Bayes Estimators for the Shape Parameter of Pareto Type I. Distribution under Generalized Square Error Loss Function

Transcription

1 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Bayes Estimators for the Shape arameter of areto Type I Distributio uder Geeralized Square Error Loss Fuctio Dr. Huda A. Rasheed * ad Najam A. Aleawy Al-Gazi ** *Dept. of Mathematics / College of Sciece / AL- Mustasiriya Uiversity Iraqaloor1@gmail.com ** Math teacher at the Miistry of Eductio Dhi Qar Iraq Abstract I this paper, we obtaied Basyia estimators of the shape parameter of the areto type I distributio usig Bayia method uder Geeralized square error loss fuctio ad Quadratic loss fuctio. I order to get better uderstadig of our Bayesia aalysis we cosider o-iformative prior for the shape parameter Usig Jeffery prior Iformatio as well as iformative prior desity represeted by Expoetial distributio. These Bayes estimators of the shape parameter of the areto type I distributio are compared with some classical estimators such as, the Maximum likelihood estimator (MLE), the Uiformly miimum variace ubiased estimator (UMVUE), ad the Miimum mea squared error (MiMSE) estimator accordig to Mote-Carlo simulatio study. The performace of these estimators is compared by employig the mea square errors (MSE s). Key words: areto distributio; Maximum likelihood estimator; Uiformly miimum variace ubiased estimator; Miimum mea squared error; Bayes estimator; Geeralized square error loss fuctio; Quadratic loss fuctio; Jeffery prior; Expoetial prior. 1. Itroductio The areto distributio is amed after the ecoomist Vilfredo areto ( ), this distributio is first used as a model for distributig icomes of model for city populatio withi a give area, failure model i reliability theory [7], ad a queuig model i operatio research [12]. A radom variable X is said to follow the two parameters of areto distributio if its pdf is give by[2]: f(t, θ) = θαθ t α, α >, θ > (1) tθ+1 Where t is a radom variable, θ ad α are the shape ad scale parameters respectively. The cumulative distributio fuctio of areto distrcibutio type I is give by [9] F(x) = 1 ( α t ) θ t α, α >, θ > (2) Therefore, the reliability fuctio is give as follows [2]: R (t) = r (T > t) = f(t, α, θ)dt t = θαθ u θ+1 du = θαθ [ 1 θt θ] t R(t) = ( α t ) θ, t α, θ >, α > (3) Ifereces of areto distributio have bee studied by may authors, Quadt (1966) [8]has obtaied differet estimators for the parameters of areto distributio usig the method of maximum likelihood, method of least 2

2 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 square ad quatile method ad discussed their properties. Charek, D. J. i (1985) [4] walked a compariso of estimatio techiques for the three parameters of areto distributio ad studied the estimator roperties. Rytgaard i (199) [9] estimated the shape parameter ad scale parameter of areto distributio by usig Maximum Likelihood method ad momets method ad used the simulatio style. The researcher foud that the maximum likelihood method is the better. Giorgi, G.M. ad Crescezi, M.(21) [6] estimated the shape parameter of areto distributio-type I by usig Bayes estimator uder squared loss fuctio ad, as prior distributios, the trucated Erlag ad the traslated expoetial oe. Set of Bayesia methods ad cocluded that the proposed methods are better tha the stadard Bayes ad also suggested a method was likig betwee the usual methods ad Bayesia. Ertefaie, A. ad arsia A.(25) [5] estimated the parameters of areto distributio by usig Bayes estimators uder LINEX loss fuctio whe the scale parameter is kow ad both scale ad shape parameters are ukow. Sigh G., B.., S. K., U., ad R.D.(211) [1] estimated the shape parameter of classical areto distributio uder prior iformatio i the form of a poit guess value whe the scale parameter is ukow. Al-Athari M.(211) [1] discussed the estimate of the parameter for Double areto distributio ad, this study cotracted with maximum likelihood, the method of momets ad Bayesia usig Jeffery's prior ad the extesio of Jeffery's prior iformatio ad, based o the results of the simulatio, the maximum likelihood ad Bayesia method with Jeffery's prior are foud to be the best with respect to MSE. 2. Some Classical Estimators of Shape arameter I this sectio, we obtai some classical estimators of the shape parameter for the areto distributio represeted by Maximum likelihood estimator, Uiformly Miimum Variace Ubiased Estimator ad Miimum mea square error estimator. Give x 1,x 2,, x a radom sample of size from areto distributio, we cosider estimatio usig method of Maximum likelihood as follows: L(θ; t 1, t 2,, t ) = f(t 1 ; θ). f(t 2 ; θ).. f(t ; θ) = f(t ; θ) L(t 1,, t θ) = θ α θ θ+1 t i i=1 = θ e θlα e = θ α θ (θ+1) lt e (θ+1) lt Takig the logarithm for the likelihood fuctio, so we get the fuctio: l L(θ; t 1,, t ) = l θ + θlα (θ + 1) l t i i=1 i=1 The partial derivative for the log-likelihood fuctio with respect to θ ad the equatig to zero we have: [l L(θ; t 1,, t )] = θ θ + lα l t i = i=1 The MLE of θ deoted by θ MLE is: θ MLE = = i=1 lt i lα l ( t i i=1 α ) (4) 21

3 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 The areto distributio belogs to the expoetial family of distributio [2] as the desity fuctio (1) ca be writte: f(x, θ) = θe θlα (θ+1)l t = θe l t e θl (t α ) Hece, a(θ) = θ, b(t) = e l t, c(θ) = θ, d(t) = l ( t α ) Therefore, statistic = l ( t i i=1 is a complete sufficiet statistic for θ. α ) It is easy to show that, statistic is distributed as Gamma distributio with parameters ad θ as follows: If T ~ areto (α, θ), the: l ( t i ) ~Expoetial(θ) = l α (t i i=1 ) ~ Gamma (, θ), with the desity α g(p) = θ Γ() p 1 e θp, p, θ > E ( i=1 1 l ( t ) = E ( 1 i α ) ) = θ p 2 e θp ᴦ() E ( 1 ) = θ 1 dp (5) Hece, 1 is ubiased estimator for θ, ad represets a complete sufficiet statistics for θ. Thus, by theorem of Lehma-Scheffe, the UMVUE of θ deoted by θ UMVUE is give by: [8] θ UMVUE = 1 (6) Miimum Mea Squared Error (MiMSE) estimator ca be foud i the class of estimators of the form Therefore MSE θ ( c ) = E [(c θ)2 ] C. = c 2 E [( 1 2] ) 2cθE ( 1 ) + θ2 Takig the partial derivative with respect to c o the both sides ad the equatig to zero we have: MSE c θ( c ) = 2cE [(1 )2 ] 2θE ( 1 ) = c = θe (1 ) E ( 1 2 (7) ) E [( 1 ) 2 ] = ( 1 )2 θ = Γ() p 1 e θp dp θ 2 ( 1)( 2) After substitutio (5), (8) ito (7), we have (8) 22

4 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 c = θ 2 1 θ 2 ( 1)( 2) Therefore, we get c = -2 ad hece θ MiMSE = 2 Is the miimum mea square error estimator for θ. Now, we'll derive the MSE for three classical estimators to compare them theoretically accordig to MSE, as follows: MSE(θ MLE ) = MSE θ ( ) = E [( θ)2 ] (9) MSE(θ MLE ) = var ( ) + [E ( ) θ] 2 (1) var ( ) = 2 var ( 1 ) var ( 1 ) = E [(1 ) 2 ] [E ( 1 )] 2 = θ 2 ( 1) 2 ( 2) (11) [E ( 2 ) θ] = [E ( 2 )] + 2θE ( ) + θ2 [E ( ) θ] 2 = θ 2 ( 1) 2 (12) Substitutig (11) ad (12) ito (1) gives MSE(θ MLE ) = θ2 ( + 2) ( 1)( 2) (13) Hece, MSE for UMVUE ad MiMSE are obtaied by the same way, as follows: Recall that, var ( 1 ) = θ 2 ( 1) 2 ( 2) The, MSE(θ UMVUE ) = E [( 1 θ) 2 ] = ( 1) 2 var ( 1 ) + [E ( 1 ) θ]2 (14) ad ( 1) is ubiased estimator for θ, MSE(θ UMVUE ) = ( 1)2 θ 2 ( 1) 2 ( 2) + [θ θ]2 (15) θ2 MSE(θ UMVUE ) = ( 2) (16) MSE(θ MiMSE ) = E [( 2 2 θ) ] = ( 2) 2 var ( ) + [E ( ) θ] MSE(θ MiMSE ) = From (13), (16), (18), we fid that: ( 2)θ2 ( 1) 2 + θ 2 ( 1) 2 = θ2 ( 1) MSE(θ MiMSE ) MSE(θ UMVUE ) MSE(θ MLE ) Now, we ca say that, Miimum Mea Squared Error (MiMSE) is the best estimator amog the Maximum (18) (17) 23

5 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Likelihood Estimator (MLE), ad the Uiformly Miimum Variace Ubiased Estimator (UMVUE), while the Maximum Likelihood Estimator is the worse amog these three estimators. 3. Stadard Bayes Estimator I this sectio, we used a two loss fuctios as followig: a) Geeralized Square Error Loss Fuctio ( GS).[11] Al-Nasser ad Saleh (26) suggested the Geeralized Square Error Loss Fuctio i estimatig the scale parameter ad the Reliability fuctio for Weibull distributio, which itroduced as follows: k L 1 (θ, θ ) = ( a j θ j )(θ θ) 2 j= Where a j, j =,1,2,3,. k is a costat. b) Quadratic Loss Fuctio (QLF).[3], k =,1,2,3,. (19) DeGroot (197) discussed differet types of loss fuctios ad obtaied the Bayes estimates uder these loss fuctios. He proposed the Quadratic loss fuctio which is asymmetric loss fuctio defied for the positive values of the parameter. If θ is a Quadratic Loss Fuctio L 2 (θ, θ ) will be, L 2 (θ, θ ) = ( θ θ 2 θ ) (2) 4. rior ad osterior Distributio: I this study we cosider iformative as well as o-iformative prior: 4.1 Bayes Estimator Usig Jeffery rior Iformatio [5] : Let us assume that θ has o-iformative prior desity defied as usig Jeffrey prior iformatio g(θ) which give by: g 1 (θ) I(θ) Where I(θ) represeted Fisher iformatio which defied as follows: I(θ) = E ( 2 lf θ 2 ) Hece, g 1 (θ) = b E ( 2 lf(t; θ) θ 2 ) (21) lf(t; θ) = lθ + θlα (θ + 1)lt lf θ = 1 + lα lt θ 2 lf θ 2 = 1 θ 2 Hece, we get: E ( 2 lf(t; θ) θ 2 ) = 1 θ 2 After substitutio ito (21) we fid that: g 1 (θ) = b θ > (22) θ The posterior desity fuctio is: g 1 (θ)l(θ; t 1,, t ) h 1 (θ t 1,, t ) = g 1 (θ)l(θ; t 1,, t ) dθ (23) 24

6 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 h 1 (θ t,, t ) = θ e θlα e (θ+1) lx c θ θ e θlα e (θ+1) lx c θ dθ θ 1 e θ = θ 1 e θ dθ Hece, the posterior desity fuctio of h 1 (θ t,, t ) = θ 1 e θ Г(), where α costat θ with Jeffery prior is: (24) The posterior desity is recogized as the desity of the Gamma distributio: θ~gamma (, ). With: E(θ) =, ver(θ) = 2 To obtai Bayes estimator uder Geeralized Square Error Loss Fuctio (GS) with Jeffery prior, Recall that, the Geeralized Square Error Loss Fuctio (GS) is: k l 1 (θ, θ) = ( a j θ j )(θ θ) 2 j=, k =,1,2,3,. l 1 (θ, θ) = (a + a 1 θ + + a k θ k )(θ θ) 2 (25) The the Risk fuctio uder the Geeralized Square Error Loss Fuctio is deoted by R GS (θ, θ)is: R GS (θ, θ) = E[l 1 (θ, θ)] = l 1 (θ, θ)h 1 (θ t)dθ = (a + a 1 θ + + a k θ k )(θ 2 2θ θ + θ 2 )h 1 (θ t)dθ a k θ 2 E(θ k t) = a θ 2 2a θ E(θ t) + a E(θ 2 t) + a 1 θ 2 E(θ t) 2a 1 θ E(θ 2 t) + a 1 E(θ 3 t) + + 2a k θ E(θ k+1 t) + a k E(θ k+2 t) Takig the partial derivative for R GS (θ, θ) with respect to θ ad settig it equal to zero yields θ = a E(θ t) + a 1 E(θ 2 t) + + a k E(θ k+1 t) a + a 1 E(θ t) + + a k E(θ k t) (26) Sice θ~г(, ) ad E(θ) =, var(θ) = 2.The, a θ + a ( + 1) ( + k)( + k 1).. ( + 1) 1 = a k k+1 a + a a ( + k 1)( + k 2).. ( + 1) k k Therefore, the Bayes estimator for θ of areto distributio uder Geeralized square error loss fuctio with Jeffery prior deoted by θ JGS is: k Г( j) j= a j θ JGS = j+1 Г() k Г( + j) j= a i j Г() (27) 25

7 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Now, we derive Bayes estimator usig Quadratic Loss fuctio, where l 2 (θ, θ) = ( θ θ 2 θ ) = (1 θ 2 θ ) The Risk fuctio uder the Quadratic Loss fuctio is deoted by R Q (θ, θ): R Q (θ, θ) = E(1 θ θ )2 = (1 θ θ )2 h 1 (θ t)dθ R Q (θ, θ) θ Let: R Q(θ,θ) θ = θ Hece, θ = E(1 θ ) = 2 ( 1 θ θ ) ( 1 θ ) h 1(θ t)dθ 1 θ 2 h 1(θ t)dθ 1 θ h 1(θ t)dθ = E( 1 θ 2) (28) E ( 1 θ ) = θ 2 e θ dθ Г() = ( 1) (29) E ( 1 θ 2) = θ 3 e θ dθ 2 = Г() ( 1)( 2) Substitutig (29) ad (3) ito (28), gives ( 1) θ JQ = 2 = 2 ( 1)( 2) It is obvious that, Bayes estimator uder Quadratic loss fuctio with Jeffery prior is equivalet to the MiMSE estimator. 4.2 Bayes Estimator uder Expoetial rior Distributio (3) Assumig that θ has iformative prior as Expoetial prior which takes the followig form: (31) >, θ, e θ = 1 (θ) g 2 Sice, the posterior distributio of θ is: L(t 1,, t θ)g 2 (θ) h 2 (θ t) = L(t 1,, t θ)g 2 (θ)dθ h 2 (θ t) = θ e (θ+1) 1 θ e θ e (θ+1) 1 e θ dθ (32) ) θ(+1 )+1 θ e + 1 ( h 2 (θ t) = (33) Г( + 1) To obtai the Bayes Estimator usig Geeralized Square Error Loss Fuctio (GS), recall that 26

8 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 θ = a E(θ t) + a 1 E(θ 2 t) + + a k E(θ k+1 t) a + a 1 E(θ t) + + a k E(θ k t) Sice θ~г ( + 1, ) ad E(θ) = + 1, var(θ) = + 1 ) ( ( E(θ k + k)( + k 1). ( + 1) ) = ( + 1 K (34) ) After substitutig, we get θ = a { + 1 ( + 2)( + 1) ( + k)( + k 1).. ( + 1) + 1 } + a 1 ( a k )k ( ) a + a 1 { + 1 ( + k)( + k 1). ( + 1) + 1 } + + a k + 1 ( k( So, the Bayes estimator of θ uder Geeralized Square Error loss fuctio deoted by θ EGS is: k Г( j) j= a j ( + 1 )j+1 Г( + 1) θ EGS = k Г( j) j= a j ( + 1 )j Г( + 1) (35) To obtai the Bayes Estimator usig Quadratic loss fuctio (GS) with Expoetial prior, E ( 1 θ ) = 1 h θ 2 (θ t)dθ 1 E ( 1 ( + θ ) = ) ( + 1 ) θ 1 e θ(+1 ) dθ Г() 1 E ( 1 ( + ) = ) θ (36) E ( 1 θ 2) = 1 θ 2 h 2 (θ t)dθ (37) Substitutig (2-18) i (2-131) we get )2 + 1 ( E ( 1 θ 2) = ( 1) )2 + 1 ( E ( 1 θ 2) = ( 1) Substitutig ito (28), we get ( + 1 ) 1 θ 2 e θ(+1 ) dθ Г( 1) 27

9 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 ) + 1 ( θ = )2 + 1 ( ( 1) So, the Bayes estimator of θ uder Quadratic loss fuctio, deoted by θ EQ is: θ EQ = 1 ( + 1 (38) ) 5. Simulatio study I our simulatio study, we geerated I = 25 samples of size = 2, 5, ad1 from areto distributio to represet small, moderate ad large sample size with the several values of shape parameter, θ =.5, 1.5 ad 2.5. We chose two values of λ for the Expoetial prior (λ =.5, 3). I this sectio, Mote-Carlo simulatio study is performed to compare the methods of estimatio usig mea square Errors (MSE s), where MSE(θ ) = I (θ i θ) 2 i=1 I The results of the simulatio study for estimatig the shape parameter (θ) of areto distributio whe the scale parameter (α) is kow, are summarized ad tabulated i tables (1), (2) ad (3) which cotai the Expected values ad MSE's for estimatig the shape parameter, ad we have observed that: 1. Table (1), shows that, the performace of Bayes estimator uder Quadratic loss fuctio with expoetial prior. =) 5) is the best estimator comparig to other estimators for all sample sizes. 2. From table (2), we otice the performace of Bayes estimator uder Geeralized Square error loss fuctio (k=1) with expoetial prior. =) 5), the Bayes estimator uder Geeralized Square (k=2) 5).. =) error loss fuctio with expoetial prior 3. Tables (3), shows that, the performace of Bayes estimator uder Geeralized Square error loss fuctio (k=2) with expoetial prior. =) 5) is the best estimator comparig to other estimators for all sample sizes. Followed by Bayes estimator uder Geeralized Square error loss fuctio (k=1) with expoetial prior,(.5 =) for all sample sizes. 4. It is observed that, MSE's of all estimators of shape parameter is icreasig with the icrease of the shape parameter value with all sample sizes. 5. I geeral, we coclude that, i situatio ivolvig estimatio of parameter of areto type I distributio uder differet loss fuctios usig expoetial prior distributio with small value of = ).5) is more appropriate tha each of usig Jeffery prior distributio or large relatively, value of = ) 3) for all sizes of samples. 6. Fially for all parameter values, a obvious reductio i MSE is observed with the icrease i sample size. Refereces [1] Al-Athari Faris M.(211), " arameter Estimatio for the Double areto Distributio" Joural of Mathematics ad Statistics 7 (4): [2] Adhami A. H. (26),"roperties of areto Distributio ad Compared Methods of Estimatig the Shape arameter", Master of Sciece i operatio research Thesis, college of Admiistratio ad Ecoomic, Baghdad. 28

10 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 [3] Bhuiya M. K., Roy M. K. ad Imam M. F.(27), Miimax Estimatio of the arameter of Rayleigh Distributio, Festschrift i hoor of Distiguished rofessor Mir Masoom Ali o the occasio of his retiremet, [4] Charek D.J.(1985), "A Compariso of Estimatio Techiques for Thicthe Three arameter areto Distributio", Sciece i Space Operatios Thesis, Faculty of the School of Egieerig,Ohio. [5] Ertefaie A. ad arsia A. (25), "Bayesia Estimatio for the areto Icome Distributio uder Asymmetric LINEX Loss Fuctio", JIRSS, Vol. 4, No. 2, pp [6] Giorgi, G. ad Crescez M., (21), "Bayesia estimatio of the Boferroi idex from a areto type I populatio", Spriger verlag, Stat. Method & App., 1: [7] Nadarajah S. ad Kotz S. (23), Reliability for areto Models, Metro Iteratioal Joural of Statistic, Vol. LXI, [8] Quadt R.E. (1964), Old ad New Methods of Estimatio of the areto Distributio, Metrika, 1, [9] Rytgaard M., (199), Estimatio i areto Distributio, Nordisk Reisurace Compay, Groiuge 25, Dk-127 Compehage K, Demark. [1] Sigh G., Sigh B.., Sigh Sajay K., Sigh U. ad Sigh R.D.(211)," Shrikage Estimator ad T Estimators for Shape arameter of Classical areto Distributio, J,Sci. R., Vol.55: [11] Saleh, Makky Akram Mohamed(26) Simulatio of The Methods of The Scale arameter ad The Reliability Fuctio Estimatio for Two arameters Weibull Distributio, Doctor of hilosophy i Mathematics, College of Educatio at Al-Mustasiriyah Uiversity. [12] Shortle, J. & Fischer, M. (25), " Usig the areto Distributio Queuig Modelig, Submitted Joural of robability ad Statistical Sciece, pp

11 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Table (1): The Expected Values ad (MSE) of the Differet Estimators for areto Distributio where θ =. 5 Estimator Criteria MiMSE BJ(GS1), K=1 BJ(GS2), K=2 BJ(Qu.) BE(GS1) K=1 BE(GS2) K=2 BE(Qu.) =. 5 = 3 =. 5 = 3 =. 5 = 3 EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX EX MSE EX MSE EX Best Estimator BE(Qu.) =.5 BE(Qu.) =.5 BE( Qu.) =.5 3

12 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Table (2): The Expected Values ad (MSE) of the Differet Estimators for areto Distributio where θ = 1. 5 Estimator Criteria MiMSE BJ(GS1), K=1 BJ(GS2), K=2 BJ(Qu.) BE(GS1) K=1 BE(GS2) K=2 BE(Qu.) =. 5 = 3 =. 5 = 3 =. 5 = 3 EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE Best Estimator BE(GS1) =.5 BE(GS1) =.5 BE(GS1) =.5 31

13 Mathematical Theory ad Modelig ISSN (aper) ISSN (Olie) Vol.4, No.11, 214 Table (3): The Expected Values ad (MSE) of the Differet Estimators for areto Distributio where θ = 3. 5 Estimator Criteria MiMSE BJ(GS1), K=1 BJ(GS2), K=2 BJ(Qu.) BE(GS1) K=1 BE(GS2) K=2 BE(Qu.) =. 5 = 3 =. 5 = 3 =. 5 = 3 EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE EX MSE Best Estimator BE(GS2) BE(GS2) BE(GS2) 32

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