Department of Statistics, Dibrugarh University, Dibrugarh, Assam, India. Department of Statistics, G. C. College, Silchar, Assam, India.

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1 A Dscrete Power Dstruto Surt Chkrort * d Dhrujot Chkrvrt Dertet of Sttstcs Drugrh Uverst Drugrh Ass Id. Dertet of Sttstcs G. C. College Slchr Ass Id. *el: surt_r@hoo.co. Astrct A ew dscrete dstruto hs ee roosed s dscrete logue of the two sded ower dstruto V Dro J. R. d Kotz S.. A ovel eteso of the trgulr dstruto d ts reter estto Jourl of the Rol Sttstcl Socet Seres D The Sttstc 5 : 63-79]. Ths rolt ss fucto d hzrd rte fucto of ths dstruto c ssue vret of shes cludg th tu rectgulr trezodl trgulr J verse J U verse U strctl decresg d strctl cresg shes. Its oet d rellt roertes log wth reter estto hve ee vestgted. Kewords: Two sded ower dstruto; zrd rte fucto; Bthtu she roc uer. INTRODUCTION The two-sded ower TSP dstruto ws frst troduced V Dro d Kotz s ltertve of the et dstruto d eteso of three-reter trgulr dstruto whch llow J-shed d U-shed fors whch re ot for the trgulr dstruto. As the TSP dstruto eteds the trgulr dstruto t herts ts tutve el d reters terretto. V Dro d Kotz hve delt wth estto of the reters of the TSP dstruto. Soe roertes of the two-reter TSP dstruto wth suort hve ee dscussed V Dro d Kotz. A rdo vrle s sd to follow TSP dstruto wth reters f ts rolt dest fucto s gve f It s deoted TSP. For the ode of the dest fucto s t d the vlue of the df t the ode s lws /. For d the

2 ode of the dest fucto s t or d f t ts odes. For f slfes to ] Ufor. For f reduces to trgulr dstruto. Fll for d f corresods to ower fucto dstruto d for d to ts reflecto. The cdf of TSP dstruto follows fro eresso s F The survvl fucto of TSP dstruto follows fro eresso s F S 3 The e d vrce of the dstruto re gve resectvel E 4 d / / Vr 5. Iterrettos of the reters The reters d re the ed ot of the suort of the dstruto s the she reter d s the threshold reter for chge the for of the df. The reters d e relted to essstc d otstc esttes of the ssocted TSP rdo vrle. Gve cotuous rdo vrle wth survvl fucto S the dscrete logue Ro 4 s defed s the ew rdo vrle = lrgest teger less or equl to. The rolt ss fucto f of s the gve Pr Pr Pr Pr S S 6 For detl lst of dscrete dstruto derved fro cotuous dstrutos see Chkrort d Chkrvrt 4.

3 3. DISCRETE TWO-SIDED POWER DISTRIBUTION The rolt ss fucto f of dscrete two sded ower dstruto derved fro the df of the TSP dstruto usg the geerl dscretzto roch equto 6 s gve Pr 7 where d re tegers d s ostve rel uer. The rdo vrle s sd to follow the. The f the equto 6 s roer f sce ] 3 ]. Prtculr cses: Whe the f equto 7 reduces to dscrete ufor dstruto wth f / For Pr Pr l l l

4 Clerl the see corresodg cells of the two rows of fgure c].. DISTRIBUTIONAL PROPERTIES.. Pf lots The lots of f for vrous vlues of the reters hve ee reseted elow. It hs ee oserved tht c ssue vret of shes cludg th tu rectgulr trezodl trgulr J verse J strctl decresg d strctl cresg. Fro fgure tht s whe d the ode of the f s t ether or - d. 5 t ts ode whle fro the fgure tht s for t hs ee see tht the ode of the f s t or - the vlue of the f t the ode s lws /. Fro the frst lot fgure t c e checked tht for ssues ufor she ] for reduces to trgulr dstruto d fll the lots corresodg cells the two rows of fgure c re reflectos of ech other. Fgure. f of 4

5 Fgure. f of Fgure c. f of.. Recurrece relto for roltes The recurrece relto for roltes of s gve 5

6 6..3 Cuultve dstruto fucto Theore. The cuultve dstruto fucto cdf of s gve Pr F..4 Moets The e d the secod oet of s resectvel } { } { }] { } { 3 E d E

7 7 }] { } { 3 } { } { } { } { } { } { } { } { ] ] 9 ece the vrce s Vr where k k / s the geerlzed hroc uer of order whose lt s ests whe. The relted su k k occurs the stud of Beroull uers the hroc uer lso ers the stud of Strlg s uers Arowtz d Stegu 97. Theore. The e d vrce of dstruto re ouded. Proof. The dscretzed verso of tht s s defed s =lrgest teger less or equl to. It c e ssued tht U whereu s the frctol rt

8 of whch s choed off fro to ot. Therefore U wll hve other cotuous rolt dstruto the suort whch s deedet of. ece E E U E E U. But E U therefore o usg equto 4 Slr rguet gves E U V V U V V U V V V U Assug deedece of d U ]. But for cotuous rdo vrle U Vr U / 4 ow o usg equto 5 / / V / /.5 ece the e d vrce s ouded ove see lso fgures d 4. Plots of the e vrce d ID for dfferet coto of reter vlues for TSP d hve ee reseted elow. Me 3 TSP Me TSP Fgure. Me of TSP d for vrg..5 Me Me Fgure 3. Me of for vrg d 8

9 Vrce TSP.5.5 Vrce TSP.5.5 Fgure 4. Vrce of TSP d for vrg.5 Vrce Vrce Mode Fgure 5. Vrce of for vrg d For the ode of the f s ether t or t. For d the ode of the f s lws ether t or t wth resectve vlue of the f equl to / d fgure ove. / s ts ode. Ths c lso e see fro. RELIABILIT PROPERTIES.. Survvl fucto Theore 3. The survvl fucto of s gve S Proof. The survvl fucto s defed S P. Cse I. 9

10 S ] ] Cse II. ] S Cog these two cses gves the desres result. The lots of survvl fucto for vrous r of retrc vlues hve ee reseted fgure 7 elow sf sf sf Fgure 7. Survvl fucto of 5 d Rerk. It s to e oted tht the survvl fucto of s se s tht of cotuous two sded ower dstruto gve equto 3. Rerk. Fro equto / S. Therefore s the th / or sl / th qutle of the dstruto rresectve of the choce of the reter. I fct wll e the ed f t s equl to / sce / / / / S Thus the reter detfes / th ercetle or sl / th qutle of dstruto.

11 .. zrd Flure rte fucto Fro equto t s sle eercse to ot the flure or hzrd rte fucto of dscrete rdo vrle followg the dstruto wth f 7 gve Pr r S The hzrd rte fucto for dfferet rs of vlues of reters hs ee grhed fgure 8 hzrd rte sf sf Fgure 8. zrd rte fucto lot of 5 Fro the ove grhs t c e see tht hs ost of hzrd rte shes cludg the Bthtu shed oe ecet ossl the strctl decresg oe..3 Estto of reters For the vlue of d c e oted fro the dt gve s these re two ed ots of the dstruto. Also s the threshold reter whch dcted the chge ot of the f d c e relstcll estted fro the dt. Therefore the ol eed s to estte the reter. Ths reter c e estted dfferet ethods s dscussed elow.3. Mu lkelhood estto MLE For rdo sle k of sze k the log lkelhood fucto of s gve log log log L

12 The the lkelhood equto s gve log log log log log log The MLE of c e oted uercll solvg the lkelhood equto or zg the log-lkelhood fucto glol otzto ethod..3. Method of oet estto MME The ukow reter c e estted solvg the equto E where s the sle e. Altertvel c e estted zg E ] wth resect to Kh et l Sulto stud for MLE d MMEs A through sulto stud hs ee crred out to scert the effectveess of the reter esttes of the geertg sles of dfferet szes for dfferet choces of the reter. ere the rdo teger fro hs ee sled frst usg the verse trsforto ethod to geerte fro cotuous dstruto d the tkg the floor of the vlues of the resultg cotuous vrte. The stes for geertg rdo teger re follows: Ste I: Cosder cotuous rdo vrle wth dstruto fucto F. Ste II: Geerte rdo uer U fro ufor dstruto U. Ste III: Coute F U. Ste IV:. The result of sulto lss hs ee reseted tles d. All the etres of the tle re the es of esttes of sles. Accurc d recso of the estto ethods hve ee checked d estlshed usg the esttes of the followg crter: I. Estte of E ˆ k / k ˆ k II. Estte of s of ˆ : Bs / k ˆ

13 k III. Estte of e squre error MSE: MSE ˆ / k ˆ E k IV. Estte of Vrce: Vr ˆ / k ˆ ˆ where ˆ s the estte of the ukow true vlue θ oted fro the th k. ere k =. sle I ddto to check the stotc orlt of the esttors the 95% cofdece tervls for the esttors usg the forul CI ˆ.96SE ˆ 95% where SE ˆ Vr ˆ hve ee couted d the ercetge of esttes ˆ elogg to the tervl oted. Tle. Results of sulto fro dstruto for. 5 Preter Sle sze 5 5 vlue esttes MLE MME MLE MME MLE MME E ˆ Bs ˆ MSE ˆ Vr ˆ % of CI Tle. Results of sulto fro dstruto for 3. 5 Preter Sle sze 5 5 vlue esttes MLE MME MLE MME MLE MME E ˆ Bs ˆ MSE ˆ Vr ˆ % of CI

14 Rerk 3. Fro the tles d t c e see tht lrger the sle sze sller s the MSE. But MME sees to e the etter oto og the two wth resect to the dfferet crter. 3. Cocluso I ths er dscrete two sded ower dstruto hs ee derved. Vrous dstrutol d rellt roertes hve ee derved. Me d vrce hs ee oted coct for. Preter estto through u lkelhood d oet ethod hs ee cosdered. It s evsged tht ths dscrete dstruto whch c odel e sutle for odelg U shed J shed d verse J shed thtu shed ss fucto d vret of flure te cout dt wll e vlule ddto to the reostor of dscrete dstruto. Further works wth ths dstruto wll e followed u lter. Refereces. Arowtz M. d Stegu I. A. 97. dook of Mthetcl Fuctos d edto Dover New ork.. Chkrort S d Chkrvrt D. 4. A Dscrete Guel Dstruto rv: th.st] 8 th Octoer Kh M. S. A. Khlque A. d Aouoh A. M O esttg reters dscrete Weull dstruto IEEE Trscto o Rellt 38 3: Ro D. 4. Dscrete Rlegh Dstruto IEEE Trscto o Rellt 53 : V Dro J. R. d Kotz S.. A ovel eteso of the trgulr dstruto d ts reter estto Jourl of the Rol Sttstcl Socet Seres D The Sttstc 5 : V Dro d Kotz : The stdrd two sded ower dstruto d ts roertes: wth lcto fcl egeerg Aerc Sttstc 56 No 9-99] 4