Inferences of Type II Extreme Value. Distribution Based on Record Values

Transcription

1 Applied Matheatical Sciece, Vol 7, 3, o 7, IKARI td, www-hikarico Ierece o Tpe II tree Value Ditributio Baed o Record Value M Ahaullah Rider Uiverit, awreceville, NJ, USA Copright 3 M Ahaullah Thi i a ope acce article ditributed uder the Creative Coo Attributio icee, which perit uretricted ue, ditributio, ad reproductio i a ediu, provided the origial work i properl cited Abtract It i kow that i the aiu or iiu o idepedet ad ideticall ditributed rado variable whe tadardized coverge a ted to iiit to o-degeerate ditributio, the it to coverge to oe the three etree value ditributio Tpe II etree value ditributio i oe o the I thi paper we will coider the record value o Tpe Ii etree value ditributio Soe ditributioal propertie o the record value o thi ditributio will be give Baed o thee propertie oe recurrece relatio o the oet ad a characterizatio o the Tpe II etree value ditributio will be preeted Itroductio Suppoe that,, i a equece o idepedet ad ideticall ditributed rado variable with cuulative ditributio uctio et Y = a i{,,, } or > We a j i a upperlower record value o {, >}, i Y j > < Y j-, j > B deiitio i a upper a well a a lower record value The idice at which the upper record value occur are give b the record tie {U}, >, where U = i{j j>u-, j > U-, >} ad U = We will deote a the idice where the lower record value occur B our auptio U = = The ditributio o U or doe ot deped o

2 357 M Ahaullah A rado variable with locatio paraeter µ ad cale paraeter i aid to have tpe II etree value ditributio i the cuulative ditributio uctio cd o i give b or e,,,,, ad the correpodig probabilit deit uctio pd i e or variou ditributioal propertie o etree value ditributio ee Reick 987, Kotz ad Nadaraja Gedeko 943, Galabo 987, iher ad Tipette 96, ad Ahaullah ad Nevzorov I thi paper we will coider record o tpe II ditributio Mai Reult We will coider i thi paper the lower record value Ma propertie o the lower record value equece ca be epreed i ter o, where = - l ad d h = ere 'l' i ued or the atural logarith I we deie a the pd o 9 or >, the we have ee Ahaullah4 = or Tpe II etree value ditributio e

3 Ierece o tpe II etree value ditributio 357 where œ<,, It ca be how that W W W d 3 where W,W,,W are idepedet ad ideticall ditributed with cd a = e ad d deote the equalit i ditributio Uig 3, we obtai,,,, 3 ad Var = [ or, k k Uig 3, we have or <, Y = e e / /

4 357 M Ahaullah weobtai, ad v g Subtituti u dudv v e v u u Y, B Thu ] [ { Cov We ca write Cov where ad Recurrece Relatio Betwee Moet We will aue without lo o geeralit ad To prove the et two theore, we ue the relatio

5 Ierece o tpe II etree value ditributio 3573 Theore or r,,, r r r r Proo r r l r l r l r Collorar r l r - [ r r ] r [ ] e, = Theore or,, ad r / r [ r r ] r or,, r [ r r ]

6 3574 M Ahaullah Proo I r r, where d I d ` or -+ I d or d Thu ] [ r r r or > +, d I I d d

7 Ierece o tpe II etree value ditributio 3575 Thu r [ r [ r ] A Characterizatio Theore et { i,, i=,, } be a equece o idepedet ad ideticall ditributed rado variable with abolutel cotiuou with repect to ebegue eaure ditributio uctio with = ad Var - i, The the ollowig two tateet are equivalet a = ep - -, >, > b Vr - -, b,, - - where b i a cotat idepedet o Proo ro 3, we have d W Thu i idepede t o ece a b et Y -, the Var , b,, I equivalet to Var Y, b,, Y U Y where b i idepedet o ad Y U ad Y U- are the upper record value ro the equece Y i, i=,, Thu Sice Y -, the Y Uk d k orall k,, b Z YU Z YU

8 3576 M Ahaullah where Z Y U Y U, = z * d * z z * * z dz ad Z Y U = 4 = z * d * z * * z dz, 5 where * - * ad *i the cd o Y i Subtitutig G = G*, we have o ipliicatio G * i the pd o Y i z * z dz ad deotig G r a the r th derivative o * z dz, G * ad G 3 * Writig 4 ad 5 i ter o G ad G r, we have, G {G r } - - { G G - } = b or all > 6 Dieretiatig 6 with repect to ad ipliig, we obtai G 3 {G } [ G - G G ] = 7 Sice G 3 or all >, we ut have {G } - G G =, 8 ie d { G G - } =, or all > 9 The olutio o 9 i G = a e -c, > where a ad c are arbitrar cotat ece * = G = ac e-c, > Sice * i a ditributio uctio o Y i with = ad Var Y i =, it ollow that * e Now or all >,

9 Ierece o tpe II etree value ditributio 3577 P P P P Y e Thi coplete the proo Reerece [] M Ahaullah, Record Value-Theor ad Applicatio, Uiverit Pre o Aerica, aha, MD, 4 [] M Ahaullah ad V B Nevzorov, Ordered Rado Variable, Nova Sciece Publiher Ic, New York, NY, [3] R A iher ad C Tipette, iitig or o the requec ditributio o the larget or allet eber o a aple Proc Cabridge Philo Soc 4 98, 8-9 [4] J Galabo, The Aptotic Theor o tree Order Statitic, Robert Krieger Publihig Co, Malabar, 987 [5] B Gedeko, Sur la Ditributio iite du Tere Maiu d'ue Serie Aletoie, A Math, , [6] S Kotz ad S Nadaraja, tree Value Ditributio- Theor ad Applicatio Iperial College Pre,

10 3578 M Ahaullah [7] S Reick, tree Value, Regular Variatio ad Poit Procee, Spriger- Verlag, New York, NY, 987 Received: March 3, 3