Record Values from Exponentiated Pareto Distribution and Associated Inference
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1 Recod Vaue o Epoetated Paeto Dtbuto ad Aocated Ieece A. I. Shaw * ad Haaa H. Abu-Zadah ** G Coege o Educato Jeddah Scetc Secto Kg Abduazz Uvet P. O. Bo 369 Jeddah 438 Saud Aaba. Abtact I th pape we code the epoetated Paeto EP dtbuto wth two hape paaete ad. We t deve the eact o o ea vaace ad covaace o ecod vaue. The au ehood etato ME' o the two hape paaete ad ae deved. Baed o the oet o owe ecod vaue we obtaed the ecea coecet o the bet ea ubaed etato BUE' o the ocato ad cae paaete o EP dtbuto. The vaace ad covaace o thee etato ae ao gve. The bet ea ubaed pedcto BUP o the utue ecod vaue ao dcued. Ke wod: owe ecod vaue; Epoetated Paeto dtbuto; Moet o ecod vaue; Mau ehood etato; he oato; Bet ea ubaed etato; Bet ea ubaed pedcto.. Itoducto Recod vaue ae atua a ea e appcato vovg data eatg to weathe pot ecooc ad e tetg tude. Ma autho have tuded ecod vaue ad aocated tattc; o eape ee Abd-E-Ha ad Suta Aod et a. 998 Baaha ad Cha 993 Suta ad Mohe Suta et a. ad Suta et a.. Gupta et a. 998 popoed to ode aue data b *tt] whee t the baee dtbuto ucto ad a potve ea ube. Th ebe eough to accoodate both ootoc a we a o- ootoc aue ate eve though the baee aue ate ootoc. Modeg uvva data b oootoc aue ate deabe o eape whe the coue o the deae uch that otat eache a pea ate oe te peod ad the ow dece. Recet Mudhoa ad Huto 996 Mudhoa ad Svatava 993 * Peaet adde: ac. o Eg. at Shouba P.O. Bo 6 E Maad 78 Cao Egpt E-a adde: ** E-a adde: haaa_abuzadah@ahoo.co.
2 - - Mudhoa et a. 995 Naa ad Ea 3 4 ad Sgh et a. 5 tuded the popoed ode wth Webu a the baee dtbuto ae the epoetated Webu a. I ae wa Gupta ad Kudu999 ab 6 Raqab 4 Raqab ad Ahauah ad Zheg tuded the popoed ode wth epoeta a the baee dtbuto whch the geeazed epoeta a o epoetated- epoeta dtbuto. I th pape we tae the popoed ode wth Paeto dtbuto a the baee dtbuto epoetated Paeto dtbuto EP wth pobabt det ucto pd ] > > >. ad cuuatve dtbuto ucto cd ] > > >. whee ad ae two hape paaete. o oe popete ee Shaw ad Abu- Zadah 6. Whe the above dtbuto coepod to the tadad Paeto dtbuto o the ecod d. et be a equece o..d. ado vaabe wth cd ad pd. Set Y a { }. We a a uppe owe ecod vaue o th equece Y > < Y >. B deto a uppe a we a a owe ecod vaue. Oe ca tao o uppe ecod to owe ecod b epacg the oga equece o { } b { } o P > o a b { }; the owe ecod vaue o th equece w coepod to the uppe ecod vaue o the oga equece. We w coe ou atteto to ut owe ecod vaue. I th pape we deve the eact o o ea vaace ad covaace o owe ecod vaue o the EP dtbuto Secto. I Secto 3 the au ehood etato ME' o the hape paaete ad ae deved. a Secto 4 we obtaed the ecea coecet o the bet ea ubaed etato BUE' o the ocato ad cae paaete o EP dtbuto ad the vaace ad covaace o thee etato baed o the oet o owe ecod vaue. I addto we dcued the bet ea ubaed pedcto BUP o the utue ecod vaue o EP dtbuto. et. Moet o owe ecod vaue... be the t owe ecod vaue o the EP dtbuto. the the ge ad doube oet ae deved a oow:
3 Sge oet The pd o ca be wtte ee Aod et a. 998] a. ] > Γ. o.. ad. the ge oet o the owe ecod vaue ae Γ ] d E µ. B ettg ] t we d that Γ ]! dt t t µ whee >. ; ;... a a a a B puttg w - t we obta >...! ad µ.. Whe we get >.! µ.3 Whe we have >.! µ.4 o.3 ad.4 the vaace o. Va µ µ σ.5. Doube oet The ot pd o two owe ecod vaue < ca be wtte ee Aod et a. 998] a ] ] Γ Γ. < < <.6
4 - 4 - o.. ad.6 the doube oet o the owe ecod vaue ad ae gve b d d E ] ] Γ Γ µ ] d I Γ Γ.7 whee. ] d I Settg w we get. ]! Γ I Puttg I.7 we obta ] ]! d A µ whee. A Γ B ettg ] t we d that. ]!!. dt t t A µ Puttg w - t we have!! µ... a > ad..8
5 - 5 - Whe we get!! > µ..9 The covaace betwee the two owe ecod vaue ad < gve b Cov µ µ µ σ.. 3. Mau ehood etato The ot det ucto o the t owe ecod vaue... o EP dtbuto ehood ucto gve b ] ]... >. 3. The og-ehood ucto o 3. gve b ] ]. 3. The oa equato becoe: v 3.3 v v v v v 3.4 whee.... v o 3.3 we obta the au ehood etato ME o a a ucto o a ˆ whee
6 - 6 - ˆ. v ] Puttg ˆ 3. we obta 3.5 g ˆ v ] v. v ] 3.6 Theeoe ME o a ˆ ca be obtaed b azg 3.6 wth epect to. It ca be how that the au o 3.6 ca be obtaed a a ed pot outo o the oowg equato; h 3.7 whee v v v h. 3.8 v v v Ve pe teatve pocedue ca be ued to d a outo o 3.7 ad t wo ve we. Oce we obta ˆ the ME o a ˆ ca be obtaed o 3.5 a ˆ ˆ ˆ. Now we tate the aptotc oat eut to obta the aptotc vaace o the etate. It ca be tated a oow: ˆ ˆ N I 3.9 R whee I the he oato at ee Ahad ad Agha ].e. I R R E E E E The eeet o the he oato at ae a oow: J J J e e e d! J J J J.
7 - 7 - J ] e e e d! ]. e e e d e! Aug that the hape paaete ow o 3. we ca obta the ME ˆ o the hape paaete a ˆ. 3. v ] o the aptotc popete o the ME t oow that ˆ ˆ E ad Va J. Now code the ME o whe the hape paaete ow. o ow the ME o a ˆ ca be obtaed b azg u v v v wth epect to. It ca be ea how that u a uoda ucto o ad ˆ whch aze u ca be obtaed a the ed pot outo o 3. whee v v v. v v o the aptotc popete o the ME t oow that ˆ ˆ E ad Va J. I Tabe we have uated baed o Mote Cao u the vaue o ba ad oot ea quae eo RMSE' o ME' o that whe ow o ad 5. o th tabe we ote that the RMSE' o the ME' o deceae a ceae. We ca ee ao the RMSE' o the ME' o ot o the te ceae a vaue o ceae o ed. Thee o potat eect o the etato whe have deet vaue.
8 - 8 - Tabe. Suated vaue o ba ad RMSE' o ME o whe ow Note: The t et the uated ba. The ecod et the uated RMSE. I Tabe we have uated baed o Mote Cao u the vaue o ba ad RMSE' o ME' o that whe ow o ad 5. o th tabe we ote that the RMSE' o the ME' o deceae a ceae. We ca ee ao the RMSE' o the ME' o deceae a vaue o ceae o ed. o ed ot o the te the RMSE' o the ME' o ceae a vaue o ceae.
9 - 9 - Tabe. Suated vaue o ba ad RMSE' o ME o whe ow Note: The t et the uated ba. The ecod et the uated RMSE. I Tabe 3 we have uated baed o Mote Cao u the vaue o ba ad RMSE' o ˆ ad ˆ whe both o the ae uow o ad 5. o th tabe we ote that the RMSE' o ˆ ad ˆ ot o the te ceae deceae that whe a ceae. We ca ee ao ot o the te the RMSE' o ˆ ceae ad the RMSE' o ˆ deceae
10 - - that whe the vaue o ceae o ed. o ed ot o the te the RMSE' o ˆ deceae ad the RMSE' o ˆ ceae a vaue o ceae. Tabe 3. Suated vaue o ba ad RMSE' o ME' o ad whe both o the uow ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ Note: The t et the uated ba. The ecod et the uated RMSE.
11 BUE' o ocato ad cae Paaete ad BUP o the utue ecod vaue The eact ad epct epeo o the t ad ecod ge oet o owe ecod vaue gve.3 ad.4 aow u to evauate the ea ad vaace o a owe ecod vaue. o the coputato o vaace ad covaace the poduct oet µ wee coputed t. The dagoa eeet µ µ ae obtaed o the eact epeo gve.4. Net the eact ad epct epeo gve.9 wa ued o the coputato o the poduct oet o two owe ecod vaue. Thee vaue wee ued to evauate the covaace.. The vaue o ea vaace ad covaace wee coputed b ug Matheatca 4.. It potat to ea hee that a ou coputato o the tadad Paeto dtbuto o the ecod d ca be obtaed whe. Aue Y... to be a te equece o d ado vaabe o the Y EP dtbuto wth pd µ ; ] µ µ σ σ σ σ 4. µ < < σ >. et Y Y... Y be the t obeved owe ecod vaue o the above equece. The... whee Y µ σ... the vecto o obeved ecod vaue o a popuato wth the tadad EP dtbuto wth pd ad cd gve. ad. epectve. The we ca wte the BUE' o µ ad σ ee Aod et a. 998] a ˆ µ a Y a Y... a Y 4. ˆ σ b Y b Y... b Y 4.3 whee a' ad b' ae the ete o the at T T C A V A A V ;
12 - - T µ T... µ µ µ... µ A V the vee o the covaace at V etato ae gve b whee σ. Vaace ad covaace o thee Va ˆ µ d σ Va ˆ σ d σ ad Cov ˆ µ ˆ σ d σ 4.4 d d T D. d A V A d Tabe 4: ea o owe ecod vaue
13 - 3 - Tabe 5: Covaace o owe ecod vaue
14 - 4 - Tabe 5: Cotued
15 - 5 - Tabe 6: Coecet o the BUE o µ
16 - 6 - Tabe 6: Cotued
17 - 7 - Tabe 7: Coecet o the BUE o σ
18 - 8 - Tabe 7: Cotued
19 - 9 - Tabe 8: Vaace ad Covaace o the BUE' o µ ad σ te o σ
20 - - Tabe 8: Cotued The ecea coecet 4. ad 4.3 equed o the BUE' o µ ad σ epectve ca be ea obtaed baed o the coputato o the ea vaace ad covaace o owe ecod vaue o the EP dtbuto. Tabe 4 peet the ea o 5 o 5 ad.5.54 to ve deca pace. Tabe 5 povde the vaace ad covaace o owe ecod vaue to ve deca pace o ad Tabe 6 ad 7 povde the coecet o the BUE' o the ocato ad cae paaete epectve o 5 5 ad The accuac o the cacuato have bee checed though the dette
21 - - a ad b. I Tabe 8 we copute the vaace ad covaace o the BUE' gve 4.4. o th tabe we ote that the vaace o the BUE' o µ ad σ deceae ad the covaace o thee etato ceae whe the vaue o o o ed ceae. o ed the vaace o the BUE' o µ ceae ad the BUE' o σ ad the covaace o thee etato deceae a the vaue o ceae. Eape: et u code the cae whee the copoet have aue te whch oow a EP dtbuto 4. wth 3 4 µ 6 ad σ. Suppoe that we obeve the oowg uated obeved aue te o EP µ σ : Theeoe we obeve the owe ecod vaue o the obeved data a oow: Hee the o the ecoded data aa wth 4 3 ad 4 µ ad V ae obtaed o Tabe 4 ad 5 epectve. The coecet 4. ad 4.3 ae peeted Tabe 6 ad 7 epectve. Theeoe the BUE' o µ ad σ ae coputed to be µˆ ad σˆ.695. The coepodg vaace ad covaace o µˆ ad σˆ ee Tabe 8 ae coputed to be Va ˆ µ.463 σ Va ˆ σ.8784 σ ad Cov ˆ µ ˆ σ.789 σ. et u code the tue popuato ea ζ E Y µ 3B33 4 σ. Had we ued ζ * Y Y Y Y 4 the ea o the obeved ecod. We woud 3 4 have ζ * ad tadad eo S.E. ζ *.398. The BUE o ζ ˆ ζ ˆ µ 3 B 33 4 ˆ σ The tadad eo o ζˆ coputed to be S.E.ζˆ.995. Theeoe the BUE' peo bette tha the ea o obeved ecod the ee o tadad eo. I the cotet o pedcto o the utue ecod obevato uppoe we obeve o the t ecoded obevato Y Y Y... Y ad the goa to pedct Y whee <. The ot we-ow pedcto the bet ea ubaed pedcto BUP ee Aod Baaha ad Nagaaa 998 o Y gve b ˆ Y ˆ ˆ T µ σ µ w Y ˆ µ ˆ V σµ
22 - - whee µ the ea o the t ecod vaue ad w the vecto o the covaace betwee the th utue ecod vaue ad the t ecoded obevato. The ea quae pedcto eo MSPE o Y ˆ oud to be ee Raqab MSPE Yˆ T T σ { E V E σ E w} whee T T T T T T E µ A V A A V w V I A A V A A V. I ou data etup we have obeved ou ecod vaue. Tabe 4-7 ae ued to copute the BUP o the utue ecod vaue Y baed o the t ou obeved ecod vaue. The vaue coputed to be Y ˆ ad the coepodg MSPE gve b ˆ MSPE Y.7539σ. 5 Acowedgeet. We ae cee thau to the eeee o the vauabe ad cotuctve coet that had poved ou oga pape. Reeece ] Abd-E-Ha N. S. ad Suta K. S.. Mau ehood etate o Webu paaete baed o ecod vaue. J. Egpt. Math. Soc ] Ahad J. ad Agha N. R.. O the he oato ecod vaue. Meta ] Aod B. C. Baaha N. ad Nagaaa H. N Recod. Joh We New Yo. 4] Baaha N. ad Cha P. S Recod vaue o Raegh ad Webu dtbuto ad aocated eece. Natoa Ittute o Stadad ad Techoog Joua o Reeach Speca Pubcato ] Gupta R. C. Gupta R. D. ad Gupta P Modeg aue te data b eha ateatve. Cou. Statt. - Theo Meth ] Gupta R. D. ad Kudu D Geeazed epoeta dtbuto. New Zeaad J. Statt ] Gupta R. D. ad Kudu D. a. Geeazed epoeta dtbuto: Deet ethod o etato. J. Statt. Coput. Su ] Gupta R. D. ad Kudu D. b. Epoetated epoeta a: A ateatve to gaa ad Webu dtbuto. Boetca Joua 7-3.
23 - 3-9] Gupta R. D. ad Kudu D. 6. O the copao o he oato o the Webu ad GE dtbuto. Joua o Stattca Pag ad Ieece I Pe. ] Mudhoa G. S. ad Huto A. D The epoetated Webu a: Soe popete ad a ood data appcato. Cou. Statt. - Theo Meth ] Mudhoa G. S. ad Svatava D. K Epoetated Webu a o aazg bath tub aue data. IEEE Ta. Reabt ] Mudhoa G. S. Svatava D. K. ad ee M The epoetated Webu a: A eaa o the bu -oto- aue data. Techoetc ] Naa M. M. ad Ea. H. 3. O the epoetated Webu dtbuto. Cou. Statt. - Theo Meth ] Naa M. M. ad Ea. H. 4. Baea etato o the epoetated Webu ode. Cou. Statt.-Theo Meth ] Raqab M. Z.. Ieece o geeazed epoeta dtbuto baed o ecod tattc. Joua o Stattca Pag ad Ieece ] Raqab M. Z. 4. Geeazed epoeta dtbuto: Moet o ode tattc. Stattc ] Raqab M. Z. ad Ahauah M.. Etato o ocato ad cae paaete o geeazed epoeta dtbuto baed o ode tattc. J. Statt. Coput. Su ] Shaw A. I. ad Abu-Zadah H. H. 6. Soe chaactezato o the epoetated Paeto dtbuto. Subtted. 9] Sgh U. Gupta P. K. ad Upadha S. K. 5. Etato o paaete o epoetated- Webu a ude tpe-ii ceog chee. Coputatoa Stattc & Data Aa ] Suta K. S. ad Mohe M. E.. Recod vaue o geeazed Paeto dtbuto ad aocated eece. Meta ] Suta K. S. Mohe M. E. ad Abd-E-Ha N. S.. Etato o paaete o oa dtbuto baed o ecod vaue. The Egpta Stattca Joua ] Suta K. S. Mohe M. E. ad Chd A.. Recod vaue o geeazed powe ucto dtbuto ad aocated eece. Joua o Apped Stattca Scece ] Zheg G.. O the he oato at tpe II ceoed data o the epoetated epoeta a. Boetca Joua
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