Ekpenyong Emmanuel John and Gideon Sunday N. x (2.1) International Journal of Statistics and Applied Mathematics 2018; 3(4): 60-64

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Itrtol Jourl of Sttstcs d Appld Mtmtcs 8; 34 6-64 ISSN 456-45 Mts 8; 34 6-64 8 Stts & Mts www.mtsjourl.

1 Itrtol Jourl of Sttstcs d Appld Mtmtcs 8; ISSN Mts 8; Stts & Mts Rcvd Accptd Ekpyo Emmul Jo Dprtmt of Sttstcs Mcl Okpr Uvrsty of Arcultur Umudk Nr Gdo Sudy N Dprtmt of Sttstcs Ab Stt Polytcc Ab Nr Bys stmto of t sp prmtr of burr typ XII dstrbuto wt rlsd squrd rror loss d l loss fuctos udr som pror dstrbutos Ekpyo Emmul Jo d Gdo Sudy N Abstrct I ts ppr w obt som clsscl d Bys stmtors of t sp prmtrs of t Burr Typ XII dstrbuto. T Bys stmto ws md udr rlzd squrd rror loss fucto GSEF d lr potl INEX loss fuctos wt o-formtv pror Jffrys pror d formtv pror Gmm pror. ywords Bys stmto pror dstrbuto postror dstrbuto loss fucto d m squr rror. Itroducto Twlv dffrt forms of dstrbuto fuctos usful modl wd r of prmtl d bolocl dt forstry frctur rouss lf tst oprtol rsk tc wr troducd by Burr 94. Amo ts dstrbutos t Burr Typs III X d XII v ovr tm d mor rcoto d rtr pplcto rsrcs d ltrturs bcus of tr flblty d comptblty wt otr dstrbutos. For stc Burr typs III d XII ppromt svrl dstrbutos t fmls of o-orml dstrbutos s Burr 973; Rodruz 977; Tdkml 98 d Hdrck t l.. Burr Typ X dstrbuto s prtculrly mportt modll procsss wt rdully crs flur rt wtout y boud. It s lso ffctv modll strt dt d lf tm Surls d Pdtt. Svrl studs d b crrd out o Burr Typ XII dstrbuto wc clud u t l 7 Mkdoom d Jfr Solm t l P d Asd Sdu d Aslm 3 Al-Sr t l 4 J 4 d Tsum t l 5.Ts ppr focusd o Bys stmto of t sp prmtr of two-prmtr Burr Typ XII dstrbuto wt rspct to Grlzd Squr Error oss Fucto GSEF d r Epotl INEX loss fucto us Jffry s pror s o-formtv pror d mm pror s formtv pror wt t m of compr t prformc of t stmtors udr t two loss fuctos.. Clsscl Estmto of t Sp Prmtr of Burr Typ XII Dstrbuto Hr w cosdr tr mtods of clsscl stmto t Mmum klood Estmto ME Mmum M Squr Error MSE d t Uform Mmum Vrc Ubsd Estmto UMVUE. t X~Burr Typ XII α β t t probblty dsty fucto pdf of X s v by Corrspodc Ekpyo Emmul Jo Dprtmt of Sttstcs Mcl Okpr Uvrsty of Arcultur Umudk Nr f ; ~6~. wr α s t sp prmtr d β t scl prmtr. It c sly b sow tt. s vld pdf d t lo-lklood fucto s v by

2 Itrtol Jourl of Sttstcs d Appld Mtmtcs I I I I I. Tus w obt t ME s ˆ ME I wr I Rsd d Njm 4 vlutd t Mmum M Squr Error MMSE t clss of stmtors rprstd s. d obtd s.3 E E.4 Rsd d Njm 4 otd tt sc. s t fmly of potl dstrbuto; t mpls tt s Gmm dstrbuto wt prmtrs d α. ~G. α. Hc E d E. Substtut ts pcttos.4 mply d tus t MMSE of s; ˆ MMSE.5 I Bsd o m-scff s torm s ubsd stmt of α sc E d s suc s complt suffct sttstc for α. Trfor t UMVUE of α s v by ˆ UMVUE.6 I Yrmommd d Pzr drvd t M Squr Error MSE of t tr clsscl stmtors d lso sowd tt MSE ˆ MSE ˆ MSE ˆ MMSE UMVUE ME 3. Bys Estmto of t Sp Prmtr of Burr Typ XII Dstrbuto Bys stmto volvs t stmto of ukow prmtr wc s rrdd s rdom vrbl from v probblty dstrbuto. Ts c b crrd out us o-formtv pror dstrbuto Jffry pror or formtv pror dstrbuto of t prmtr of trst cosdrto of obsrvd dt sy... Howvr Jffry pror d mm pror dstrbuto wr cosdrd s o-formtv d formtv pror dstrbutos of t sp prmtr of Burr Typ XII dstrbuto d stmtos wr md udr GSEF d INEX oss Fucto. 3. T Postror Dstrbutos T o-formtv pror dstrbuto for α us Jffry pror s v by I wr s t Fsr formto dfd s 3. If I E 3. Tk t turl lo of. w v ~6~

3 ~6~ Itrtol Jourl of Sttstcs d Appld Mtmtcs I I I I If 3.3 T scod ordr prtl drvtv of 3.3 wt rspct to α vs If 3.4 Substtut 3.4 to 3. mpls I wc furtr mpls tt ; 3.5 obt t postror dsty fucto of α v Jffry s pror s ;... d d I I I I Tus t postror dstrbuto of α v Jffry s pror s Gmm wt prmtrs d. If t pror dstrbuto of s mm wt prmtrs d t t probblty dsty fucto pdf of s v by 3.7 Us 3.7 w obt t postror dstrbuto of s ;... d d I I I I Tus t postror dstrbuto of α v Gmm pror s Gmm wt prmtrs d. 3. Bys Estmto udr GSEF Rsd d Al Gz 4 v t Grlzd Squr Error oss fucto GSEF lθ θ s ˆ ˆ j j ;....3 d obtd stmtor for us t corrspod rsk fucto s θ = Eθ t+ Eθ t+ + Eθ + t + Eθ t+ + Eθ t 3.9 Substtut α for θ d for t 3.9 α = Eα + Eα + + Eα + + Eα + + Eα 3.

4 Itrtol Jourl of Sttstcs d Appld Mtmtcs T rsult 3.6 mpls tt Eα = d w Eα = + c substtut ts momts 3. vs t GSEF w stmtor for α us Jffry s pror α GSEFJ = w + + w + + +k+k + w k+ + w + + +k +k + w k = Γ++j j=o j I+ β j+ Γ Γ+j j=o j I+ β j Γ 3. From 3.8 Eα = +θ d +λ Eα = +θ+θ+ c t GSEF stmtor for α us mm pror +λ α GSEF = +θ w+λ + +θ++θ +θ+k+θ+k +θ++θ w+λ + + w+λ k+ +θ + w+λ + + +θ+k +θ+k +θ++θ w k = Γ+θ++j j=o j I+ β +λ j+ Γ+θ Γ+θ+j j=o j I+ β +λ j Γ+θ Bys Estmto udr INEX Rsd d Sult 5 ppld INEX loss fucto of t form ˆ b ˆ ˆ 3.3 wr b d obtd stmtor for θ us t rsk fucto s E l ˆ 3.4 Substtut α for θ 3.4 w v E l ˆ 3.5 Apply t rsult 3.6 w obtd stmtor for α udr INEX loss fucto wt Jffry s pror s follows; E.. d d 3.6 Substtut w v ˆ INEX j l l l l 3.7 Gv t postror dstrbuto 3.8 t INEX stmtor for α us mm pror s obtd s follows.. d E ~63~ d 3.8

5 Itrtol Jourl of Sttstcs d Appld Mtmtcs INEX l l l ˆ Cocluso I ts work w v b bl to drv t Bys stmtor of t sp prmtr of t Burr typ IX dstrbuto udr squrd rror loss d INEX fuctos wt Jffry s d Gmmα β prors. 5. Rfrcs. A-Sr AY Brt A Mous SA. Mrsl Olk Etdd Burr Typ XII Dstrbuto. Itrtol Jourl of Sttstcs d Probblty Burr I. Cumultv Frqucy Fuctos. Al of Mtmtcl Sttstcs. 94; Burr I. Prmtrs for Grl Systm of Dstrbuto to Mtc Grd of α 3 d α 4 Commucto Sttstcs-Tory d Mtods. 973; Hdrck TC Pt MD S Y. O Smult Uvrt d Multvrt Burr Typ III d Typ XII Dstrbutos. Appld Mtmtcl Scc. ; J DH. Bys Estmto of Burr Typ XII Bsd o Grl Prorssv Typ II Csor. Appld Mtmtcl Sccs. 4; Mkdoom I Jfr A. Bys Estmtos o Burr Typ XII Dstrbuto us Groupd d Uroupd. Austrl Jourl of Bsc d Appld Sccs. ; P H Asd S. Alyss of t Typ II Hybrd Csord Burr Typ XII Dstrbuto udr INEX oss Fucto. Appld Mtmtcl Sccs. ; Rsd HA AAlwy Al-Gz NA. Bys Estmto for t Rlblty Fucto of Prto Typ I Dstrbuto udr Grlzd Squr Error oss Fucto. Itrtol Jourl of Er d Iovtv Tcoloy IJEIT. 4; Rsd HA Al-Gz NA. Bys Estmtors for t Sp Prmtr of Prto Typ I Dstrbuto udr Grlzd Squr Error oss Fucto. Mtmtcl Tory d Modl. 4; Rsd HA Sult AJ. Bys Estmto of t Scl Prmtr for Ivrs Gmm Dstrbuto udr INEX oss Fucto. Itrtol Jourl of Advcd Rsrc. 5; Rodruz RN. A Gud to Burr Typ XII Dstrbutos Bomtrk. 977; Sdu TN Aslm M. Estmto of t Burr Typ VIII Dstrbuto trou Bys Frmwork Advcs Arts Socl Sccs d Educto Rsrc. 3; Solm AA Abd Ell AH Abou-Al NA. Bys Ifrc d Prdcto of Burr Typ XII Dstrbuto for Prorssv Frst Flur Csord Smpl. Itllt Iformto Mmt. ; Surls JG Pdtt J. Ifrc for Rlblty d Strss-Strt for Scld Burr Typ X Dstrbuto ftm Dt Alyss. ; Tdkml PR. A look o t Burr d Rltd Dstrbuto. Itrtol Sttstcl Rvw. 98; AlBldw TH Rsd HA Jsm SH. Us Grlzd Squr oss Fucto to Estmt t Sp Prmtr of t Burr Typ XII Dstrbuto. Itrtol Jourl of Advcd Rsrc. 5; u SJ C YJ C CT. Sttstcl Ifrc Bsd O Prorssvly Csord Smpls wt Rdom Rmovls from t Burr Typ XII Dstrbuto. Jourl of Sttstcl Computto d Smulto. 7; Yrmommd M Pzr H. Mm Estmto of t Prmtr of t Burr Typ XII Dstrbuto. Austrl Jourl of Bsc d Appld Sccs. ; ~64~

A note on Kumaraswamy Fréchet distribution

A note on Kumaraswamy Fréchet distribution AENSI Jourls Austrl Jourl of Bsc d Appld Sccs ISSN:99-878 Jourl hom pg: wwwswcom A ot o Kumrswmy Frécht dstruto Md M E d 2 Ad-Eltw A R Dprtmt of Sttstcs Fculty of Commrc Zgzg Uvrsty Egypt 2 School of Busss