Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:99-878 EISSN: 239-844 Joural home page: www.ajbasweb.com Bayesia iferece for Parameter ad Reliability fuctio of Iverse Rayleigh Distributio Uder Modified Squared Error Loss Fuctio Huda A. Rasheed ad 2 Raghda, kh. Aref AL-Mustasiriyah Uiversity Collage of Sciece, Dept. of Math. 2 AL-Mustasiriyah Uiversity Collage of Sciece, Dept. of Math. Address For Correspodece: Huda, A. Rasheed, AL-Mustasiriyah Uiversity Collage of Sciece, Dept. of Math. A R T I C L E I N F O A B S T R A C T Article history: Received September 26 Accepted November 26 Published 28 November 26 Keywords: Iverse Rayleigh, MLE, Bayes estimator, Modified squared error loss fuctio, Jefferys prior ad Gamma prior, Mea squared error, Itegrated mea squared errors. I this study, obtaied some Bayes estimators based o Modified squared error loss fuctio as well as Maximum likelihood estimator for scale parameter ad reliability fuctio of Iverse Rayleigh distributio. I order to get better uderstadig of our Bayesia aalysis, we cosider o-iformative prior for the scale parameter usig Jefferys prior iformatio as well as iformative prior desity represeted by Gamma distributio. Based o Mote-Carlo simulatio study, the behavior of Bayes estimates of the scale parameter of iverse Rayleigh distributio have bee compared depedig o the mea squared errors (MSE s), while the estimates of the reliability fuctio have bee compared depedig o the Itegrated mea squared errors (IMSE s). I the curret study, we observed that, the performace of Bayes estimator for the scale parameter ad reliability fuctio uder Modified squared error loss fuctio with is better tha the correspodig estimators with Jefferys prior, for all cases. INTRODUCTION The iverse Rayleigh distributio is itroduced by Voda. (972). He studies some properties of the MLE of the scale parameter of iverse Rayleigh distributio which is also beig used i lifetime experimets. Gharraph (993) derived five measures of locatio for the Iverse Rayleigh distributio. These measures are the mea, harmoic mea, geometric mea, mode, ad the media. He, also, estimated the ukow parameter usig differet methods of estimatio. A compariso of these estimates was discussed umerically i term of their bias ad root mea square error. I (2) Solima ad other researchers studied the estimatio ad predictio from Iverse Rayleigh distributio based o lower record values, Bayes estimators have bee developed uder squared error ad zero oe-loss fuctios. I (22) Dey discussed the Bayesia estimatio of the parameter ad reliability fuctio of a Iverse Rayleigh distributio usig differet loss fuctio represeted by Square error, LINEX loss fuctio. I (23) Tabassum ad others studied the Bayes estimatio of the parameters of the Iverse Rayleigh distributio for left cesored data uder Symmetric ad asymmetric loss fuctios. I (24) Muhammad Shuaib Kha obtaied the Modified Iverse Rayleigh Distributio as a special case of Iverse Weibull, which is extesio to it. I (25) Guobig discussed Bayes estimatio for Iverse Rayleigh model uder differet loss fuctios represeted by squared error loss, LINEX loss ad etropy loss fuctios. I (25) Rasheed ad others compared betwee some classical estimators with the Bayes estimators of oe parameter Iverse Rayleigh distributio uder Geeralized squared error loss fuctio. I (26) Rasheed ad Aref obtaied ad discussed the Bayesia approach for estimatig the scale parameter of Iverse Rayleigh distributio uder differet loss fuctio. Fially, i (26) Rasheed ad Aref obtai Reliability Estimatio i Iverse Rayleigh Distributio usig Precautioary Loss Fuctio. Ope Access Joural Published BY AENSI Publicatio 26 AENSI Publisher All rights reserved This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). http://creativecommos.org/liceses/by/4./ To Cite This Article: Huda A. Rasheed ad Raghda, kh. Aref., Bayesia iferece for Parameter ad Reliability fuctio of Iverse Rayleigh Distributio Uder Modified Squared Error Loss Fuctio. Aust. J. Basic & Appl. Sci., (6): 24-248, 26
242 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248. Oe parameter Iverse Rayleigh distributio: The probability desity fuctio (pdf) of the Iverse Rayleigh distributio with scale parameter θ is defie as follows: (Shawky, A.I. ad M.M. Badr, 22) f(x; θ) = 2θ exp ( θ x 3 x2) x >, θ > () The correspodig cumulative distributio fuctio (C.D.F), is give by: F(x; θ) = exp ( θ x2) ; x >, θ > (2) The Reliability fuctio of iverse Rayleigh distributio is give by: R(x; θ) = F(x; θ) = exp ( θ x2) x >, θ > (3) The Hazard fuctio of Iverse Rayleigh distributio is give by: f(x; θ) h(x; θ) = R(x; θ) After substitutio () ad (3) ito h(x; θ), yields: h(x; θ) = 2θ x 3 exp( θ x 2) exp( θ x 2) = 2θ x 3 (exp( θ x 2) ) (4) 2. Estimatio of parameter ad the Reliability fuctio I this sectio, we estimate the scale parameter θ ad the Reliability fuctio R(t) usig Maximum Likelihood Estimator i additio of some Bayesia estimators, as follows. Maximum likelihood estimator: The Maximum likelihood estimator of the scale parameter θ ad the reliability fuctio R(t) of iverse Rayleigh distributio have bee derived as oe of the classical estimators. The maximum likelihood estimator θ ML of the parameter that maximizes the likelihood fuctio is defied as: i= x 3 i L(x, x 2,, x ; θ) = 2 θ exp ( θ i= x 2) (5) i Takig the partial derivatives for the atural log-likelihood fuctio, with respect to θ ad the, equatig to zero we have: l L(x i ; θ) = θ θ = x 2 i i= Hece, the MLE of θ deoted by θ ML is: θ ML = = i=x 2 T i Where T = i= x 2 i Sice the Maximum likelihood estimator is ivariat ad oe to oe mappig (Sigh, S.K., et al., 2), hece the Maximum likelihood estimator of reliability fuctio will be: R ML (t) = exp ( θ ML t 2 ) (7) 2. Bayes estimators: We provide Bayesia estimatio method icludig iformative ad o-iformative priors, uder Modified squared error loss fuctio to estimate scale parameter ad reliability fuctio of Iverse Rayleigh distributio. 2.. Jefferys prior iformatio: Assume that θ has a o-iformative prior desity defied as usig iformatio g (θ) which is give by (Rasheed, H. A. ad Khalifa, Z. N. 26) g I(θ) Where I(θ) represeted Fisher iformatio, defied as, follows: I(θ) = E [ 2 l f(x;θ) θ 2 ] (6)
243 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 Therefore, g (θ) = b E ( 2 l f(x;θ) θ 2 ), b is a costat (8) Now, takig the secod partial derivative of log f(x; θ) with respect to θ, gives 2 l f(x i ; θ) θ 2 = θ 2 Hece, E ( 2 l f (x i ; θ) θ 2 ) = θ 2 After substitutio ito (8), we get g (θ) = b, θ > (9) θ The posterior desity fuctio is defied as: g(θ)l(θ; x,, x ) h(θ x,, x ) = g(θ)l(θ; x,, x ) h (θ x,, x ) = b θ 2 θ b θ 2 θ i= x 3 i i= x 3 i exp ( θ exp ( θ i= x 2) i i= x 2 i ) h (θ x,, x ) = θ e θt θ e θt x 2 i, T = i=, θ > Hece, the posterior desity fuctio of θ with ca be writte as h (θ x,, x ) = T θ e θt () Γ It is clear, h (θ x,, x ) is recogized as the desity of the Gamma distributio, i.e., θ x~gamma(, T), with E(θ) = T, Var(θ) = T 2 2.2. distributio: Assumig that θ has iformative prior as which takes the followig form g 2 (θ) = βα θ α e θβ Γα ; θ >, α >, β > () Where, β, α are the shape ad the scale parameters respectively. From Bayesia theorem the posterior desity fuctio of deoted by h 2 (θ x) ca be obtaied as g 2 (θ)l(θ; x x 2.. x ) h 2 (θ x) = g 2 (θ)l(θ; x x 2.. x ) Now, combiig (5) ad (), gives h 2 (θ x) = θ α + e θ(t+β) θ α + e θ(t+β) So, the posterior desity fuctio of θ with is: h 2 (θ x) = Pα+ θ α + e θp Γ(α + ), θ > (2) Where, P = T + β Notice that: θ x~gamma(α +, P), with, E(θ) = α+ P, Var (θ) = α+ P 2 2.3.Bayes estimator for θ uder Modified squared error loss fuctio: The modified squared error loss fuctio ca be defied as follows (Al-Baldawi, T. H., 23)
244 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 L(θ, θ) = θ r (θ θ) 2 The Risk fuctio uder the Modified squared error loss fuctio which is deoted by R MS (θ, θ) is R MS (θ, θ) = E[L(θ, θ)] R MS (θ, θ) = θ r (θ θ) 2 h(θ x) (3) R MS (θ, θ) = θ 2 E(θ r x) 2θ E(θ r+ x) + E(θ r+2 x) Takig the partial derivative for R MS (θ, θ) with respect to θ ad settig it equal to zero, gives the Bayes estimator relative to Modified square error loss fuctio which is deoted by θ MS as θ Ms = E(θ r+ x) E(θ r x) (i) With Jefferys prior iformatio: Accordig to the posterior desity fuctio (), the Bayes estimator for θ uder Modified squared error loss fuctio ca be derived as follows: E(θ m x) = θ m h (θ x) E(θ m x) = E(θ m x) = Γ(+m) θm+ e θt T Γ θm+ e θt T +m Γ T m Γ(+m) E(θ m x) = Γ(+m) Γ Tm (5) We get the Bayes estimator for the scale parameter of iverse Rayleigh distributio uder Modified square loss fuctio with deoted by θ MSJ, θ MSJ2 with r =, 3 respectively, are: θ MSJ = E(θ+ x) E(θ x) θ MSJ2 = E(θ+3 x) E(θ 3 x) = + T = + 3 T (ii) With iformatio: Accordig to the posterior desity fuctio (2), the Bayes estimator of θ of Iverse Rayleigh distributio uder Modified squared error loss fuctio we substitute two values of r, r =, 3 respectively ito (4) as follows: E(θ m x) = θ m h 2 (θ x) P α+ θ α+ e θp Γ(α+) θm+α+ e θp P α++m Γ(α+) P m Γ(α++m) = θ m E(θ m x) = Γ(α++m) E(θ m x) = Γ(α++m) Γ(α+) Pm (8) the Bayes estimator for the scale parameter of iverse Rayleigh distributio uder Modified square loss fuctio with deoted by θ MSG, θ MSG2 with r =, 3 respectively, are: θ MSG = E(θ+ x) α + + = (9) E(θ x) P θ MSG2 = E(θ+3 x) E(θ 3 x) = α + + 3 P 2.4.Bayes estimator for R(t) Uder Modified squared error loss fuctio: We ca fid the Bayes estimator for the reliability fuctio R(t) by usig the probability desity fuctio for θ. Accordig to (4), the Bayes estimator for R(t) uder Modified squared error loss fuctio, will be : (4) (6) (7) (2)
245 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 R (t) MS = E((R(t))r+ t) E((R(t)) r (2) t) (i) Bayes estimator for R(t) based o iformatio: To derive the Bayes estimator for the R(t) uder Modified squared error loss fuctio (MSLF) with deoted by R (t) MSJ, we substitute two values of r, r =, 3 respectively ito (2) which required to obtai E(R(t) t), E((R(t)) 2 t), E((R(t)) 3 t) ad E((R(t)) 4 t) as follows: E(R(t) t) = R(t)h (θ t) (22) Sice R(t) = exp ( θ t 2) E(R(t) t) = ( exp ( θ t 2) ) E(R(t) t) = ( Tt2 Tt 2 + ) T θ e θt Γ (23) By the same way, we ca fid E((R(t)) 2 t), E((R(t)) 3 t) ad E((R(t)) 4 t) so, E((R(t)) 2 t) = 2 ( Tt2 Tt 2 + ) + ( Tt2 Tt 2 + 2 ) E ((R(t)) 3 t) = 2 3 ( Tt2 t 2 T + ) + ( Tt2 t 2 T + 2 ) (24) (25) E((R(t)) 4 t) = 2 4 ( Tt2 Tt 2 + ) + 2 ( Tt2 Tt 2 + 2 ) (26) Hece, from (24), (23) we get the Bayes estimator for the R(t) of iverse Rayleigh distributio uder Modified square error loss fuctio with Jefferys prior with r =, which is deoted by R (t) MSJ as follows 2 ( Tt2 Tt 2 + ) + ( Tt2 Tt 2 + 2 ) R (t) MSJ = ( Tt2 (27) Tt 2 + ) Now, from (26) ad (25) we obtai the Bayes estimator for the R(t) of iverse Rayleigh, distributio uder Modified squared error loss fuctio with Jefferys prior with r = 3, which is deoted by R (t) MSJ2 R (t) MSJ2 = 2 4 ( Tt2 Tt 2 + ) + 2 ( Tt2 Tt 2 + 2 ) 2 3 ( Tt2 Tt 2 + ) + ( Tt2 Tt 2 + 2 ) (28) (ii) Bayes estimator for R(t) based o iformatio: To derive the Bayes estimator for the R(t) uder Modified squared error loss fuctio (MSLF) with Gamma prior, that is deoted by R (t) MSG, we'll derive E(R(t) t), E((R(t)) 2 t), E((R(t)) 3 t) ad E((R(t)) 4 t) as follows: E(R(t) t) = R(t)h 2 (θ t) (29) sice R(t) = exp ( θ t 2) E(R(t) t) = ( exp ( θ t 2) ) α+ P α+ θ α + e θp Γ(α+) E(R(t) t) = ( Pt2 Pt 2 + ) By same way we ca fid E((R(t)) 2 t), E((R(t)) 3 t) ad E((R(t)) 4 t). Hece, E ((R(t)) 2 t) = 2 ( Pt2 α+ Pt 2 + ) + ( Pt2 Pt 2 + 2 ) α+ (3) (3)
246 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 E((R(t)) 3 t) = 2 3 ( Pt2 α+ t 2 P + ) + ( Pt2 t 2 P + 2 ) E((R(t)) 4 t) = 2 4 ( Pt2 α+ Pt 2 + ) + 2 ( Pt2 Pt 2 + 2 ) α+ α+ (32) (33) From (3), (3) we ca get the Bayes estimator for the R(t) usig Modified squared error loss fuctio based o with r =, which is deoted by R (t) MSG as follows 2 ( Pt2 α+ Pt 2 + ) + ( Pt2 α+ Pt 2 + 2 ) R (t) MSG = ( Pt2 α+ (34) Pt 2 + ) Now, from (33), (32) we get the Bayes estimator for the R(t) uder Modified squared error loss fuctio based o with r = 3, that is deoted by R (t) MSG2 R (t) MSG2 = 2 4 ( Pt2 Pt 2 + ) α+ + 2 ( Pt2 Pt 2 + 2 ) α+ 2 3 ( Pt2 Pt 2 + ) α+ + ( Pt2 Pt 2 + 2 ) α+ (35) 4. Simulatio Study: I our simulatio study, the process have bee repeated 5 times (L=5). We geerated samples of sizes =, 25, 5, ad from Iverse Rayleigh distributio with θ =.5,.5 ad 3. The values of the parameters are chose to be β =.2,3, α =.3,.8. The expected values ad mea squared errors (MSE's) for all estimates of the parameter θ are obtaied, where: MSE(θ) = L (θ i θ) 2 i= ; i =, 2, 3,, L L ad itegral mea squares error (IMSE) for all estimates of the reliability fuctio of Iverse Rayleigh distributio which is defied as distace betwee the estimate value of the reliability fuctio ad actual value of reliability fuctio that is give as follows: t IMSE (R (t)) = j= MSE(R i (t j )), i =, 2,, L, t the radom limits of t i L IMSE(R (t)) = L [ (R i(t j ) R(t j )) 2 ] t i= t t j= The results were summarized ad tabulated i the followig tables for each estimator ad for all sample sizes as follows: Table : Expected Values ad MSE s of the Differet Estimates for the Iverse Rayleigh Distributio whe θ=.5 Uder Modified Squared Error Loss Fuctio Estimates Criteria MLE r = r = 3 r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 EXP.5533.6863.7929.5827.6782.5284.5579.6859.784.6293.64532 MSE.427.5926.438.4294.5.2452.2854.8442.9846.4772.5648 25 EXP.5279.54266.5844.53493.545.5529.5258.5756.58578.55447.56427 MSE.282.57.226.349.477.74.54.992.227.53.673 5 EXP.52.5242.5483.57.5225.5762.5256.5377.5422.5274.53235 MSE.568.622.793.587.67.523.543.74.79.634.674 EXP.5536.54.5252.588.532.542.5669.5886.5237.546.5665 MSE.265.278.32.27.278.255.26.39.32.283.293 Table 2: Expected Values ad MSE s of the Differet Estimates for the Iverse Rayleigh Distributio whe θ=.5 Uder Modified Squared Error Loss Fuctio Estimates r = r = 3 MLE Criteria r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 EXP.6599.82589 2.5788.53852.666.2292.27494.8789.8789.437.494 MSE.37858.53335.294.892.267.4937.286.42355.42355.299.669 25 EXP.56536.62797.7532.5272.55624.3788.4438.64335.67238.48298.598 MSE.534.365.2343.964.9652.738.737.2464.3752.6856.779 5 EXP.5364.5625.62247.5368.52844.43598.44998.5727.58745.4997.5596 MSE.5.5594.737.456.472.474.3986.5432.576.3962.434 EXP.567.5323.5656.587.555.46834.47559.53784.49733.54529.5458 MSE.2385.255.2882.2256.2296.22.2.2483.22.2567.223
247 Huda A. Rasheed ad Raghda, kh. Aref, 26 Australia Joural of Basic ad Applied Scieces, (6) November 26, Pages: 24-248 Table 3: Expected Values ad MSE s of the Differet Estimates for the Iverse Rayleigh Distributio whe θ=3 Uder Modified Squared Error Loss Fuctio Estimates r = r = 3 Criteria MLE r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 EXP 3.398 3.6579 4.3575 2.625 2.73645.8289.999 3.843 3.226 2.577.656 MSE.5433 2.3342 4.759.5265.48655.4672.2867.53698.656.84222 2.23266 25 EXP 3.37 3.25594 3.564 2.8488 2.9296 2.37333 2.4845 3.6543 3.96 2.5538.3382 MSE.4636.5463.8374.29282.28973.52.4752.3685.3382.34774 2.59893 5 EXP 3.628 3.225 3.24495 2.9293 2.954 2.6463 2.6792 3.3585 3.6432 2.74929.777 MSE.2442.22377.28546.6333.6278.2332.2479.72.777.763 2.7759 EXP 3.324 3.6246 3.23 2.96269 2.97732 2.8326 2.8273 3.29 3.358 2.8688.892 MSE.9542.8.529.853.858.285.9853.8752.892.879 2.88268 Table 4: IMSE's of the Differet Estimates for R(t) of Iverse Rayleigh Distributio where θ =.5, R(t) =.544 Uder Modified Squared Error Loss Fuctio MLE r = r = 3 r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8.77.29.7.65.94.2.6.488.679.9.245 25.59.66.37.59.64.49.5.8.49.949. 5.27.28.952.27.28.24.25.93.962.87.9.3.3.87.3.3.2.2.86.876.832.847 Table 5: IMSE's of the Differet Estimates for R(t) of Iverse Rayleigh Distributio where θ =.5,R(t) =.53583 Uder Modified Squared Error Loss Fuctio MLE r = r = 3 r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8.7.75.397.39.45.497.45.242.2767.97.73 25.265.275.2942.28.23.24.98.2385.2539.626.75 5.25.27.256..2..7.2299.2379.892.962.6.6.2386.56.57.56.55.2259.2299.247.285 Table 6: IMSE's of the Differet Estimates for R(t) of Iverse Rayleigh Distributio where θ = 3, R(t) =.2834688 Uder Modified Squared Error Loss Fuctio MLE r = r = 3 r = r =3 β =.2 β = 3 β =.2 β = 3 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8 α =.3 α =.8.68.9.3888.853.74.2683.234.334.586.254.262 25.434.426.374.375.349.84.757.939.282.86.94 5.29.28.2742.94.87.334.3.26.224.42.486.2.2.2587.98.96.36.29.2292.2335.866.95 Discussio:. The results of the simulatio study for estimatig the scale parameter (θ) of Iverse Rayleigh distributio show that: From table (), whe the θ=.5, the performace of Bayes estimator uder Modified squared error loss fuctio with (β=3, α=.3 ad r=) is the best estimator comparig to the other estimators for all sample size. From table (2), whe the θ=.5, it is a clear that, the performace of Bayes estimator uder Modified squared error loss fuctio with (β=3, α=.3 ad r=2) is the best estimator comparig to the other estimators for all sample size except the sample (). From table (3), we observed that, the performace of Bayes estimator uder Modified squared error loss fuctio with (β=.2, α=.8 ad r=) is the best estimator comparig to the other estimators for all sample size. 2. The results of the simulatio study for estimatig the reliability fuctio R(t) of Iverse Rayleigh distributio show that From table (4), otice that, the performace of Bayes estimator uder Modified squared error loss fuctio with (β=3, α=.3 ad r=) is the best estimator comparig to the other estimators for all sample size. From table (5), we observed that, the performace of Bayes estimator uder Modified squared error loss fuctio with (β=3, α=.3 ad r=) is the best estimator comparig to the other estimator for all sample size except the sample (). From table (6), it is a clear that, the performace of Bayes estimator uder Modified squared error loss fuctio (MSELF) with (β=.2, α=.8 ad r=) is the best estimator comparig to the other estimator for all sample size except the sample (). I geeral, we coclude that, i situatio ivolvig estimatio of scale parameter(θ) ad reliability fuctio R(t) of Iverse Rayleigh distributio uder Modified squared error loss fuctio usig for all samples sizes. REFERENCES Al-Baldawi, T., 23, "Compariso of Maximum Likelihood ad some Bayes Estimators for Maxwell Distributio based o No-iformative Priors", Baghdad Sciece Joural, (2): 48-488. Dey, S., 22. "Bayesia estimatio of the parameter ad reliability fuctio of a iverse Rayleigh distributio", Malaysia Joural of Mathematical Scieces, 6(): 3-24.
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