Iteratioal Mathematical Forum, Vol., 3, o. 3, 3-53 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.9/imf.3.335 Double Stage Shrikage Estimator of Two Parameters Geeralized Expoetial Distributio Alaa M. Hamad Departmet of Mathematics Ib Al-Haitham College of Educatio, Baghdad Uiversity, Baghdad, Iraq Copyright 3 Alaa M. Hamad. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract This paper is cocered with double stage shrikage estimator (DSSE) for lowerig the mea squared error of classical estimator (MLE) for the shape parameter () of geeralized Expoetial (GE) distributio i a regio (R) aroud available prior kowledge ( ) about the actual value () as iitial estimate i case whe a scale parameter (λ) is kow as well as to reduce the cost of experimetatios. I situatio where the experimetatios are time cosumig or very costly, a double stage procedure ca be used to reduce the expected sample size eeded to obtai the estimator. This estimator is show to have smaller mea squared error for certai choice of the shrikage weight factor ψ( ) ad for acceptace metioed regio R. Expressios for Bias, Mea square error (MSE), Expected sample size [E(/,R)], Expected sample size proportio [E(/,R)/], probability for avoidig the secod sample [p( ˆ R)] ad percetage of overall sample saved ˆ for the proposed estimator are derived. [ p( R) ] Numerical results ad coclusios are established whe the cosider estimator (DSSE) are estimator of level of sigificace Δ. Comparisos with the classical estimator ad with the last studies show the usefuless of the proposed estimator
Alaa M. Hamad Keywords: Geeralized Expoetial Distributio; double stage shrikage estimator; Maximum Likelihood Estimator; mea squared error. Itroductio Gupta ad Kudu (999) proposed the geeralized expoetial (GE) distributio as a alterative to the well kow Weibull or gamma distributios. It is observed that the proposed two-parameter GE distributio has several desirable properties ad i may situatios it may fit better tha the Weibull or gamma distributio. Extesive work has bee doe sice the to establish several properties of the geeralized expoetial distributio. The readers are referred to the recet review article by Gupta ad Kudu (7) for a curret accout of it.[].geeralized expoetial (GE) distributio has bee proposed ad studied quite extesively recetly by Gupta ad Kudu [, 3,, 5, 6]. The readers may be referred to some of the related literature o (GE) distributio.[7].the two-parameters (GE) distributio has the followig distributio fuctio: ( λx) F(x; λ, ) = [ e ] for x >, >, λ > () Thus, the probability desity fuctio (p.d.f.) of (GE) distributio is λ ( x) λ ( x) λxe ( e ) for x>,, λ > f(x; λ, ) =..() o.w. Where, ad λ are the shape ad scale parameters respectively. I this paper we itroduce the problem of estimatig of the shape parameter () of GE distributio with kow scale parameter (λ) whe some prior iformatio ( ) regardig the actual value () available due past experieces such a prior estimate may arise for ay oe of a umber of reasos [], e.g., we are estimatig ad; i. We believe is close to true value of, or ii. We fear that may be ear the true value of, i.e.; somethig bad happes if = ad we do ot kow about it. I such a situatio it is atural to start with a estimator ˆ (e.g. MLE) of ad modify it by movig it closer to, so that the resultig estimator, though perhaps biased, has smaller mea square error tha that of ˆ i some iterval aroud. This method of costructig a estimator of that icorporates the prior value leads to what is kow as a shrikage estimator. It is a importat aspect of estimatio that oe should be able to get a estimator quickly usig miimum experimetatio. This also ecoomizes cost of experimetatio. To achieve this, double stage shrikage estimator were itroduced. A double stage shrikage estimator procedure is defied as follows:
Double stage shrikage estimator 5 Let x i ; i =,,, be a radom sample of from GE distributio ad ˆ be a "good" estimator of based o these observatio. Costruct a prelimiary test regio R i the parameter space based o ad a appropriate criterio. If ˆ R shrik ˆ towards by shrikage weight factor ψ ( ˆ ) = e ad use the shrikage estimator ψ ( ˆ) ˆ ˆ + ( ψ ( )), for estimate. If ˆ R, obtai x i ; i =,,,, a additioal sample of size ad use a pooled estimator ˆ p of based o combied sample of size = +, ˆ + ˆ i.e.; ˆ p =. Thus, the double stage shrikage estimator (DSSE) of will be: ψ( ˆ) ˆ + ( ψ ˆ ˆ ( )),if R = % ˆ ˆ p,if R.(3) The motivatio of this study was provided by the work of [9], [] ad []. The aim of this paper is to employ the double stage shrikage estimator (DSSE) % defied by (3) for estimate the shape parameter () of two parameters geeralized Expoetial (GE) distributio whe the scale parameter (λ) is kow. The expressio of Bias, Mea squared error (MSE), Relative Efficiecy [], Expected sample size, Expected sample size proportio, probability for avoidig the secod sample ad percetage of overall sample saved are derived ad obtaied for the estimator %. Numerical results ad coclusios due metioed expressios icludig some costats are performed ad displayed i aexed tables. Comparisos betwee the proposed estimator with the classical estimator ( ˆ ) ad with some of the last studies are demostrated.. Ubiased - Maximum Likelihood Estimator of I this sectio, we cosider the maximum likelihood estimator (MLE) of GE distributio with shape ad scale parameter ad λ respectively i.e. GE (,λ). Assume x, x,, x be a radom sample of size from GE(,λ) the the log-likelihood fuctio L (,λ) ca be writte as: λ ( x i ) + λ+ λ i i= i= L( λ, ) = l l ( ) l( e ) x.() I this paper we take λ = (λ is kow). L x So, = + l( e i ) =..(5) i= The, the MLE of, say ˆ is MLE
6 Alaa M. Hamad ˆ = MLE i=...(6) x i l( e ) Note that, if x i iid GE (, ), the x i l( e ) G (, ), see []. i= i.e.; E( ˆ ) = ad var( ˆ ) =. MLE MLE ( ) ( ) Usig (6), a ubiased estimator ˆ of ca be easily obtaied as: ˆ = ˆ =... (7) MLE xi l( e ) E( ˆ ) = ad i= var( ˆ ˆ ) = MSE( ) =..() 3. Double Stage Shrikage Estimator (DSSE) % I this sectio, we cosider the (DSSE) % which is defied i (3) usig ˆ defied by (7), whe ˆ ψ ( ) = k is a costat weight factor ( k = e ) for estimate the shape parameter of GE distributio whe λ =. k ˆ + ( k) ˆ,if R = % ˆ ˆ + ˆ = ˆ p,if R.. (9) Where R is a pretest regio for testig the hypothesis H : = vs H A : ( ) with level of sigificace (Δ) usig test statistic fuctio T( ˆ / ) = ˆ ( ) ( ) i.e.; R =, b a..() Where a( = X Δ /, ) ad b( = X Δ /, )... () are the lower ad upper (Δ/) percetile poit of chi-square distributio with degree of freedom ( ) respectively. The expressio for Bias of DSSE ( % ) is defied as below
Double stage shrikage estimator 7 Bias( % /, R) = E( % ) ˆ = ˆ R ˆ = ˆ R [ ] = k( ˆ ) + ( ) f( ˆ )f( ˆ )dˆ d ˆ + ˆ f( ˆ )f( ˆ )dˆ dˆ p Where R is the complemet regio of R i real space ad i + (i ) (i ) ˆ i e ˆ ˆ = for >, >.() o.w. i f( i ) ˆ i Γ( i) (i ) We coclude, Bias( % /, R) = (K + )( )J (a*, b*) + [( )J (a*, b*) J (a*, b*)] (3) + u b* y l y e Where J l (a*, b*) = y dy; l =,,,,,() Γ( ) a* ( ) Also y =..(5) ˆ The Bias ratio [B ( )] of DSSE ( % ) is defied as: Bias( % /, R) B( % ) =.(6) The expressio of Mea squared error [MSE ( )] of % derived as below:- MSE( % /,R) = E( % ) ˆ = ˆ R ˆ = ˆ R [ ] = k( ˆ ) + ( ) f( ˆ )f( ˆ )dˆ d ˆ + ˆ f( ˆ )f( ˆ )dˆ dˆ p Ad by simple computatios, oe ca get:
Alaa M. Hamad k ( ) J (a*, b*) ( )J (a*, b*) + J (a*, b*) + k( ) ( )J (a*, b*) J (a*, b*) + ( ) J (a*, b*) + MSE( /, R) [( ) % = J (a*,b*) + ( )J (a*,b*) J (a*,b*) + + u u J (a*,b*) + u u... (7) Now, the Efficiecy of % relative to ˆ deote by R.Eff ( % /, R) is defied by: MSE( ˆ ) R.Eff ( % /,R) =...() MSE( % /,R) [E( /,R / ] Where E(/,R) is the Expected sample size, which is defied as: u E( /,R) = J (a*,b*) (9) + u See for example [] ad []. As well as the Expected sample size proportio E (/, R)/ equal to u J (a*,b*)...() + u Also, we have to defie the percetage of the overall sample saved (p.o.s.s.) of % as: p.o.s.s. = J (a*, b*)..() Ad, fially, P( ˆ R) represet the probability of a voidig the secod sample (stage).. Coclusios ad Numerical Results The computatios of Relative Efficiecy [] ad Bias Ratio [], Expected sample size [E(/,R)], Expected sample size proportio [E(/,R)/], Percetage of the overall sample saved (p.o.s.s.) ad probability of a voidig the secod sample [ P( ˆ R) ] were used for the estimator %. These computatios were performed for =,, 6, u (= / ) =, 6,,, (= /) =.5(.5), Δ =.,.5, ad k = ψ ( ˆ ) = e Some of these computatios are give i the tables ()-(). The observatio metioed i the tables lead to the followig results:
Double stage shrikage estimator 9 i. The Relative Efficiecy [] of % are adversely proportioal with small value of Δ especially whe =, i.e. Δ =. yield highest efficiecy ii. The Relative Efficiecy [] of % has maximum value whe = (=), for each, Δ, ad decreasig otherwise ( ). This feature show the importat usefuless of prior kowledge which give higher effects of proposed estimator as well as the importat role of shrikage techique ad its philosophy. iii. Bias ratio [B ( )] of % are reasoably small whe = for each, Δ, ad icreases otherwise. This property show that the proposed estimator % is very closely to ubiasedess property especially whe =. iv. The Effictive iterval of % [the value of % which makes of % greater tha oe] is [.5,.5]. v. Bias ratio [B ( )] of % are icreases with small value of u. vi. R.Eff( % ) is decreasig fuctio with icreasig of the first sample size, for each Δ ad. vii. The expected values of sample size of % are close to, especially whe ad start far-away otherwise. viii. Percetage of the overall sample saved J (a*,b*) is icreasig value with icreasig value of u (u = / ) ad. ix. R.Eff( % ) is a icreasig fuctio with respect to u. This property show the effective of proposed estimator usig small relative to (or large ) which give higher efficiecy ad reduce the observatio cost. x. The cosidered estimator % is better tha the classical estimator especially whe, this will give the effective of % relative to ˆ ad also give a importat weight of prior kowledge, ad the augmetatio of efficiecy may be reach to tes times. xi.the cosidered estimator % is more efficiet tha the estimators itroduced by [], i the sese of higher efficiecy. Refereces [] D. Kudu ad R.D.Gupta, Absolute cotiuous bivariate geeralized expoetial distributio, AStA Advaces i Statistical Aalysis, (95) (), 69 5. [] R.D.Gupta ad D. Kudu, Geeralized expoetial distributios, Australia ad New Zealad Joural of Statistics, () (999), 73-. [3] R.D.Gupta ad D. Kudu, Expoetiated expoetial distributio: alterative to gamma ad Weibull distributios, Biometrical Joural, (3) (), 7-3.
5 Alaa M. Hamad [] R.D.Gupta ad D. Kudu, Geeralized expoetial distributios: differet methods of estimatios, Statistical Computatios ad Simulatios, (69) (), 35-33. [5] R.D.Gupta ad D. Kudu, Geeralized expoetial distributios: statistical ifereces, Statistical Theory ad Applicatios, () (), -. [6] R.D.Gupta ad D. Kudu, Discrimiatig betwee Weibull ad geeralized expoetial distributios, Computatioal Statistics ad Data Aalysis, 3(3), 79-96. [7] M.Z.Raqab, Iferece for geeralized expoetial distributio based o record statistics, Statistical Plaig ad Iferece, () (), 339-35. [] J.R., Thompso, Some shrikage techiques for estimatig the mea, America Statistical Associatio, 63(96), 3-. [9] S.K. Katti, Use of some a prior kowledge i the estimatio of meas from Double Samples, Biometrics, (96), 39-7. [] V.B Waikar, F.J. Schuurma ad T.E Raghuatha, O a Two-Stage shrikage testimator of the mea of a ormal distributio, Commuicatios i Statistics Theory ad Methods, 3 (9), 9-93. []A.N. Al-Joboori,.Pre-Test sigle ad double stage shruke estimators for the mea of ormal distributio with kow variace, Baghdad Joural for Sciece, 7()(),3-. Table () Show Bias ratio [B ( )] ad R.E.ff of % w.r.t Δ, ad whe u = R.Eff. Δ Bias. 6.5 6.5.75.5.75.55 -.5.9-6.39*^-.956-3.9*^-.965.73 -.7.9-3.9*^-6.956.965.36 -.95. -.9.76 -.73. -.9.59 -.55.9 -.3.7 -.5.93-3.9*^-3 6.7 -.35. -.37.6 -.3. -. 3.5 -.65.66 -.5.57 -.5.975.6.57.57.97.66..76.76.735.7.6.977.55.6.6.3.6.6.793.56.97.7.7.59.77.569.7.5.9.53.56.59.97.35.6.59.7.77.59.97.59.39..5.5.73.55.95.53
Double stage shrikage estimator 5 Table () Show Bias ratio [B ( )] ad R.E.ff of % w.r.t Δ, ad whe u = 6 R.Eff. Δ Bias. 6.5 6.5.75.5.75.3 -.93.93-5.63*^-.9-3.5*^-.95.69 -..956-3.53*^-6.9.95.5 -.6.3 -..5 -.6.66 -.5.7 -.3.65 -..7 -.3. -3.365*^-3.9 -.5 7.95 -.6.6 -.5.93 -..97 -. 3.7 -..9 -..73-6.765*^-3..57.9.63.7.73.9.73.9.9.77.569.5.66.96.65.5.799..95..3.7..6.76.369.9.5.6.37..3.6.39.7.6.59..59.59..6.9..57..55 Table (3) Show Bias ratio [B ( )] ad R.E.ff of % w.r.t Δ, ad whe u = Δ R.Eff. Bias..5 6 6.5.75.5.75.6 -.6.95-5.*^-.97-3.39*^-.99.66 -.3.97-3.37*^-6.97.99.9 -.56.65 -.3.37 -.6. -..6 -..5 -.7.65 -..6-3.9*^-3 36. -9.59*^-3 7.6 -..3-9.37*^-3.75-7.76*^-3. -..56 -..75-6.79*^-3.5 -.35*^-3..59.5.6.75.7.759.75.3.96.6.573.655.63.7.69.5..39.95.7.3.39..396.765.33.97.337.66.36.6.5.69.7.73.5.5.7.59.9.3.76..97.5..55
5 Alaa M. Hamad Table () Show Bias ratio [B ( )] ad R.E.ff of % w.r.t Δ, ad whe u = R.Eff. Δ Bias. 6.5 6.5.75.5.75.7 -.5.95-5.393*^-.99-3.36*^-.99.5 -..975-3.9*^-6.99.99.956 -.5.53 -..37 -.6.37 -..9 -.9.9 -.5.63 -.. -3.5*^-3.5 -.7*^-3 6.73 -.*^-3.5-7.75*^-3.67-6.5*^-3 3. -.5.7 -..63-5.77*^-3.55-3.63*^-3.7.53.73.65.76.75.73.75.769.9.565.57.5.63.6.63.9..3.95.5.3.5..37.766.9.97.3.67.33.7.7.7..73.5.5.6.59...65...5..56 Table (5) Show Probability of a Voidig Secod Sample w.r.t Δ, u, ad u Δ.5.75.5.75..6.93.95.96.99.99.5.6.66.3.937.95.95 6. 53.6.5.976.9.99.5 3 6.3.6.93.9.95 6. 7.7.5.9.93.99.5.9.6.3.99.95..55.9.93.9.99.5 7 3.59.79.9.95 Table (6) Show Expected Sample Size of % w.r.t Δ, u, ad whe = u Δ.5.75.5.75..7.59.3..75..5.793 6.655 5..53.3. 6.. 6.57 5.3.35.6..5 7.3.965. 5.5 5.53 5.
Double stage shrikage estimator 53 Table (7) Show Expected Sample Size Proportio w.r.t Δ, u, ad u Δ.5.75.5.75..56.5.365.3.3.3.5.93.555.5.375.365.367 6..6.7.6.53.5.5.73.75.6.9.6 6..66.96.3.6..5.97.76.6.55.36..99.63..95.6.5.993.53.7.5.3 Table () Show Percetage of Overall Sample Saved w.r.t Δ, u, ad u Δ.5.75.5.75.. 59.55 63.9 65.76 66.39 66..5.73.5 5.3 6.6 63.63 63.333 6..5 53.57 7.976 3.657.75.57.5 76 5.69 5.93 77.396.95. 6. 53 3. 5.397 6. 9. 9.999.5 35.6.3 75.7.56 6.363. 9 5 5.397 37.73 5.93 9.537 9.35.5.667.655 73. 5. 7.69 Received: March 5, 3