Preliminary Test Single Stage Shrinkage Estimator for the Scale Parameter of Gamma Distribution

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1 America Jural f Mathematics ad Statistics, (3): 3-3 DOI:.593/j.ajms.3. Prelimiary Test Sigle Stage Shrikage Estimatr fr the Scale Parameter f Gamma Distributi Abbas Najim Salma,*, Aseel Hussei Ali, Mua Daud Salma Departmet f Mathematics, Cllege f Educati fr Pure Sciece/Ib-Al-Haitham, Uiversity f Baghdad Cmputer Ceter - Uiversity f Baghdad Abstract I this paper, prelimiary test sigle stage shrikage (PTSSS) techiques was used fr estimati the scale parameter f Gamma distributi whe the shape parameter α was kw as well as a prir kwledge abut was available i the frm f iitial estimate f. It is prpsed t estimate by a testimatr that is based up the result f a test f the hypthesis H : = agaist the hypthesis H A : with level f sigificace. If H is accepted we used = ψ ˆˆ ()( ) +, where the weightig factr ψ ˆ () is a fucti f the test statistic fr testig H r may be cstat such that ψ ˆ (). Hwever if H is rejected we used = ψ ˆˆ ()( ) +, where ψ ˆ () ad ˆ is the classical estimatr f (MLE r MVUE). Chsig the weightig factr ψ () ˆ, (i =,) apprpriately, a expressi fr the Mea Squared Errr (MSE) ad Bias Rati [B(.)] f were derived ad cmpariss were made with classical estimatr ( ˆ ) i the sese f efficiecy ad with sme related earlier studies. Keywrds Gamma Distributi, Maximum Likelihd Estimatr, Prelimiary Test Sigle Stage Shrikage Estimatr, Bias Rati, Mea Squared Errr ad Relative Efficiecy i. Itrducti The tw-parameter gamma distributi has bee used quite extesively i reliability ad survival aalysis, particularly whe the data are t cesred. The tw-parameter Gamma distributi has e shape ad e scale parameter. The radm variable X fllws a Gamma distributi with the shape ad scale parameters as α > ad >, respectively, if it has the fllwig prbability desity fucti (PDF): x α x e f (x; α, ) fr x > ; α>, > α = Γ( α ) (.) therwise ad it will be deted by Gamma (α,). Here Γ(α) is the Gamma fucti ad it is expressed as α x ( ) x e dx Γα = (.) * Crrespdig authr: abbasajim@yah.cm (Abbas Najim Salma) Published lie at Cpyright Scietific & Academic Publishig. All Rights Reserved It is well kw that the (PDF) f Gamma (α,) ca take differet shapes but it is always uimdal. The Hazard fucti f Gamma (α,) ca be icreasig, decreasig r cstat depedig α >, α < r α =, respectively. The mmets f X ca be btaied i explicit frm; fr example E(X) = α ad Var(X) = α (.3) The Gamma, r Pears [] Type III, distributi has bee used t mdel a wide rage f data types i may disciplies, especially i the ctext f reliability mdelig, life testig ad fatigue testig. Fr example, Birbaum ad Sauders [7] itrduced the gamma distributi fr mdelig the life-legth f certai materials, ad the use f this distributi fr varius reliability prblems is ted by bth Herd [] ad Dreick [9]. Gupta ad Grll [] discuss acceptace samplig based this distributi, ad they derive the peratig characteristic fucti, prducer s risk, failure rates ad miimum sample sizes fr this prblem. Empirical applicatis f the Gamma distributi arise i a diverse rage f fields. Fr example, Wei ad Baveja [] applied this distributi i aalyses f huma figerprit data. Segal et al. [7] used it fr matchig scres i the ctext f DNA figerprit getypig f tuberculsis, ad Keat [3] adpted it fr a ivetry ctrl prblem. The Gamma distributi has als bee applied i a umber f studies i the fields f sigal prcessig (e.g.,

2 3 Abbas Najim Salma et al.: Prelimiary Test Sigle Stage Shrikage Estimatr fr the Scale Parameter f Gamma Distributi Marti [5], Jese et al. [], ad Kim ad Ster []), hydrlgy (e.g., Asky [], ad Bhuya et al. []) ad meterlgy (e.g., Simps []). The aim f this paper is t estimate the scale parameter f Gamma distributi whe the shape parameter is kw usig prelimiary test sigle stage shrikage estimati (PTSSS) techiques via study the perfrmace f Bias, Mea Squared Errr ad Relative Efficiecy expressis f the prpsed estimatr whe we set up a selecti f shrikage weight factr ϕ() ˆ ad suitable regi R ad create cmpariss f the umerical results with ˆ ad with sme existig studies. A umerical study is carried ut t appraise these effects f prpsed estimatrs. Shrikage techique was itrduced by Thmps [9] as fllws (.) = ϕ()( ˆˆ ) + where is a prir estimate(iitial value) abut () frm the past experieces ad ϕ() ˆ, represets a shrikage weight factr specifyig the degree f belief i ˆ ad - ˆ ϕ() specifyig the degree f belief i. We used the frm (.) abve t estimate the scale parameter f Gamma distributi i case ϕ() ˆ is chse as fllws:- ( ˆˆ ˆ ψ ), if R ϕ() = (.5) ( ˆˆ ψ ), if R where R is the prelimiary test regi fr acceptace f size fr testig the hypthesis H : = agaist the hypthesis ˆ H A : usig the test statistic ˆ α Τ(/) = ad ˆ is the classical estimatr f (MLE r MVUE), the the estimatr which is defied i (.) will be writte as belw ψ ˆˆ ( ) +, if R = ˆˆ ψ ( ) +, if R (.) where ψi (ˆ ), i =,; ψ ˆ i () represets as shrikage weight factrs which may be a fuctis f ˆ r may be cstats. The resultig estimatr (.) is kw as prelimiary test sigle stage shrikage estimatr (PTSSSE). Several authrs had studied the estimatr defied i (.) fr special distributi fr differet parameters ad suitable regis (R) as well as fr estimate the parameters f liear regressi mdel. Fr example see [], [], [3], [], [5] ad [9].. Maximum Likelihd Estimatr ˆ Let t, t,, t be a radm sample f size frm the tw parameter Gamma distributi with scale parameter ad shape parameter α. i.e. ; ti G(α,) fr,,3,... Ad assume that the shape parameters α is kw ( say α = α ) The maximum likelihd fucti L (t i ; α, ) is defied as belw : - L (t i, α, )= f (t, α, ), i =,,..., i Where, f(.) is defied i equati(.). Therefre, The maximum likelihd fucti will be [ Γ()] L(t i, α, ) = ( t i )EXP( ) α Ad the lgarithm f L (t i, α, ) is:- Lg L(t, α, ) α ti. (.7a) (.7b) i t i = - lgγ () - α lg +( α -) ti - i = (.) The partial derivative f Lg L(t i, α, ) with respect t (w.r.t) is as belw:- t Lg L(t i, α, ) α = + i (.9) Ad equatig the equati (.9) t zer, we btai the maximum likelihd estimatr fr as belw t ^ i = (ad smetimes symblized by α Nted that ˆ Gamma (α,/α ). It is easy f te that ˆ is ubiased estimatr. i.e.; E ( ˆ ) = ad MSE ( ˆ ) = Var ( ˆ ) = /α. 3. Prelimiary Test Sigle Stage Shrikage Estimatr (PTSSSE) ^ mle ) (.) Usig the frm (.), we prpsed the prelimiary test sigle stage shrikage estimatrs fr estimate the scale

3 America Jural f Mathematics ad Statistics, (3): parameter f Gamma distributi whe a prir ifrmati available abut with kw shape parameter α as belw:-, if ˆ R = (.) k( ˆˆ ) +, if R, i.e. ψ ˆ ( ) = ad ψ ˆ ( ) = k i equatis (.5). Ad the regi R which is defied i (.5) will be as belw: R =[ Χ, Χ ] α /, α /,α α (.) Fr simplicity, assume that R = [a,b], a < b i.e.; a = Χ /, α, b = Χ /,α α α (.3) where Χ /,α ad Χ /,α are respectively the lwer ad upper ( /) percetile pit f Chi-square distributi with (α ) degree f freedm. The expressis fr Bias f the estimatr is as fllws:- Bias(, R) = E( ) where R is the cmplemet regi f R i real space ad = [ ] f() ˆˆˆˆˆ d + [k( ) + ] f() d R R ˆ f () is a (PDF) f ˆ which has the fllwig frms ˆˆα ˆ [] exp[ α / ] f ()=, α > ο Γ( α )( / α ) α fr < ˆ < (.) we cclude, { λ a* b* λ } (.5) Bias (, R) =( )( k) k[ j (, ) j (a*, b*)] where, b* α + y (.) a* j ( a*, b*) = y e dy, fr =,, ( α) Γ( α) λ = /, a* = λ The bias rati B ( ) f the estimatr is defied belw Χ α/,α,b* = λ α/,α Ad, the expressi fr mea square errr (MSE) f is give as belw: Χ ad y= ˆ / α (.7) B ( ) = Bias(, R) / (.) MSE (, R) = { k [ + ( ) ] + ( ) k( ) k [J (a, b ) * * λ λ λ α * * * * * * * * λj (a,b ) + λ J (a,b )] K( λ )[J (a,b ) λj (a,b )] } As fr the value f k is fud by miimizig the mea squared errr f as fllw MSE(,,R) = k k * * * * * ( λ ) + ( λ )[J (a, b ) λj (a, b )] = * * * * * * α + λ λ +λ ( ) ( ) [J (a,b ) J (a,b ) J (a,b )] (.9) (.)

4 3 Abbas Najim Salma et al.: Prelimiary Test Sigle Stage Shrikage Estimatr fr the Scale Parameter f Gamma Distributi T be sure that the value f k * [, ] as a shrikage weight factr we put k as fllws:- * if k * * k = k if < k < * if k (.) The Relative Efficiecy f estimatr w.r.t. the classical estimatr ( ˆ ) is defied as belw:- MSE(ˆ ) / α R.Eff (, R) = = MSE( ) MSE( ) See fr example [], [], [3] ad [9].. Cclusis ad Numerical Results (.) The cmputatis f Relative Efficiecy [R.Eff( )] ad Bias Rati [] fr the equatis (.) ad (.9) were used fr the estimatrs. These cmputatis (usig Mat. LAB prgrams) were perfrmed fr =.,.5,., λ =.5(.5) ad =,,. These cmputatis are give i three aexed tables. The bservati metied i the tables lead t the fllwig results:. R.Eff ( ) were adversely prprtial with small value f α ad specially whe clse t.. R.Eff ( ) are maximum whe = (λ = ) fr all, α ad. This feature shw the imprtat usefuless f prir kwledge which give higher effects f prpsed estimatr as well as the imprtat rle f shrikage techique ad its philsphy. 3. Bias Rati [B ( Β ias ( ) )] [Β() = ] were reasably small whe, therwise start t be maximum fr all ad. This prperty shw that the prpsed estimatr is very clsely t ubiased ess especially whe =.. Bias Rati [B ( )] were at mst decreasig fucti with fr all ad λ 5. Effective Iterval [the values f λ that makes R. Eff. greater tha e] fr was [.5, ] fr all α, ad. Here the pretest criteri is very imprtat fr guaratee that prir ifrmati is very clsely t the actual value ad prevet it faraway frm it, which get ptimal effect f the csidered estimatr t btai high efficiecy.. The suggested estimatr is mre efficiet tha the classical estimatr ˆ specially whe is very clse t which is give the effective f whe give a imprtat weight f prir kwledge. Ad the augmetati f efficiecy may be reach t tes times. Als mre efficiet tha the estimatr itrduced by [9] i the sese f Mea Squared Errr ad Relative Efficiecy. 7. Whe the shape parameter f Gamma distributi equal t e [α = ], the distributi becme a Expetial distributi, thus the suggested estimatr i this case is mre efficiet tha the estimatr itrduced by [3]. Table (). Shw Bias Rati [B ( )] ad Relative Efficiecy [R. Eff. ( )] f whe α =..5. λ Bias R.Eff E 5.75 E E 3.5 E + 9. E.75 E E. E E E

5 America Jural f Mathematics ad Statistics, (3): Table (). Shw Bias Rati [B ( )] ad Relative Efficiecy [R. Eff. ( )] f whe α =..5. λ Bias R.Eff E 5 7. E E Table (3). Shw Bias Rati [B ( )] ad Relative Efficiecy [R. Eff. ( )] f whe α = λ Bias R.Eff E f Educati,, -75. REFERENCES [] Al-Jbri, A.N., (), "O Shruke Estimatrs fr the Parameters f Simple Liear Regressi Mdel", Ib-Al-Haitham J. fr Pure ad Applied Scieces, 5,(A), -7. [] Al-Jbri, A.N., (), "Pre-Test Sigle ad Duble Stage Shruke Estimatrs fr the Mea f Nrmal Distributi with Kw Variace", Baghdad Jural fr Sciece, 7,, 3-. [3] Al-Jbri, A.N., Khalaf, B.A. ad Hamza, S., ()," Estimate the Scale Parameter f Expetial Distributi Via Mdified Tw Stage Shrikage Techique", Jural Cllege [] Al-Jubri, A.N. (), "O Sigificace Testimatr fr the Shape Parameter f Geeralized Rayleigh Distributi ". Jural f AL-Qadisyia. 3,, [5] Al-Jubri, A.N. (), "Estimate the Reliability Fucti f Expetial Distributi Via Pre-Test Sigle Stage Shrikage Estimatr". Accepted t publish, Tikrit f Mathematics ad Cmputer Jural. [] Asky, H. (), Use f gamma distributi i hydrlgical aalysis, Turkish Jural f Egieerig ad Evirmetal Sciece,, 9-. [7] Birbaum, Z. W. ad S. C. Sauders (95), A statistical mdel fr life legths f materials, Jural f the America Statistical Assciati, 53, 53-.

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