Record Values from Size-Biased Pareto Distribution and a Characterization

Transcription

1 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 Recod Values om Sze-Based Paeto Dtbuto ad a Chaactezato Shakla Bash, Mu Ahmad Asstat Poesso, Kad College o Wome, Lahoe Poesso, Natoal college o Busess Admtato & Ecoomcs (NCBA&E), Lahoe E-mal- shaklabash5@gmalcom Abstact I th pape uppe ecod values om the sze-based Paeto dtbuto(s-bpd) ae studed Seveal dtbutoal popetes o uppe ecod values om the sze-based Paeto dtbuto, cludg pobablty desty ucto(pd), cumulatve dtbuto ucto(cd), momets, etopy, vese/egatve momets, elatos betwee egatve ad postve momets, meda, mode, jot ad codtoal pds, codtoal mea ad vaace, have bee deved The elablty measues o the uppe ecod values om the S-BPD such as suvval ucto, hazad ate ucto, cumulatve hazad ate ucto ad evesed hazad ate ae also dcussed A chaactezato o the S-BPD based o the codtoal epectato o ecod values gve Keywods S-BPD; dtbuto ucto; momets; ecod values; hazad ucto; etopy; mg; cd; pd; chaactezato INTRODUCTION Chadle (95) toduced ecods as sequece o adom vaables such that adom vaable at th place lage (smalle) tha vaable at ( )thplace He called adom vaables as uppe (lowe) ecods a adom sample o sze om some pobablty dtbuto Ate the toducto o the eld, umbe o eseache jumped to th aea o stattcs Shoock (973) has compehesvely dcussed about ecod values ad ecod tme a sequece o adom vaables Ahsaullah (979) has chaactezed the epoetal dtbuto by usg the ecod values Ahsaullah (99) has also deved the dtbutoal popetes o ecods by usg the Loma dtbuto Some momet popetes o the ecods have bee gve by Ahsaullah (99)Balakha ad Ahsaullah (994) have establhed some ecuece elatos sated by the sgle ad double momets o uppe ecod values om the stadad om o the geealzed Paeto dtbuto Ahsaullah (997) deved some popetes ad a chaactezato o uppe ecod values om the classcal Paeto dtbuto Sulta ad Moshe () have obtaed the best lea ubased estmates o the locato ad scale paametes o ecod values om the geealzed Paeto dtbuto Ahsaullah () cosdeed seveal dtbutoal popetes o the uppe ecods om the epoetal dtbuto Based o these dtbutoal popetes, some chaactezatos o the epoetal dtbuto ae also peseted th pape Ahsaullah et al (3) dcussed a ew chaactezato o powe ucto dtbuto based o lowe ecod valuesthe pd o uppe ecod values U R, () Thejot pd o U j ad U y j R R R, j, y j j The codtoal pd o U j U y, y () Ry R y U j j U j! j y F, y (3) Fo j wwwjegsog

2 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 y y U, y (4) F SIZE-BIASED PARETO DISTRIBUTION Whe a vestgato ecods a obsevato by atue accodg to a ceta stochastc model the ecoded obsevato wll ot have the ogal dtbuto uless evey obsevato gve a equal chace o beg ecoded Patl ad Rao (978) eamed some geeal models leadg to weghted dtbutos wth weght uctos ot essetally estcted by uty The esults wee appled to the aalys o data elatg to huma populatos ad wldle maagemet Suoj ad Maya (6) toduced some udametal elatoshps betwee weghted ad uque vaables the cotet o mataablty ucto ad veted epa ate Futhemoe, some chaactezato theoems o specc models such as powe, epoetal, Paeto II, beta, ad Peaso system o dtbutos usg the elatoshps betwee the ogal ad weghted adom vaables was also establhed M ad Ahmad (9) toduced some sze-based pobablty dtbutos ad the geealzatos These dtbutos oe a jog appoach o the poblems whee the obsevatos all the o-epemetal, o- eplcated, ad oadom categoesthey toduced some o the possble uses o sze- based dtbuto theoy to some eal le data A umbe o papes have bee appeaed dug the last te yeas mplctly usg the cocepts o weghted ad sze-based samplg dtbutos w The pobablty desty ucto o weghted Paeto dtbuto obtaed by applyg the weght m m m m,, (5) Whee m o, these specal cases ae amed as sze-based o legth- based dtbuto ad aea-based dtbuto, espectvely We dee the Sze-Based Paeto dtbuto, whe w The pobablty desty ucto o the sze-based Paeto Dtbuto,, (6) The cumulatve dtbuto ucto o momet Paeto dtbuto F (7) I th pape, uppe ecod values om S-BPD have bee deved ad dcussed vaous popetes cludg chaactezato Pevously, o eseach wok has bee doe o weghted dtbutos the cotet o ecod values So t hoped that dgs o th pape wll be useul o eseaches deet elds o appled sceces UPPER RECORD VALUES FROM SIZED-BIASED PARETO DISTRIBUTION (S-BPD),,, Let U U U deote the uppe ecod values ag om the d sze-based Paeto vaables, the usg equatos (6) ad (7), the pobablty desty ucto o the th uppe ecod U gve by l,, () wwwjegsog

3 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 Chat Ttle Chat Ttle (), =, β= (), =, β=3 (), =, β=4 (), =, β=5 (),=3,β= (),=3,β=4 (),=3,β=3 (),=3,β=5 (a),,,3,4, 5 (b), 3,,3,4, 5 Chat Ttle Chat Ttle (), =5, β= (),=5, β=3 (),=5,β=4 (),=5,β=5 (),=7,β= (),=7,β=4 (),=7,β=3 (),=7,β=5 (c), 5,,3,4, 5 (d), 7,,3,4, 5 Fg pd plots o uppe ecod values om S-BPD PROPERTIES I th secto some dtbutoal popetes o the uppe ecod values om the S-BPD have bee deved MOMENTS The th momet o the th uppe ecod value U by usg (), ae E () The mea ad vaace o the uppe ecod values om the S-BPD ae, espectvely Mea = (3) Vaace = 3 3 (4) 3 wwwjegsog

4 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 The mode o the uppe ecod values om sze-based Paeto dtbuto ep mod e ( ) (5) The vese/egatve momets o th uppe ecod values om the sze-based Paeto dtbuto ae E (6) By usg the equato () ad (6), the elato betwee egatve ad postve momets o the th uppe ecod values om S-BPD (7) The momet geeatg ucto o the uppe ecod values om sze-based Paeto dtbuto ENTROPY The etopy o the th ecod values M U om the S-BPD t k t (8) k k! k H l k k k l l k k 4 SURVIVAL AND HAZARD RATE FUNCTION The suvval ucto o uppe ecod values om the S-BPD 4 wwwjegsog (9) 3 CUMULATIVE DISTRIBUTION FUNCTION The cumulatve dtbuto ucto o the uppe ecod values om the S-BPD a whee a s e d, the uppe complete gamma ucto s F, l ()

5 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 S, l () The Hazad ate ucto h l, l () The cumulatve hazad ate ucto H, l l (3) The evese hazad ate ucto l, l (4) S(),= S(), =3 S(), =5 S(), = S(), = S(), =3 S(), =5 S(), =7 (a),,,3,5, 7 (b), 5,,3,5, 7 Fg Suvval Plots o uppe ecod values om S-BPD 5 wwwjegsog

6 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN h(),β=, = h(),β=, =3 h(),β=, =5 h(),β=, = h(),β=5,= h(),β=5,= 3 h(),β=5,= 5 h(),β=5,= 7 (a),,,3,5, 7 (b), 5,,3,5, 7 Fg3 Hazad ate plots o uppe ecod values om S-BPD 3 JOINT AND CONDITIONAL DENSITY FUNCTIONS The jot pobablty desty ucto o U ad U j om S-BPD j l l l y, y y, j, y j j (3) Thus the codtoal pd y j o U j U gve by j y j y y l j, y (3) The mea o the codtoal pd y j E E y U j Vaace o the codtoal pd y U j j j j (33) va j U j U va j y (34) j j 3 j 6 wwwjegsog

7 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN CHARACTERIZATION Usg codtoal pd o L gve L E L L, 3 as gve equato (4), t ca be show that P, 3 L The ollowg theoem gves a chaactezato o the S-PBD usg the above esult Theoem 4 Let, be d adom vaables havg absolutely cotuous (wth espect to Lebesgue measue) cd F wthout loss o geealty F ad F We assume uthe that o 3,, the We wll assume F twce deetal ad let, E I ad P, E L L L, 3 (4) 3 PooUsg Equato (4), we get y y dy 3 F (4) Deetatg both sdes o equato (4), we get y y dy 4F 3 3 (43) Now takg d devatve o both sdes o equato (43), we have Substtutg y F y, y 4F F (44),, the equato (44) educes to y 4y 5 4y The equato (45) the well-kow Eule type equato It has soluto o the om, (45) y whee must saty the equato wwwjegsog

8 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 The oots o, om the above equato ae ad 4 So the solutos ae o the type c ae costats Hece we assume Whee & c y c ad 4 c E ets ad y F F y (46), so we have lm F, lm F (47) The soluto y c sates the codto o equato (47) 4 y c sates the codto o equato (47) 4 F, whch cotadcts the assumpto that, whch cotadcts the assumpto 4,, We must have, ad 5 CONCLUSION I th pape, we developed the dtbuto o uppe ecod values om the sze-based Paeto dtbuto The gaphs show that the dtbuto o uppe ecod values postvely skewed Fo lage values o ad the pd showg peaked ad ght tal loge whle o smalle values o ad the pd latteg We deve the postve ad egatve momet o the uppe ecod values om the sze-based Paeto dtbuto ad developed a elato betwee them The assocated cd, suvval ucto, hazad ucto, etopy, mg, meda, mode, skewess ad kutos have bee deved We deve the jot ad codtoal pobablty dtbuto uctos o th ad jth uppe ecod values om the sze-based Paeto dtbuto ad d out codtoal mea ad vaace o t The cumulatve hazad ate ucto ad evese hazad ate ucto o the ecod values om sze-based Paeto dtbuto have also bee deved The plot o suvval ucto shows that the suvval ucto ceasg o small & ad deceasg ad showg bathtub shape o lage Plot o hazad ucto shows ceasg ted wth = whle deceasg ucto whe cease We hope th pape wll cotbute a valuable cotbuto o the ehacemet o eseach the theoy o ecod values REFERENCES: [] Adamc, LA () Zp, Powe-laws ad Paeto-a akg tutoal Iteet Ecologes aea, eo Palo Alto Reseach Cete, Palo Alto, CA 9434 ( [] Ahsaullah, M (979) Chaactezato o epoetal dtbuto by ecod values, Sakhya, Vol 4, 6 [3] Ahsaullah, M (988) Itoducto to Recod Values G Pess, Needham Heghts, Massachusetts [4] Ahsaullah, M (99) Recod values o Loma dtbuto, Statt Nedeladca, Vol 4(), 9 [5] Ahsaullah, M (99) Recod values o depedet ad detcally dtbuted cotuous adom vaables, Pak J Statt Vol 8(), 9 34 [6] Ahsaullah, M (995) Recod Stattcs, Nova Scece Publhes, USA [7] Ahsaullah, M (997) O the ecod values o the classcal Paeto dtbuto Pak J Statt, Vol, 3(), 9-5 [8] Ahsaullah, M () Cocomtats o Uppe Recod Stattcs o Bvaate Pseudo Webull Dtbuto, J Appl Math Vol 5(), [9] Ahsaullah, M () Some chaactezatos o epoetal dtbuto by uppe ecod values, Pak J Statt, Vol 6(), wwwjegsog

9 Iteatoal Joual o Egeeg Reseach ad Geeal Scece Volume, Issue 4, Jue-July, 4 ISSN 9-73 [] Ahsaullah, M, Shakla, M, ad GolamKba, B M (3) A chaactezato o powe ucto dtbuto based o lowe ecod values PobStat Foum, Vol, 6, 68-7 [] Aold, BC, Balakha, N, ad Nagaaja, HN (99) A Ft Couse Ode Stattcs, Joh Wley ad Sos, New Yok [] Balakha, N ad Balasubamaa, K (995) A chaactezato o geometc dtbuto based o ecod values, J Appl Statt Scece, Vol (), [3] Cadle, KN (95) The dtbuto ad equecy o ecod values, J Roy Statt Soc, Vol4, 8 [4] Gustaso, G & Fasso, A (5) The use o the Paeto dtbuto o actue tasmsvty assessmet, Hydogeology Joual, Vol 4, 5- [5] Johso, NL, Kotz, S, ad Balakha, N (995) Cotuous Uvaate Dtbutos, Vol, Secod edto, Joh Wley & Sos, New Yok [6] M, K A, ad Ahmad, M (9) Sze-based dtbutos ad the applcatos, Pak J Statt, Vol 5(3), [7] Patl, G P, ad Rao, C R (978) Weghted Dtbutos ad Sze-Based Samplg wth Applcatos to Wld le Populatos ad Huma Famles, Bometcs, Vol 34, [8] Sulta, K S, ad Moshe, M E () Recod values om geealzed Paeto dtbuto ad assocated eece, Metka, Vol 5, 5-6 [9] Sulta, KS (7) Recod Values om the Moded Webull Dtbuto ad Applcatos, Iteatoal Mathematcal Foum, Vol,(4), Suoj, S M, ad Maya, S S (6) Some Popetes o Weghted Dtbutos the Cotet o Repaable Systems, Commucatos Stattcs Theoy ad Methods, Vol 35, wwwjegsog