Al- Mustansiriyah J. Sci. Vol. 24, No 5, 2013

Size: px

Start display at page:

Download "Al- Mustansiriyah J. Sci. Vol. 24, No 5, 2013"

Polly Walton
5 years ago
Views:

1 Al- Mustasiriyah J. Sci. Vol. 24, No 5, 23 Usig Etropy Loss Fuctio to Estimate the Scale Parameter for Laplace Distributio Huda A. Rasheed, Akbal J. Sulta ad Nadia J. Fazah Departmet of Mathematics, college of Sciece, AL Mustasiriya Uiversity Received 2/4/23 Accepted 5/9/23 الخالصة يهدف هذا البحث الى ايجاد أفضل مقدر لمعلمة القياس لتوزيع البالس باستخدام دالة Etropy للخسارة مع دوال اسبقية معلوماتية وغير معلوماتية. لقد تمت المقارنة بين أداء هذه المقدرات مع أداء مقدرات بيز تحت دالة الخسارة التربيعية المعدلة مع نفس دالتي األسبقية تبعا لمتوسط مربعات الخطأ (MSE's). أظهرت النتائج أن مقدر بيز تحت دالة Etropy للخسارة مع دالة اسبقية معكوس كاما كان األفضل عند احجام العينات المتوسطة والكبيرة بينما المقدر تحت التربيعية المعدلة مع دالة اسبقية كاما كان األفضل مع احجام العينات الصغيرة في حالة كون معلمة القياس ذات قيمة صغيرة. ABSTRACT The object of the preset paper is fidig the best estimator for the scale parameter of Laplace distributio usig Etropy loss fuctio with iformative ad oiformative priors are preseted ad compared with bayes estimators uder Modified quadratic loss fuctio with the same two priors. The compariso was made o the performace of these estimators with respect to the mea square error (MSE). The results showed that the Bayes' estimator uder Etropy loss fuctio with Iverted Gamma is the best estimator with a moderate ad large sample sizes while the estimator uder Modified quadratic loss fuctio was better with small sample sizes ad whe the scale parameter has a small value. INTRODUCTION I Bayesia aalysis the ukow parameter is regarded as beig the value of a radom variable from a give probability distributio, with the kowledge of some iformatio about the value of parameter prior to observig the data x, x 2 x. The object of the preset paper is to obtai Bayesia estimates of the scale parameter for Laplace distributio usig Etropy loss fuctio with iformative ad o-iformative priors. The compariso was based o a Mote Carlo study. The efficiecy for the estimators was compared accordig to the mea square error (MSE). Bayes' Estimators Let x, x 2,, x be a radom sample of size, the items have a idepedet ad idetically Laplace distributio, with probability desity fuctio give by [] : f(x a, b) = 2b x a exp [ ] < x < () b < a <, b > Where a is the locatio parameter ad b is the scale parameter. Bayes' estimators for the scale parameter b was cosidered with Etropy loss fuctio ad Modified quadratic loss fuctio with iformative loss 253

2 Usig Etropy Loss Fuctio To Estimate The Scale Parameter For Laplace Distributio Huda, Akbal ad Nadia fuctio represeted by Iverted Gamma prior ad o-iformative prior which represeted by Jeffrey prior.. Bayes estimator uder etropy loss fuctio Etropy loss fuctio was first itroduced by James ad Stei for the estimatio of the Variace-Covariace (i.e., Dispersio) matrix of the Multivariate ormal distributio. Dey et al. [3],[7] cosidered this loss fuctio for simultaeous estimatio of scale parameters ad their reciprocals, for p idepedet gamma distributios. Rukhi ad Aada cosidered the estimatio problem of the variace of a Multivariate Normal vector uder the Etropy loss ad Quadratic loss.[7] We cosider the etropy loss fuctio of the form:[6] L(b, b ) = w [( b ) l (b ) ], w > o (2) b The the Bayes estimator of b is:[6] b b = [E ( b X) ] (3) Where E ( b X) = b h(b X)db h(b X) is the posterior distributio. Now, accordig to Etropy loss fuctio we will estimate the scale parameter for Laplace distributio usig iformative ad o iformative priors as follows: (i) Posterior distributio usig Iverted Gamma prior (IG) Assumig that b has iformative prior as Iverted Gamma prior which takes the followig form [4] : g (b) = αβ. Γβ b β+ e-(α b), α, β, b> (4) So, the posterior distributio for the parameter b give the data (x, x 2, x ) is: f(x i b)g (b) h (b X) = ( = ( π i= π i= f(x i b)g (b)db b (+β+))e b i= b (+β+))e b i= x i a +α x i a +α db b i= = x i a +α +β. /b (+β+). e Γ( + tβ) x i a +α +β ( + β)b (+β+) e The the posterior distributio became as follows: b i= x i a +α x i a +α db 254

3 Al- Mustasiriyah J. Sci. Vol. 24, No 5, 23 h (b X) = ( x i a + α) +β. e b i= x i a +α b (+β+) Γ( + β) Now, otice that b~ig( + β, ( x i a ) + α ) Let T = b Thus T~ G( + β, /( x i a ) + α ) ad E(T X) = [ (5) +β ] ( i= x i a )+α Accordig to the Etropy loss fuctio, the correspodig Bayes' estimator for b is such that: b = [E(T X)] = [ b = i= x i a + +β ] ( i= x i a ) = i= x i a + +β + β (ii) posterior distributio usig Jeffery prior. Followig the form of Jeffrey prior iformatio [] : g 2 (θ) = k θ c, with k a costat, c R+ (7) I the same way, the posterior distributio with Jeffrey prior iformatio will be as follows: h 2 (b X) = b +c e b ( i= x i a ) b +c e b ( i= x i a ) db ( i= x i a ) +c. = b +c e b ( i= Γ( + c ) ( x i a ) +c Γ( + c )b+c. e ( i= x i a ) +c. h 2 (b X) = b +c e b ( i= Γ( + c ) b~ig( + c ), ( i= x i a ) Let T = b Thus, T~G ( + c, ) ( i= x i a ) (6) x i a ) b ( i= x i a ) db x i a ) Hece, accordig to the Etropy Loss Fuctio we get: b 2 = ( i= x i a ) (9) + c 2. Bayes estimator uder Modified Quadratic Loss Fuctio For the estimatio of the scale parameter of Laplace distributio, a modified form of this loss fuctio may be defied as follows:[] L(θ, θ ) = ( θ θ θ )2 () (8) 255

4 Usig Etropy Loss Fuctio To Estimate The Scale Parameter For Laplace Distributio Huda, Akbal ad Nadia b = E ( b X) E ( b 2 X) () (i) Posterior distributio usig Iverted Gamma prior (IG) To estimate the scale parameter for Laplace distributio usig Iverted Gamma prior uder modified quadratic loss fuctio we use h (b X) where: b~ig( + β, ( x i a ) + α ) Ad we foud that: T~G ( + β, ), where T = b i= x i a + β E(T) = (2) ( x i a )+ + β var(t) = ( i= x i a + ) 2 Hece: E(T 2 + β ) = x i a + ) 2 + ( + β) 2 x i a + ) 2 (3) ( i= ( i= ( E(T 2 + β)( + + β) ) = ( i= x i a + ) 2 Accordig to (): b 3 = E(T) E(T 2 (4) ) Substitutig (2) ad (3) i (4), we get: + β b 3 = i= x i a + ( + β)( + + β) x i a + ) 2 ( i= b 3 = i= x i a + (5) + β + (ii) Posterior distributio usig Jeffery prior. The correspodig Bayes estimator for b with posterior distributio h 2 (b x) comes out as: T~G ( + c, ( ), where T = b i= x i a ) + c E(T) = ( (6) i= x i a ) + c var(t) = ( i= x i a ) 2 Thus: E(T 2 ) = (+c )(+c) ( i= x i a ) 2 (7) Substitutig (6) ad (7) i (3), we fid: 256

5 Al- Mustasiriyah J. Sci. Vol. 24, No 5, 23 b 4 = ( i= x i a ) + c (8) Simulatio Results I this sectio, Mote Carlo simulatio study is performed to compare the methods of estimatio by usig mea square Errors (MSE s) as follows: MSE(b ) = R (b i b) 2 i= R Where R is the umber of replicatios. We geerated R=3 samples of size =, 2, 5, ad 3 to represet small, moderate ad large sample sizes from Laplace distributio with the scale parameter b =, 2. I order to compare the Bayes' estimators uder two differet loss fuctios ad two priors, we chose the values of Jeffrey costats; (c =.5, 2, 3) ad for the Iverted Gamma prior (α=.5, 3) with β = 2. The results were summarized ad tabulated i the followig tables for each estimator ad for all sample sizes. Table-: Expected values ad MSE of the differet estimators for Laplace distributio whe b= ad β =2 criteria b b 2 b 3 b 4 α =.5 α = 3 c=.5 c=2 c =3 α =.5 α = 3 c=.5 c=2 c EXP. MSE EXP MSE EXP MSE EXP MSE Table-2: Expected values 6 ad MSE 4 of the differet 3 estimators 9 for Laplace 3 4 distributio whe b=2 ad β =2 Criteria b b 2 b 3 b 4 α =.5 α = 3 c=.5 c=2 c =3 α =.5 α = 3 c=.5 c= EXP MSE EXP MSE EXP MSE EXP MSE

6 Usig Etropy Loss Fuctio To Estimate The Scale Parameter For Laplace Distributio Huda, Akbal ad Nadia RESULTS AND DISCUSSIONS From table () It appears that i small samples ( =, 2) b 3 is the best estimator which represeted bayes estimator with iverted gamma prior uder modified quadratic loss fuctio wheα = 3, while b 3 ad b which represeted bayes estimator with iverted gamma prior uder Etropy loss fuctio, are closed i MSE's with small value of α (α =.5) also we ca say that the estimators uder Etropy loss fuctio with Iverted Gamma prior become the most efficiet with the moderate ad large sample sizes. The results i table (2) showig that MSE s icreases for all estimators whe the scale parameter (b) icrease. We ca see clearly that the bayes estimator with iverted gamma prior uder Etropy loss fuctio b became the best with all sample sizes. I geeral, we ca say that, the Bayes' estimator uder Etropy loss fuctio with Iverted Gamma is the best estimator with a moderate ad large sample sizes while the estimator uder Modified quadratic loss fuctio was better with small sample sizes ad whe the scale parameter has a small value. REFERENCES. Al- Noor. N. H. ad Rasheed, H. Abdullah(22)," Miimax Estimatio of the Scale Parameter of the Laplace Distributio uder Quadratic Loss Fuctio",Iteratioal Joural for Scieces ad Techology, Vol (7), No(3), pp Abbasi, N. (2),"Compariso of Bayes' estimator ad Maximum Etropy estimator for discrete Laplace distributio" It. J. Cotemp. Math. Scieces: Vol.6, No. 9, pp Dey, D.K.; Ghosh, M. ad Sriivasa, C. (987). Simultaeous estimatio of parameters uder etropy loss, J. Statist. Pla. Iferece, 5, Esfahai, M. Nasr, Nematollahi, N.(29), " Admissible ad Miimax Estimators of a Lower Bouded Scale Parameter of a Gamma Distributio uder the Etropy Loss Fuctio", Joural of Mathematical Extesio: Vol. 4, No. (29), Julia,O. ad Vives-Rego, J. (28), "A microbiology applicatio of the skew-laplace distributio". SORT Vol. 32, No. 2, pp Padey, H., & Rao, A. K. (29). Bayesia estimatio of the shape parameter of a geeralized Pareto distributio uder asymmetric loss fuctios. Hacettepe Joural of Mathematics ad Statistics. Vol, 38 (), pp Sigh, S. Kumar, Sigh, U. Ad Kumar, D. (2)," BAYESIAN ESTIMATION OF THE EXPONENTIATED GAMMA PARAMETER AND RELIABILITY FUNCTION UNDER ASYMMETRIC LOSS FUNCTION", REVSTAT Statistical Joural: Vol. 9, No. 3, pp

International Journal of Mathematical Archive-5(7), 2014, Available online through ISSN

International Journal of Mathematical Archive-5(7), 2014, Available online through ISSN 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