Iteratioal Joural of Mathematical Archive-5(7), 214, 11-117 Available olie through www.ijma.ifo ISSN 2229 546 USING SQUARED-LOG ERROR LOSS FUNCTION TO ESTIMATE THE SHAPE PARAMETER AND THE RELIABILITY FUNCTION OF PARETO TYPE I DISTRIBUTION Huda, A. Rasheed* Al-Mustasiriya Uiversity, Collage of Sciece, Dept. of Math., Iraq. Najam A. Aleawy Al-Gazi Math teacher at the Miistry of Eductio Dhi Qar Iraq. (Received o: 15-6-14; Revised & Accepted o: 16-7-14) ABSTRACT I this paper, we derived Bayes estimators for the shape parameter ad the reliability fuctio of the Pareto type I distributio uder Squared-Log error 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 prior. Accordig to Mote-Carlo simulatio study, the performace of these estimators is compared depedig o the mea square Errors ( s). Key words: Pareto distributio, Reliability fuctio, Maximum Likelihood Estimator, Bayes estimator, Squared-Log error loss fuctio, Jeffery prior ad Expoetial prior. 1. INTRODUCTION The Pareto distributio is amed after the ecoomist Vilfredo Pareto (1848-1923), this distributio is first used as a model for the distributio of icomes a model for city populatio withi a give area, failure model i reliability theory [1] ad a queuig model i operatio research [5]. A radom variable X, is said to follow the two parameter Pareto distributio if its pdf is give by: θα θ ; x α, α >, θθ > (1) f(x; α, θ) = xθ+1, otherwise where α ad θ are the scale ad shape parameters respectively. The cumulative distributio fuctio (CDF) i its simplest form is give by: θ F(x; α, θ) = 1 α x, x α; α, θ >, otherwise So, the reliability fuctio is: R(t) = α t θ (2) (3) Correspodig author: Huda, A. Rasheed* Al-Mustasiriya Uiversity, Collage of Sciece, Dept. of Math., Iraq. Iteratioal Joural of Mathematical Archive- 5(7), July 214 11
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. I this paper, for the simplificatio we ll assume that α = 1 2. MAXIMUM LIKELIHOOD ESTIMATOR Give x 1, x 2,, x a radom sample of size from Pareto distributio, we cosider estimatio usig Maximum likelihood method as follows: L(x 1,, x θ) = f(x ; θ) L(x 1,, x θ) = θ (θ+1) lx e The Log- likelihood fuctio is give by l L(x 1,, x θ) = l θ (θ + 1) l x i Differetiatig the log likelihood with respect to θ: [l L(x 1,, x θ)] = θ θ l x i Hece, the MLE of θ is: θ ML = l x i θ ML = T, where T = l x i (4) Usig the ivariat property, the MLER ML (t) for R(t) may be obtaied by replacig θ by its MLEθ ML i (3) [6] R ML (t) = 1 t θ ML (5) 3. BAYES ESTIMATOR UNDERSQUARED-LOG ERROR LOSS FUNCTION Bayes estimators for the shape parameter θ ad Reliability fuctio were cosidered uder squared-log error loss fuctio with o-iformative prior which represeted by Jeffrey prior ad iformative loss fuctio represeted by Expoetial prior where the Squared-log error loss fuctio is of the form: L θ, θ = l θ l θ 2 Which is balaced with LLLLLL L θ,θ as θ or. A balaced loss fuctio takes both error of estimatio ad goodess of fit ito accout but the ubalaced loss fuctio oly cosiders error of estimatio. This loss fuctio is covex for θ θ e ad cocave otherwise, but its risk fuctio has a uique miimum with respect to θ. [3] Accordig to the above metioed loss fuctios, we drive the correspodig Bayes' estimators for θ usig Risk fuctio R θ θ which miimizes the posterior risk R θ θ = E L θ, θ = l θ l θ 2 h(θ x 1.. x )dθ = (l θ ) 2 2 lθ E(l θ x) + E((l θ) 2 x) Rsik θ = 2 l θ 1 θ 2 E((l θ) x) θ By lettig, R θ θ = θ The Bayes estimator for the parameter θ of Pareto distributio uder the squared-log error loss fuctio is: θ = Exp[E(lθ x)] (6) 214, IJMA. All Rights Reserved 111
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. Accordig to the Squared-Log error loss fuctio, the correspodig Bayes' estimator for the reliability fuctio will be: R (t) = Exp[E(lR(t) t)] (7) E(lR(t) t) = lr(t) h(θ t)dθ We have R(t) = 1 t θ Hece, E[lR(t)] = l 1 E[θ] (8) t Substitutig (8) i (7), we get: R (t) = Exp l 1 E[θ] (9) t 4. PRIOR AND POSTERIOR DISTRIBUTIONS I this paper, we cosider iformative as well as o-iformative prior desity for θ i order to get better uderstadig of our Bayesia aalysis as follows: (i) Bayes Estimator Usig Jeffery Prior Iformatio Let us assume that θ has o-iformative prior desity defied by usig Jeffrey prior iformatio g 1 (θ) which give by: g 1 (θ) I(θ) where I(θ) represets Fisher iformatio which defied as follows: I(θ) = E 2 lf θ 2 g 1 (θ) = C E 2 lf θ2 (1) lf(x; θ) = lθ (θ + 1)lx lf θ = 1 θ lx 2 lf θ 2 = 1 θ 2 E 2 lf θ 2 = 1 θ 2 After substitutio i (1) we fid that: g 1 (θ) = c θ So, the posterior distributio for θ usig Jeffery prior is: L(x 1,, x θ)g 1 (θ) h 1 (θ x) = L(x 1,, x θ)g 1 (θ)dθ θ e (θ+1) lx c θ = θ e (θ+1) lx c dθ θ = T θ 1 e θt Г() This posterior desity is recogized as the desity of the gamma distributio: (11) 214, IJMA. All Rights Reserved 112
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. θ~gamma, l x i, with: E(θ) = Now,, ver(θ) = l x i ( l x i ) T E(lθ x) = Г() l θ θ 1 e θt dθ 2 Let y = θt Hece, T E(lθ x) = Г() = T 1 l y T y T dy y e T Г()T [l y l T]y 1 e y dy = l y y 1 e y dy l T y 1 e y dy Г() Г() E(lθ x) = φ() l T (12) Where, φ() = Г() is the digamma fuctio [5] Г() Substitutig (12) i (6), we get θ J = Exp φ() l T (13) Now, usig (9) to estimate Reliability fuctio we reach to: = Exp T l 1 (14) t We ca otice that is equivalet to the Maximum Likelihood Estimator for R(t). (ii) Posterior Distributio Usig Expoetial Prior Distributio Assumig that θ has iformative prior as Expoetial prior, which takes the followig form: g 2 (θ) = 1 λ e θ λ, θ, λ > So, the posterior distributio for the parameter θ give the data (x 1, x 2, x ) is: π f(x i θ)g 2 (θ) h 2 θ X = π f(x i θ)g 2 (θ)dθ The the posterior distributio became as follows: T + 1 ג +1 θ e θ T+1 ג h 2 (θ t) = Г( + 1) (15) This posterior desity is recogized as the desity of the gamma distributio where: θ~gamma + 1, 1 + l x λ i, With: + 1 + 1 E(θ) = 1 + l x, ver(θ) = λ i ( 1 + l x λ i) The Bayes estimator uder Squared-Log error loss fuctio will be: θ = Exp lθ h E 2 (θ t)dθ 2 214, IJMA. All Rights Reserved 113
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. = lθ T + 1 +1 θ e θ T+ λ Г( + 1) 1 λ dθ (16) Let y = θ T + 1 λ Substitutig i (16), we have: E(lθ x) = T + 1 +1 λ y y l Г( + 1) T + 1 T + 1 λ λ By simplificatio, we get: E(lθ x) = φ( + 1) + l T + 1 λ e y dy T + 1 λ = ExP φ( + 1) + l T + 1 (17) λ Now, the correspodig Bayes estimator for with posterior distributio (15), come out as: = ExP ( + 1)l (1 ) t T + 1 λ 5. SIMULATION RESULTS I our simulatio study, we geerated I = 25 samples of sizes = 2, 5, ad1 from Pareto type I distributio to represet small, moderate ad large sample size with the shape parameter θ =.5, 1.5, 2.5 ad takig t = 1.5, 3. 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 by usig mea square Errors ( s) as a idex for precisio to compare the efficiecy of each of estimators, where: (θ ) = I θ i θ 2 I The results were summarized ad tabulated i the followig tables for each estimator ad for all sample sizes. 6- NUMERICAL VALUES OF ESTIMATOR (θθ ) The expectatios ad s for θθ are schedule i tables (1, 2, ad 3) accordig to the sequece of tables as follows: Table - 1: Expected Values ad s of the Parameter of Pareto Distributio with θ =.5 N 2 5 1 Bayes(Jeffery) θ J.5161.15625.54876.5465.5249.2633 λ=.5.51388.13726.54564.5226.52345.2578.537547.17743.513291.576.5661.276 214, IJMA. All Rights Reserved 114
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. Table - 2: Expected Values ad s of the Parameter of Pareto Distributio with θ = 1.5 N 2 5 1 θ J 1.548297.14624 1.514626.49183 1.57228.2371 λ=.5 1.395296.91596 1.454416.41884 1.477195.21876 1.583429.143131 1.529289.49711 1.514685.23851 Table - 3: Expected Values ad s of the Parameter of Pareto Distributio with θ = 2.5 7. DISCUSSION N 2 5 1 θ J 2.585.39623 2.524381.13662 2.51247.65835 λ=.5 2.125352.295328 2.332812.122543 2.41421.62163 2.592182.359498 2.531424.132948 2.515959.6516 From tables (1, 2, 3) whe θ=.5, 1.5, 2.5, the simulatio results show that with λ=.5 was the best i performace, followed by θ J (which equivalet to R ML (t)) for differet size of samples. ad we ca otice that 's icreases with icreases of λ (λ =3). Fially for all sample sizes, a obvious icrease i is observed with the icrease of the shape parameter values. 8. NUMERICAL VALUES OF ESTIMATOR RR (tt) The umerical results are schedule i tables (4, 5, 6, 7, 8, ad 9) accordig to the sequece of tables as follows: Table - 4: Expected Values ad s of the Reliability Fuctio of Pareto Distributio with θ =.5, t = 1.5, (R(t) t=1.5 =.816497) 2 5 1.8798.1732.813546.62.814982.29 λ=.5.88736.153.813665.577.81511.284.8113.1967.81787.6373.81364.299 Table - 5: Expected Values ad s of the Reliability Fuctio of Pareto Distributio with θ =.5, t = 3, (R(t) t=3 =.577353) 2 5 1.564311.572.572896.2122.57528.141 λ=.5.56554.5117.57375.235.57574.12.55155.6282.56768.2211.57243.164 214, IJMA. All Rights Reserved 115
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. Table - 6: Expected Values ad s of the Reliability Fuctio of Pareto Distributio with θ = 1.5, t = 1.5, (R(t) t=1.5 =.5443319) 2 5 1.531299.6138.53989.2299.5429.1131 λ=.5.56397.4498.55324.236.54868.162.52392.6157.536717.236.54388.1134 Table - 7: Expected values ad s of the Reliability Fuctio of Pareto Distributio with θ = 1.5, t = 3, (R(t) t=3 =.192451) 2 5 1.18899.4813.19161.198.191916.13 λ=.5.117.2969.79374.888.71787.38.18183.4547.188568.1931.19364.991 Table - 8: Expected Values ad s of the Reliability Fuctio of Pareto Distributio with θ = 2.5, t = 1.5, (R(t) t=1.5 =.3628883) 2 5 1.352656.6952.359528.2742.3618.1371 λ=.5.41937.748.387663.2868.375469.14.3553.6452.358448.2661.3655.1351 Table - 9: Expected Values ad s of the Reliability Fuctio of Pareto Distributio with θ = 2.5, t = 3, (R(t) t=3 =.71278) 2 5 1.66859.1538.65596.621.64821.312 λ=.5.117.2969.79374.888.71787.38.65278.1385.64983.594.64521.35 9. DISCUSSION From tables (4, 5, 6, 7) whe θ =.5, 1.5 ad t = 1.5, 3 the simulatio results shows that with all sample sizes, with λ=.5 was better i performace tha each of (which equivalet to MLE) ad with. Also tables (4, 5, 6, ad 7) shows that the results of estimators (icluded MLE) are closed especially with large sample sizes. 214, IJMA. All Rights Reserved 116
Huda, A. Rasheed* ad Najam A. Aleawy Al-Gazi / Usig Squared-Log Error Loss Fuctio To Estimate The Shape Parameter ad The Reliability Fuctio of Pareto Type I Distributio / IJMA- 5(7), July-214. Tables (8, 9) showig that, with a large value of θ, (θ = 2.5) s is decreases with icreasig of λ (λ = 3) for the Bayes estimator with expoetial prior so, we ca say that with is better tha each of the estimator with Jeffery prior (MLE) ad with λ=.5. I geeral, we coclude that i situatios ivolvig estimatio of parameter Reliability fuctio of Pareto type I distributio uder Squared-Log error loss fuctio, usig expoetial prior with small value of λ (λ =.5) is more appropriate tha usig Jeffery prior (or MLE) whe t,θ are small relatively (t=1.5, θ =.5). Otherwise usig expoetial prior with large value of λ (λ = 3) is better tha usig Jeffery prior (or MLE). REFERENCES [1] Nadarajah, S. & Kots, S., Reliability i Pareto models, Metro, Iteratioal, Joural of statistic, vol. LXI, No.2, (23), 191-24. [2] Podder, C.K., Compariso of Two Risk Fuctios Usig Pareto distributio, pak. J. States. Vol. 2(3), (24), 369 378. [3] Dey, S., Bayesia Estimatio of the Parameter of the Geeralized Expoetial Distributio uder Differet Loss Fuctios, Pak. J. Stat. Oper. Res., Vol.6, No.2, (21), 163-174. [4] Setiya, P. & Kumar, V. BAYESIAN ESTIMATION IN PARETO TYPE-I MODEL", Joural of Reliability ad Statistical Studies; ISSN (Prit): 974-824, (Olie):2229-5666 Vol. 6, Issue 2 (213), 139-15. [5] Shortle, J & Fischer, M., Usig the Pareto distributio i queuig modelig, Submitted Joural of probability ad Statistical Sciece, (25), 1 4. [6] Sigh, S. K., Sigh, U. ad Kumar, D. Bayesia Estimatio of the Expoetiated Gamma Parameter ad Reliability Fuctio uder Asymmetric Loss Fuctio. REVSTAT Statistical Joural, vol. 9, o.3, (211), 247 26. Source of support: Nil, Coflict of iterest: Noe Declared [Copy right 214 This is a Ope Access article distributed uder the terms of the Iteratioal Joural of Mathematical Archive (IJMA), which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited.] 214, IJMA. All Rights Reserved 117