 Γ( (0 [ where Γ ] (. s the th dervatve of the gamma fucto. the frst two momets ca e gve the followg relatoshps ( ( µ ( r+ r+ r ψ ( ψ( < < < ( ( ( r µ + r r + < < < ψ ( + ψ ( ψ( ψ( + ψ ( ψ ( + ( Where ψ ( z s PolGamma [z] the logarthmc ( dervatve of the Gamma fucto ad µ s the k th k r : momet of the r th k ( k d o.s. ad M M : ( t k r dt. Corollar. or the case of a sample of IID r.v.,s arsg from SGL II dstruto, the momet geeratg fucto of the r th o.s. ( r s gve ( r + M ( t t ( β( t, t ( r r + ad the k th momet ecomes 4. The Momets of the r th No Idetcall (o.s from Stadard tpe II Geeralzed Logstc Dstruto ( k µ : ( r+ r r+ r k k [ l] [ k l] Γ ( t + Γ ( t l l 0 Γ ( t 0 rom (7 ad (9 the k th momet of the r th o.s. ca e Specal cases: from (: wrtte as ( ( r + µ ( [ ( ( ] r ψ ψ r + ( ( r + µ ( ( ( ( ( ψ + ψ ψ ψ + ψ ( + ψ ( r r + (3 (4 5. Numercal Applcatos The followg examples are compute whe k. Example (: Let ad,,3,4,5 ad 5 (3. Tale shows the results: Tale. The momets µ : arsg from IID SGL tpe II r.v.,s µ : or example whe 3,, M : ti, ti

4 4 Z. AL-Saar: Order Statstcs from No-Idetcal Stadard Tpe II Geeralzed Logstc Varales ad Applcatos at Momets I ( k β( t, t Γ( t. Γ( t Γ( I β( t, t Γ( t. Γ( t Γ( t Γ( t. Γ( t t Γ( t. Γ( t M:( t Γ( Γ( ` : : Γ ( t +. Γ( t tγ ( t +. Γ( t Γ( Γ( µ ( t M ( Example (: Set, ad ( 5, ( 5 Eq (, we ota results Tale : ( µ ψ ψ ( ( : ψ ( ψ( < ψ( ψ( ψ( + ψ( + ( µ ψ ψ ( ( : ψ ( ψ( < ψ( ψ( ψ( + ψ( + Tale. The mea µ of all order statstcs from INID SGL tpe II r.v.,s Example (3: Let X, X, X3 SGLtp II( 0.5,, 3.5 Theorem (. gves rom ( 0 we get: :3 M:3 ( t t ( I 3 ti ti 3 3 M ( t t β( t, t t β( t, t M ( t :3 < 3 < < 3 3 t β( t, + t + t β( t, + t + t β( t, + t 3 3 t β( t, + + t 3 t Γ( t Γ ( + t t Γ( t Γ ( + t t Γ( t Γ ( + t Γ ( + ε Γ ( + Γ ( t Γ( t Γ ( t Γ ( M ( t :3 Γ ( t + Γ ( + t Γ ( t + Γ ( + t Γ ( t + Γ ( + t Γ ( + Γ ( + Γ ( Γ ( t + Γ ( t Γ ( µ ( t M ( ` :3 :3 3 3 µ ψ( ψ( + ψ( + + ψ( + + ψ( + ψ( 3:3 3 3 ψ ψ ψ ψ ( + (.5 (3 ( Tale 3. The mea µ of all order statstcs from INID SGL tpe II r.v.,s. r Cocluso The momet geeratg fucto techque s strogl recommeded for dervato of recurrece relatos of momets of order statstcs from d r.v. s for a other cotuous dstrutos wth cdf the form: (x - ( x e λ. Comparso etwee relatve techques avalale the lterature s also recommeded for future stud. Ackowledgemet I would lke to thak Kg Adulazz Uverst for ecouragg facult staff to partcpate scetfc researches ad motvate them to pulsh scetfc ourals, whch resulted wrtg ths paper. Refereces [] ABDELKADER, Y. (004 Computg the momets of order statstcs from odetcall dstruted Gamma varales wth applcatos It J Math Game Theo Algera, 4. [] ABDELKADER, Y. (008 Computg the momets of order statstcs from depedet odetcall dstruted Beta radom varales. Stat Pap, 49, [3] BALAKRISHNAN, N. (994 Order statstcs from odetcall expoetal radom varales ad some applcatos. Comput. Statstcs Data-Aal, 8, [4] BALAKRISHNAN, N. & HOSSAIN, A. (007 Iferece for the tpe II geeralzed logstc dstruto uder progressve tpe II cesorg. Joural of statstcal computato ad smulato, 77, [5] BAPAT, R. B. & BEG, M. I. (989 Order statstcs from odetcall dstruted varales ad permaets. Sakh a, A, 5, [6] BARAKAT, H. & ABDELKADER, Y. (003 Computg the momets of order statstcs from odetcal radom varales Stat Meth Appl, 3, 5-6.

5 Amerca Joural of Theoretcal ad Appled Statstcs 05; 4(: -5 5 [7] BARAKAT, H. M. & ABDELKADER, Y. H. (000 Computg the momets of order statstcs from odetcall dstruted Weull varales. J. Comp. Appl. Math, 7, [8] CHILDS, A. & BALAKRISHNAN, N. (006 Relatos for order statstcs from o-detcal logstc radom varales ad assessmet of the effect of multple outlers o as of lear estmators. J. Statst. Pla. Iferece, 36, [9] DAVID, H. A. & NAGARAJA, H. N. (003 Order Statstcs, Thrd Edto. Wle, New York. [0] JAMJOOM, A. A. (006 Computg the momets of order statstcs from depedet odetcall dstruted Burr tpe XII radom varales. oural of Mathematcs ad Statstcs,, [] JAMJOOM, A. & AL-SAIARY. A. (0 Momet Geeratg ucto Techque for Momets of Order Statstcs from Nodetcall Dstruted Radom varales. Iteratoal oural of Statstcs ad Sstems, 6, [] MOHIE ELIDIN, M., MAHMOUD, M., MOSHRE, M. & MOHMED, M. (007 O depedet ad odetcal order statstcs ad assocated ferece. Departmet of mathematcs. Caro, A-Azhar Uverst. [3] VAUGHAN, R. J. & VENABLES, W. N. (97 Permaet expressos for order statstcs destes. J. R. Statst. Soc, 34, 08-0.