Recurrence Relations for the Product and Single Moments of k th Lower Record Values of Inverse Weibull Distribution

Transcription

1 ISSN Jonal o Statitic Volme 7 2 pp Recence Relation o the Podct Single Moment o th owe Recod Vale o Invee Weibll Ditibtion Abtact Mhammad Aleem In thi pape we deal with the ecence elation o the podct atio ingle moment o the th lowe ecod vale om Invee Weibll Ditibtion. Kewod Moment Recence elation th owe ecod vale Invee Weibll ditibtion. Intodction Fo a ied we deine the eqence l l2 --- o the th lowe ecod time o the eqence a ollow: l = o n= n min j n : with n. : n : J i : n The eqence { n n>=} i called the eqence o the th ecod vale o the above eqence ee Bienie Sznal 22.Note that o = we have n= :n =R n ae lowe ecod vale o the eqence i i n= :n =ma --- n. The aim o thi pape i to give a geneal pocede o obtaining ecence elation o ingle podct moment o th lowe ecod vale Z n. We obtain geneal elation o the moment o the th lowe ecod vale om Invee Weibll Ditibtion e thee to give ecence elation o the moment. The pobabilit denit nction o an Invee Weibll Ditibtion i given b: K Depatment o Statitic Ilamia Univeit Bahawalp Paitan daleemib@hotmail.com

2 48 Aleem m = ep. m m whee > m>. And the coeponding ditibtion nction i: F = ep - m.2 whee > m> I we pt m= it edce to the Invee Eponential Ditibtion. I we pt m=2 it edce to the Invee Raleigh Ditibtion. Some wo ha been done on Invee Raleigh Ditibtion b Ghaaph 993 Majee Mait 996 Majee Saen 984 Voda 972 o the podct ingle moment o lowe ecod vale o Invee Weibll Ditibtion ee Aleem 25 o Recence Relation o Qotient Moment o the Weibll Ditibtion b ecod vale ee hang 27. m et p = ing thi elation in..2 we obtain: m F =..3 p 2. Podct Moment The th lowe ecod vale ae epeented b l l2 --- ln then the joint pd o > i given a:. = S ] whee = -!! = - In F < F < d h = - d ] < < < h F] - 2.

3 Recence Relation o the Podct Single Moment o K th owe Recod Vale o Invee Weibll Ditibtion 49 I g i a Boel meaable nction om R 2 to R then: E } { g = g ] ] h F] - dd 2.2 Theoem 2. Fo the Ditibtion nction F in.2 we have: E } { g = } { E - } { E } { E 2.3 whee. p g = Poo: Uing.3 in 2.2 we have: E } { g = ] ] F] - d.d 2.4 Upon integating the R..S. o 2.4 we get: d d F h 2 ] ] - d d F h 2 ] ] - - d d F h ] ] E } { g = } { E - } { E } { E 2.5 ence the theoem.

4 5 Aleem Theoem 2.2 Fo the Ditibtion nction F in.2 the Recence Relation o the podct moment o Invee Weibll Ditibtion i given b: J m J m J m J m ] 2.6 J m whee g. p = Poo: Now m p g = J. Thi give: m m J m Now ptting in Theoem 2. we get the eqied J m Recence Relation 2.6. oolla 2.2 B epected application o the Recence Relation 2.5 we obtain o. j=2. v=2 - J J m v j m m v p J m p j m m J m J p ] J ] 2.7 Rema: Fo = the Recence Relation 2.5 & 2.6 become identical to Aleem 25 Note o the atio let = -j then 3. Single Moment J J J E J The lowe ecod vale ae epeented b N. The pd o n n 2 i:

5 Recence Relation o the Podct Single Moment o K th owe Recod Vale o Invee Weibll Ditibtion n n ] F 3. n! n 5 whee = -nf d h = - d <F< I g i a Boel meaable nction om R 2 to R then: E { g n } = n n g n ] F d 3.2 whee n n! Theoem 3. Fo the Ditibtion nction F in.2 we have: E { g n } = E{ n } KE{ n } 3.3 Poo: Uing.3 in 3.2 we have: E { g n } = K n n Upon integating the R..S. o 3.4 we get: n2 = n ] d - n n ] { F } d 3.4 ] n d E { g n } = E{ n } E{ n } 3.5 ence the theoem. Theoem 3.2 Fo the Ditibtion nction F in.2 the Recence Relation o the ingle moment o Invee Weibll Ditibtion i given b:

6 52 Aleem J m J m J m n = n n 3.6 J m Poo: Now m J p g m n = thi give: m J m n = ptting in 3. we get the Recence Relation. J m oolla 3.2 B epeated application o the Recence Relation 3.2 we obtain o n J=2 v=2 n-2 jm n v J m j V nv j m P n p 3.7 p Rema: i Thi Recence Relation in Theoem 2.2 between the moment o atio o two th lowe ecod vale Qai-ange joint moment geneating nction chaacteitic nction can be obtained b etting epectivel g eqal to: Reeence J K T Y it Y e e. ii Fo = the Recence Relation become identical to Aleem 25.. Aleem M. 25. Podct atio ingle moment o lowe ecod vale o invee Weibll ditibtion. Jonal o Statitic Bienie M. Sznal D. 24. On the actional ecod vale. Pobabilit Mathematical Statitic hang S. 27. Recence elation o qotient moment o the Weibll ditibtion b ecod vale. Jonal o Applied Mathematic ompting Ghaaph M. K ompaion o etimato o location meae o an invee Raleigh ditibtion. The Egptian Statitical Jonal

7 Recence Relation o the Podct Single Moment o K th owe Recod Vale o Invee Weibll Ditibtion 5. Majee S. P. Maiti S.S.996. A pecentile etimato o the invee Raleigh paamete. IAPR Tanaction Majee S. P. Saen. K Bivaiate invee Raleigh ditibtion in eliabilit tdie. Jonal o Indian Statitical Aociation Voda V. Gh On the invee Raleigh ditibted om vaiable. Repot in Statitical Applied Reeach JUSE