Bayesian and E- Bayesian Method of Estimation of Parameter of Rayleigh Distribution- A Bayesian Approach under Linex Loss Function

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1 Iteratioal Joural of Statistics ad Systems ISSN Volume 12, Number 4 (217), pp Research Idia Publicatios Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh Distributio- A Bayesia Approach uder Liex Loss Fuctio Isha Gupta Departmet of Statistics; Uiversity of Jammu, Jammu, Idia. Abstract I this paper, Bayesia ad E Bayesia method of estimatio are proposed for estimatig the parameter of Rayleigh distributio. The Bayes estimate of the parameter is derived uder the assumptio that the prior distributio is iformative i.e. gamma prior usig Liex loss fuctio. Further, compariso betwee the E-Bayes estimators with the associated Bayes estimators have bee carried out through simulatio study usig MATLAB software. Keywords: Rayleigh distributio, Liex Loss fuctio, Bayes ad E-Bayes estimators, Gamma prior. 1. INTRODUCTION The Rayleigh distributio is a cotiuous probability distributio servig as a special case of the well-kow Weibull distributio. This distributio has log bee cosidered to have sigificat applicatios i fields such as survival aalysis, reliability theory ad especially commuicatio egieerig. The Rayleigh distributio provides a populatio model which is useful i several areas of statistics (Feradez, 2). I the literature, may researcher studied properties of the Rayleigh distributio, particularly i life testig ad reliability. Whe cosiderig the complete Rayleigh model, the probability desity fuctio is give by f (x, θ) = 2 θ x e θx2, x, θ > ; (1) usig the parameterizatio of the distributio as proposed by Bhattacharya ad Tyagi

2 792 Isha Gupta (199), ad is deoted by X Rayleigh(θ). The parameter θ is a scale parameter, ad characterizes the lifetime of the object uder cosideratio i applicatio. May authors have studied Rayleigh distributio e.g. Ferreira et al. (216) have proposed Bayes estimators icludig shrikage estimators of the ukow parameter of the cesored Rayleigh distributio usig Al-Bayyati loss fuctio cosiderig differet objective prior distributio. Dey (212) has obtaied Bayes estimator of parameter ad reliability fuctio of iverse Rayleigh distributio uder two loss fuctios ad also obtaied associated risk fuctios of the Bayes estimator. Saat et al. (216) have proposed Bayes estimate of parameter of Rayleigh distributio usig Quasi prior uder differet loss fuctios. Ahmed et al. (213) have obtaied Bayes estimate of parameter of Rayleigh distributio usig Jeffrey s ad extesio of Jeffrey s prior uder Squared error ad Al-Bayyati s loss fuctio. The mai objective of this paper is to itroduce a statistical compariso betwee the Bayesia ad Expected- Bayesia procedures for estimatig the parameter of Rayleigh distributio. The resultig estimators are obtaied by usig Liex loss fuctio. The layout of the paper is as follow. I Sectio 2, Bayes estimate of parameter have bee obtaied usig cojugate prior uder Liex loss fuctio. I Sectio 3, E- Bayes estimate have also bee obtaied usig three differet prior distributios. Fially, compariso betwee Bayes ad E-Bayes estimates have bee made usig simulatio study i Sectio 4. Some cocludig remarks have bee give i Sectio BAYESIAN ESTIMATION. I this sectio, Bayes estimate of parameter of Rayleigh distributio is obtaied by usig Liex loss fuctio. Let X 1, X 2, X 3..be a sequece of radom variables from Rayleigh distributio, whose desity fuctio is give by (1), the the likelihood fuctio is give by L( x, θ ) = (2θ) i=1 x i e θ i=1 (x i )2 (2) We use the gamma cojugate prior desity for the parameter θ ad the pdf of gamma prior desity with scale parameter r is give by g( θ / r) = e θ θ r 1 ; r >, θ > (3) Г(r) O combiig (2) ad (3), ad usig Bayes theorem, the posterior desity of θ give x is give by P ( θ x) α L ( x, θ) g( θ / r) = (2θ) i=1 x i e θ i=1 (x i )2 e θ θ r 1 Г(r) ; r >, θ >, x >

3 Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh 793 = k θ +r 1 e θ [ i=1 (x i )2 + 1 ] ; where k is idepedet of θ Thus, Posterior desity is give by P ( θ x) = e θ [ (x i ) 2 i=1 + 1 ] θ +r 1 [ i=1 (x i ) 2 + 1] +r Г (+r) Bayesia estimatio of θ uder liex loss fuctio. Zeller represet Liex ( i.e. liear expoetial ) loss fuctio as L (θ, θ ) = a {exp[b(θ θ) b(θ θ) 1]} ; ; x >, θ >, r > (4) where a >, b ; a is scale of loss fuctio ad b determies its shape. Without loss of geerality, we assume a = 1 ad obtai Bayes estimate of θ. Here, E[L (θ, θ )] = L (θ, θ ) θ [ e i=1 (x i )2 + 1 ] θ +r 1 [ i=1 (x i ) 2 + 1] +r Г (+r) P ( θ x) dθ = [ e [b( θ θ)] b(θ θ) 1] = e bθ [ dθ i=1 (x i ) b + (x i ) 2 +r ] bθ + i=1 + 1 b ( + r ) Thus Bayes estimator of θ uder Liex loss fuctio is give by θ BL = +r + 1 i= (x i ) 2 log b [b + i=1 (x i )2 ] (5) i=1 (x i ) EXPECTED - BAYESIAN ESTIMATE OF PARAMETER USING LINEX LOSS FUNCTION Ha (27) itroduced a ew method, amed E- Bayesia method to estimate failure probability. The E Bayes estimate of θ i.e. expectatio of the Bayes estimate of θ is give by θ EB = E ( θ x) = θ BE π(θ, r) dr ;. Q where Q is the domai of r for which the prior desity is decreasig i θ. θ BE is the Bayes estimate of θ uder the Liex loss fuctio. I order to obtai E-Bayes estimates of θ, we have to choose prior distributio of hyper parameter r. These distributios are used to study the impact of differet prior distributios o E-Bayes estimatio of θ. The followig distributios of r are give by π 1 (θ, r) = 2 (c r) c 2 ; < r < c, (6)

4 794 Isha Gupta π 2 (θ, r) = 1 ; < r < c, (7) c π 3 (θ, r) = 2 r c2 ; < r < c, (8) E-Bayesia Estimatio of θ uder Liex loss fuctio : E-Bayesia estimate of θ relative to Liex loss fuctio based o π 1 (θ, r), is deoted by θ EBL1 ad is obtaied by usig (5) ad (6). c θ EBL1 = θ BE π 1 (θ, r) dr c = { + r b log [b + i=1 (x i) i=1 (x i ) ]} 2 (c r) c 2 dr = [ 3+c ] log 3b [b + i=1 (x i )2 + 1 ] (9) i=1 (x i ) E-Bayesia estimates of θ based o π 2 (θ, r), is deoted by θ EBL2 ad is obtaied by usig (5) ad (7). θ EBL2 = [ 2 + c 2b ] log [ b + i=1 (x i )2 + 1 ] (1) i=1 + 1 E-Bayesia estimates of θ based o π 3 (θ, r), is deoted by θ EBL3 ad is obtaied by usig (5) ad (8). θ EBL3 = [ 3 +2 c 3b (x i ) 2 ] log [ b + i=1 (x i )2 + 1 ] (11) i=1 + 1 (x i ) 2 4. SIMULATION STUDY I order to compare the performace of Bayes ad E-Bayes techiques of estimatio, a simulatio study was coducted usig Matlab software for differet sample sizes ad for differet values of loss parameter. The followig steps were coducted: a. For give value of the prior parameter c, we geerate samples to fid value of r from uiform priors (6-8) respectively. b. For give value of r, we geerate θ from the gamma prior desity (3), c. For kow values of θ, we geerate sample from Rayleigh distributio with pdf (1), ad the Bayes ad Expected Bayes estimates usig Liex loss fuctio are computed from (5), (9), (1) ad (11) respectively. d. The above steps are repeated 5 times ad the mea square error of the Bayes ad E-Bayes estimates are computed ad the results are show i table 1.

5 Bayesia ad E- Bayesia Method of Estimatio of Parameter of Rayleigh 795 Table 1: Averaged values of MSE for Bayes ad E-Bayes estimates of the parameter θ. (sample size) c = 2, r =.5, b =.75 c = 2, r =.5, b = -.75 θ BL θ EBL1 θ EBL2 θ EBL3 θ BL θ EBL1 θ EBL2 θ EBL CONCLUSION I this paper, Bayes ad Expected-Bayes methods are used for estimatio of parameter of Rayleigh distributio usig Liex loss fuctio. It has bee oticed from the results of simulatio study, that the E-Bayes estimates have smaller Mea Square Error as compared with the associated Bayes estimate. It has also bee observed that E-Bayes estimate, i most cases, ted to be more efficiet tha Bayes estimate except (for = 35, b=.75 ad = 3, 4,b= -.75) where Bayes ad E-Bayes are equally efficiet. REFERENCES [1] Ahmed, A., Ahmad S. P. ad Reshi, J. A. (213). Bayesia aalysis of Rayleigh distributio. Iteratioal Joural of Scietific ad Research Publicatios, Vol. 3, 1-9. [2] Bhattacharya ad Tyagi (199). Bayesia survival aalysis based o the Rayleigh model. Trabajos de Estadistica, 5(1), [3] Dey, S. (212). Bayesia estimatio of the parameter ad reliability fuctio of a Iverse Rayleigh distributio. Malaysia Joural of Mathematical Scieces, 6(1), [4] Feradez, A. J. (2). Bayesia iferece from type II cesored Rayleigh data. Statistics ad Probability Letters, 48, [5] Ferreira, J. T., Bekker, A. ad Arashi M. (216). Objective Bayesia estimators for the right cesored Rayleigh distributio evaluatig the Al-

6 796 Isha Gupta Bayyati loss fuctio. Revstat- Statistical Joural, Vol. 14, No. 4, [6] Ha, M., (27). E-Bayesia estimatio of failure probability ad its applicatio. Mathematical ad Computer Modellig, 45, [7] Saat. P ad Srivastava, R. S. (216). Bayesia aalysis of Rayleigh distributio uder quasi prior for differet loss fuctios. Iteratioal Joural of Sciece ad Research, Vol. 5, [8] Zeller, A. (1986). Bayesia Estimatio ad Predictio Usig Asymmetric Loss Fuctios, Jour. Amer. Statist. Assoc., 81,