Non-Bayes, Bayes and Empirical Bayes Estimators for the Shape Parameter of Lomax Distribution

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1 No-Bayes, Bayes ad Empirical Bayes Estimators for the Shape Parameter of Lomax Distributio Dr. Nadia H. Al-Noor* ad Shahad Saad Alwa *Dept. of Mathematics / College of Sciece / AL- Mustasiriya Uiversity /Baghdad / Iraq. adialoor@yahoo.com Abstract Poit estimatio is oe of the core topics i mathematical statistics. I this paper we cosider the most commo methods of poit estimatio: o-bayes, Bayes ad empirical Bayes methods to estimate the shape parameter of Lomax distributio based o complete data. The maximum likelihood, momet ad uiformly miimum variace ubiased estimators are obtaied as o-bayes estimators. Bayes ad empirical Bayes estimators are obtaied correspodig to three iformative priors "gamma, chi-square ad iverted Levy" based o symmetric "squared error" ad asymmetric "LINEX ad geeral etropy" loss fuctios. The estimates of the shape parameter were compared empirically via Mote Carlo simulatio study based upo the mea squared error. Amog the set of coclusios that have bee reached, it is observed that, for all sample sizes ad differet cases, the performace of uiformly miimum variace ubiased estimator is better tha other o-bayes estimators. Further that, Mote Carlo simulatio results idicate that the performace of Bayes ad empirical Bayes estimator i some cases are better tha o-bayes for some appropriate of prior distributio, loss fuctio, values of parameters ad sample size. Keywords: Lomax distributio; maximum likelihood estimator; momet estimator; uiformly miimum variace ubiased estimator; Bayes estimator; empirical Bayes estimator; iformative prior; squared error loss fuctio; LINEX loss fuctio; geeral etropy loss fuctio; mea squared error. 1. Itroductio Lomax distributio "also called Pareto type-ii distributio or Pearso type-vi distributio [2]" itroduced ad studied by Lomax (1954) [19]. He used this distributio to aalyze busiess failure data. The probability desity fuctio ad cumulative distributio fuctio of two parameters Lomax distributio, Lomax (, β), are give by [7]: f(t;, β) = β (1 + t β ) ( +1) F(t;, β) = 1 (1 + t β ) ; t > 0 ;, β > 0 (1) ; t > 0 ;, β > 0 (2) The r th momet about the origi of the Lomax distributio is: E(t r Г(r + 1) Г(λ r) r )= λ β ; λ > r ; r = 1,2, (3) Г(λ + 1) Where, ad β are shape ad scale parameters, respectively. Lomax distributio has bee received greatest attetio from theoretical ad statisticias primarily due to its use i reliability ad lifetime testig studies [13][22]. Dubey (1970) [12] showed that Lomax distributio ca be derived as a special case of a particular compoud gamma distributio. Bryso (1974) [10] argued that Lomax distributio provide a very good alterative to commo lifetime distributios like expoetial, Weibull, or gamma distributios where the experimeter presumes that the populatio distributio may be heavy-tailed. Tadikamalla (1980)[28] related the Lomax distributio to the Burr family of distributios. Ahsaullah (1991) [4] studied the record statistics of the Lomax distributio with some distributioal properties. Balakrisha ad Ahsaullah (1994) [9] obtaied some recurrece relatios betwee the momets of record values from Lomax distributio. Sara ad Pushkara (1999) [24] established some recurrece relatios for both sigle ad product momets of order statistics from a doubly trucated Lomax distributio. Habibullh ad Ahsaullah (2000) [15] addressed the problem of estimatig the parameters of Lomax distributio based o geeralizatio order statistics. Al-Awadhi ad Ghitay (2001) [5] used the Lomax distributio as a mixig distributio for the Poisso parameter ad derived the discrete Poisso-Lomax distributio. Petropoulos ad Kourouklis (2004) [23] 17

2 cosidered the estimatio of a quatile of the commo margial distributio i a multivariate Lomax distributio with ukow locatio ad scale parameters. Nadarajah (2005) [20] derived several properties of the logarithm of the Lomax radom variable. Abd Ellah (2006) [3] addressed the problem of Bayesia estimatio alog with maximum likelihood estimatio of the parameters, reliability ad hazard fuctios i the cotext of record statistics values. Abd-Elfattah et al. (2007) [1] cosidered the Lomax distributio as a importat model of lifetime models ad derived the o-bayesia "maximum likelihood estimator" ad Bayesia estimators of sample size i the case of type I cesored samples. Kozubowski et al. (2009) [17] cosidered the problem of maximum likelihood estimatio of the parameters. Abd-Elfattah ad Alharbey (2010) [2] applied the geeralized probability weighted momets method for estimatig the parameters. Ashour ad Abd-Elfattah (2011) [7] provided the maximum likelihood estimators for the ukow parameters of Lomax distributio ad their variace covariace matrix uder hybrid cesored sample. Giles et al. (2011) [14] evaluated some of the smallsample properties of the maximum likelihood estimators. Nasiri ad Hosseii (2012) [21] obtaied maximum likelihood, momet ad Bayes estimators of oe parameter Lomax distributio based o record values. Bayes estimatios are calculated for both iformative ad o-iformative priors based o records for quadratic ad squared error loss fuctios. El-Di et al. (2013) [13] discussed o- Bayes, Bayes ad Empirical Bayes estimates for the parameters of Lomax model based o progressively type-ii cesored samples. Okasha (2014) [22] used Bayes ad empirical Bayes approaches for obtaiig the estimates of the ukow shape parameter ad some other life time characteristics such as the reliability ad hazard fuctios of Lomax distributio based o type-ii cesored data. 2. Differet Estimatio Methods 2.1 No-Bayes Estimators of λ I this subsectio, the o-bayes estimators for the shape parameter,, have bee obtaied i based o maximum likelihood estimatio, momet estimatio, ad uiformly miimum variace ubiased estimatio. Maximum Likelihood Estimator: Let t = (t 1, t 2,.., t ) be the life time of a radom sample of size draw idepedetly from the Lomax distributio defied by (1). The the likelihood fuctio for the give sample observatios is: L(λ, β t) = f(t i λ, β) i=1 L(λ, β t) = β e ( +1) l(1+ t i i=1 β ) (4) The maximum likelihood estimator of λ, deoted by λ ML, yields by takig the derivative of the atural loglikelihood fuctio with respect to λ ad settig it equal to zero as: λ ML = l (1 + t = i i=1 β ) W ; w = l (1 + t i β ) (5) i=1 Momet Estimator: The momet estimator of the shape parameter of Lomax distributio, λ, deoted by λ MO, yields by equatig the first populatio momet to the first sample momet as: λ MO = 1 + β (6) t Uiformly Miimum Variace Ubiased Estimator: The probability desity fuctio of Lomax distributio is belogs to expoetial family. Therefore, W = l (1 + t i i=1 ) is a complete sufficiet statistic for λ. The, β depedig o the theorem of Lehma-Scheffe [16], the uiformly miimum variace ubiased estimator of λ, deoted by λ UMVU is: λ UMVU = i=1 1 l (1 + t i β ) = 1 W ; w = l (1 + t i β ) i=1 (7) 2.2 Bayes Estimators of λ Previously, we have obtaied the o-bayes poit estimators for parameter of iterest. I o-bayes approach, the parameter of iterest had a fixed but ukow value. I this subsectio, Bayesia cotext, the parameter is a radom variable with posterior desity fuctio. Prior ad Posterior Desity Fuctios: From Bayes' rule the posterior probability desity fuctio of the parameter give t, π( t), ca be expressed as: 18

3 L( t) g( ) π( t) = (8) L( t) g( ) d The posterior distributios of the parameter have bee obtaied uder the assumptio of three iformative priors: Gamma Prior [26]: ba g 1 ( ) = (a) a 1 e b ; > 0 ; a, b > 0 (9) Chi-Square Prior, see [8]: b a 2 g 2 ( ) = 2 a 2 ( a a 2 2 ) 1 e b Iverted Levy Prior [27]: g 3 ( ) = b 1 2π 2 e b 2 ; > 0 ; a, b > 0 (10) 2 ; > 0 ; b > 0 (11) Via Bayes theorem, combiig the likelihood fuctio (4) with the desity fuctio of gamma prior (9), chisquare prior (10) ad iverted Levy prior (11), results the first, secod ad third posterior desity fuctios of respectively as: (W + b)+a π 1 ( t) = +a 1 (W+b) e (12) ( + a) which implies that:( t) Gamma ( + a, π 2 ( t) = (W + b + a 2 ) 2 ( + a 2 ) a e (W+b 2 ) 1 W+b ) which implies that:( t) Gamma ( + a 2, 1 W+ b 2 ) 19 (13) π 3 ( t) = (W + b ) 2 ( e (W+b 2 ) (14) 2 ) which implies that:( t) Gamma ( + 1 2, 1 W+ b 2 ) Loss Fuctios: The Bayes estimatio of a parameter is based i miimizatio of a Bayesia loss (risk) fuctio, L(, ), defied as a average cost-of-error fuctio [29]: Risk( ) = E[L(, )] = L(, ) π( t) d There are two types of loss fuctio: symmetric ad asymmetric. The symmetric loss fuctio associates equal importace to the losses due to overestimatio ad uderestimatio of equal magitude. We have bee adopted the two types of loss fuctio: squared error loss fuctio as a symmetric loss fuctio as well as LINEX ad geeral etropy loss fuctios as asymmetric loss fuctios. Squared Error Loss Fuctio: The squared error loss fuctio ca be expressed as [6]: L(, ) = ( ) 2 (15) The Bayes estimator of based o this loss fuctio, deoted by BS, ca be obtaied as: BS = E π ( t) (16) LINEX Loss Fuctio: The LINEX loss fuctio ca be expressed as [18]: L(, ) = d [e c( ) c( ) 1 ] ; c 0, d > 0 (17) The Bayes estimator of based o this loss fuctio, deoted by BL, ca be obtaied as: BL = 1 c l[e π(e c t)] (18) Provided that the posterior expectatio with respect to the posterior desity of parameter, E π (e c t), exists ad is fiite.

4 Geeral Etropy Loss Fuctio: This loss fuctio ca be expressed as [11]: L(, ) = d [( p ) p l ( ) 1 ] ; p 0, d > 0 (19) The Bayes estimator of based o this loss fuctio, deoted by BG, ca be obtaied as: BG = [E π ( p t)] 1 p (20) Provided that the posterior expectatio with respect to the posterior desity of, E π ( P t) exists ad is fiite. Without ay loss of geerality it ca be assumed that d = 1. Now, the Bayes estimators of the shape parameter for Lomax distributio based o squared error, LINEX ad geeral etropy loss fuctios correspodig to differet prior distributios are show i table (1): 2.3 Empirical Bayes Estimators of The Bayes estimators i previous subsectio are see to deped o the hyper-parameter b. Whe b is ukow, we may use the empirical Bayes approach to get its estimate from likelihood fuctio ad probability desity fuctio of prior distributio [25]. Now, from likelihood fuctio (4) ad gamma, chi-square ad iverted Levy prior distributios (9, 10, 11), we calculate the margial pdf of T, with desities: Assume that a is kow, the based o f(t b) we obtai a estimate, b of b. The maximum likelihood estimators of b uder the assumptio of gamma, chi-square ad iverted Levy priors, deoted by b 1, b 2, b 3 respectively, are calculatig by takig the derivative of the atural log for (21), (22) ad (23) ad settig it equal to zero. The empirical Bayes estimators of the shape parameter for Lomax distributio based o squared error, LINEX ad geeral etropy loss fuctios correspodig to differet prior distributios are show i table (2): 20

5 3. Mote Carlo Simulatio Study ad Results I this sectio, Mote Carlo simulatio study has bee coducted to assess the behavior of differet estimators for the ukow shape parameter of Lomax distributio. The simulatio desig cosists of four basic steps which are: Step (1): Set the default values (true values) for the parameters of Lomax distributio which are varied ito six cases to observe their effect o the estimates whe λ > β, λ = β, λ < β. Parameter Cases I II III IV V VI λ β The umbers of sample size used are ( = 10, 30 ad 50) to represet small, moderate, ad large dataset. The default values of the hyper-parameters of prior distributios (a, b) chose to be (9, 3). The values of LINEX loss fuctio costat (c) ad geeral etropy loss fuctio costat (p) used are (c = 0.8, 0.8) ad (p = 0.5, 0.5). The umber of sample replicated (L) chose to be (3000). Step (2): Geerate data distributed as Lomax distributio with parameters (λ, β), through the adoptio of iverse trasformatio method, by usig the formula: t i = F 1 (U i ) = β [(1 U i ) 1 1] ; i = 1,2,, (24) Where U is a radom variable distributed as uiform distributio for the period (0,1). Step (3): Calculate the o-bayes, Bayes ad empirical Bayes estimators of the ukow shape parameter of Lomax distributio accordig to the formulas that have bee obtaied. Step (4): After the shape parameter is estimated, mea squared error (MSE) is calculated to compare the estimatio methods, where: MSE(λ ) = L (λ j λ) 2 j=1 (25) L λ j : is the estimate of λ at the j th replicate (ru). The simulatio program is writte by usig MATLAB (R2011b) program. The simulatio results of MSE are tabulated i tables (3) (8) 4. Coclusios ad Recommedatios The most importat coclusios of Mote-Carlo simulatio results for estimatig the shape parameter of Lomax distributio, with assumptio the scale parameter is kow, are: Amog o-bayes estimators, table (3), the performace of the uiformly miimum variace ubiased estimator (UMVU) is higher tha that of other estimators "maximum likelihood (ML) ad momet (MO) estimators". As well as the performace of ML estimator is better tha that of MO estimator for all differet cases ad all sample sizes, i.e. MSE values for (UMVU<ML<MO) 21

6 Whe λ = 2.1 ad > 10, table (4), Bayes estimator correspodig to iverted Levy prior based o squared error loss fuctio represet the best Bayes estimator comparig to other Bayes estimators for all differet values of β. Whe λ = 3, Bayes estimator correspodig to gamma prior based o LINEX (c = 0.8) loss fuctio is represet the best Bayes estimator comparig to other Bayes estimators for all differet values of β ad all sample sizes as well as whe λ = 2.1 for = 10. Amog the empirical Bayes estimators, table (5), the performace of empirical Bayes estimator correspodig to iverted Levy prior based o LINEX (c = 0.8) loss fuctio is better tha that of other estimators for all differet cases ad all sample sizes. For all cases ad all sample sizes, LINEX (c = 0.8) record full appearace as best loss fuctio associated with Bayes estimates correspodig to gamma ad chi-square priors as well as with empirical Bayes estimates correspodig to all differet priors. It is importat to metio that LINEX (c = 0.8) ad squared error record appearace as best loss fuctios associated with Bayes estimates correspodig to iverted Levy prior. Accordig to results of β = 1, MSE values of o-bayes ad empirical Bayes "correspodig to differet priors" estimators of shape parameter are icreasig as the shape parameter value icrease from λ = 2.1 up to λ = 3 for all sample sizes. As well as that is true for Bayes estimator correspodig to chi-square ad iverted Levy priors for all sample sizes ad correspodig to gamma prior for 30. For all sample sizes, the MSE values associated with o-bayes ad empirical Bayes estimators are icreased with icreasig the shape ad scale parameters (λ, β) together from (2.1) up to (3). As well as that is true for the MSE values associated with Bayes estimators correspodig to chi-square ad iverted Levy priors for all sample sizes ad correspodig to gamma prior for 30. For all differet cases ad all sample sizes, the performace of Bayes estimators based o LINEX ad geeral etropy loss fuctios with positive value of c ad p respectively is better tha that with egative value correspodig to gamma ad chi-square priors while that is true for empirical Bayes correspodig to all differet priors. With some cases, loss fuctios ad sample sizes, the performace of chi-square prior is better tha that of gamma ad iverted Levy priors. I spite of that chi-square prior does't record ay appearace as best prior. The MSE values of the empirical Bayes estimators correspodig to all differet priors based o squared error loss fuctio are idetical. The MSE values associated with each o-bayes, Bayes ad empirical Bayes estimates correspodig to each prior ad every loss fuctio reduces with the icrease i the sample size. Also, the results show a covergece betwee most of the estimators to icrease the sample sizes. For all sample sizes, tables (6,7,8), the MSE values for the best Bayes estimators with differet cases are less tha MSE values for the best empirical Bayes estimators which i tur are less tha o-bayes estimators, i.e., MSE values for the best (Bayes < empirical Bayes < o-bayes) estimators. I the light of the coclusios that have bee obtaied for estimatig the shape parameter of Lomax distributio, some recommeded have bee put forward: Usig UMVU estimator as a o-bayes estimator. As Bayes estimators, usig Bayes estimator correspodig to iverted Levy prior based o squared error loss fuctio whe λ = 2.1 for > 10 ad usig Bayes estimator correspodig to gamma prior based o LINEX (c = 0.8) loss fuctio whe λ = 3. As empirical Bayes estimators, usig empirical Bayes estimator correspodig to iverted Levy prior based o LINEX (c = 0.8) loss fuctio. Refereces [1] Abd-Elfattah, A. M.; Alaboud, F. M. ad Alharby, A. H. (2007), O Sample Size Estimatio for Lomax Distributio, Australia Joural of Basic ad Applied Scieces, Vol.1, No.4, PP [2] Abd-Elfattah, A. M. ad Alharbey, A. H. (2010), Estimatio of Lomax Parameters Based o Geeralized Probability Weighted Momet, JKAU: Sci., Vol.22, No.2, PP [3] Abd Ellah, A. H. (2006), Compariso of Estimates Usig Record Statistics from Lomax Model: Bayesia ad No Bayesia Approaches, J. Statist. Res. Ira, Vol. 3, PP [4] Ahsaullah, M. (1991), Record Values of Lomax distributio, Statistica Nederladica, Vol.41, No.1, PP [5] Al-Awadhi, S. A. ad Ghitay, M. A. (2001), Statistical Properties of Poisso Lomax Distributio ad its Applicatio to Repeated Accidets data, J. Appl. Statist. Sci., Vol.10, PP

7 [6] Ali, S. ; Aslam, M. ; Abbas, N. ad Kazmi, S. (2012), Scale Parameter Estimatio of the Laplace Model Usig Differet Asymmetric Loss Fuctios, Iteratioal Joural of Statistics ad Probability, Vol.1, No.1, PP [7] Ashour, S. K. ad Abdelfattah, A. M. (2011), Parameter Estimatio of the Hybrid Cesored Lomax Distributio, Pak.j.stat.oper.res., Vol.VII, No.1, PP [8] Aslam, M. ad Abdul Haq (2009), O the Prior Selectio for Variace of Normal Distributio, Iterstat.Statjourals, PP [9] Balakrisha, N. ad Ahsaullah, M. (1994), Relatios for Sigle ad Product Momets of Record Values from Lomax Distributio, Sakhya: The Idia Joural of Statistics, Vol.56, Series B, Pt.2, PP [10] Bryso, M. C. (1974), Heavy-Tailed Distributios: Properties ad Tests, Techometrics, Vol.16, No.1, PP [11] Calabria, R. ad Pulcii, G. (1996), Poit Estimatio uder Asymmetric Loss Fuctios for Left-Trucated Expoetial Samples. Comm. Statist. Theory Methods, Vol.25, No.3, PP [12] Dubey, S. D. (1970), Compoud Gamma, Beta ad F Distributios, Metrika, Vol.16, PP [13] El-Di, M. M; Okasha, H. M. ad Al-Zahrai, B. (2013), Empirical Bayes Estimators of Reliability Performaces usig Progressive Type-II Cesorig from Lomax Model, Joural of Advaced Research i Applied Mathematics, Vol.5, Issue 1, PP [14] Giles, D. E.; Feg, H. ad Godwi, R. T. (2011), O the Bias of the Maximum Likelihood Estimator for the Two-Parameter Lomax Distributio, Ecoometrics Workig Paper EWP1104, Departmet of Ecoomics, Uiversity of Victoria, PP [15] Habibullah, M. ad Ahsaullah, M. (2000), Estimatio of Parameters of a Pareto Distributio by Geeralized Order Statistics, Commuicatio i Statistics, Theory ad Methods, Vol.29, Issue 7, PP [16] Hogg, R. V.; McKea, J. W. ad Craig, A. T. (2005), Itroductio to Mathematical Statistics, Sixth Editio, Pearso Pretice Hall. [17] Kozubowski, T. J.; Paorska, A. K.; Qeada, F.; Gershuov, A. ad Romiger, D. (2009), Testig Expoetiality versus Pareto Distributio via Likelihood Ratio, Commuicatios i Statistics-Simulatio ad Computatio, Vol.38, PP [18] Li, X.; Shi, Y.; Wei, J. ad Chai, J. (2007), Empirical Bayes Estimators of Reliability Performaces usig LINEX Loss uder Progressively Type-II Cesored Samples, Mathematics ad Computers i Simulatio, Vol.73, No.5, PP [19] Lomax, K.S. (1954), Busiess Failures: Aother Example of the Aalysis of Failure Data, J. Amer. Statist. Assoc., Vol.49, PP [20] Nadarajah, S. (2005), Expoetiated Pareto distributios, Statistics, Vol.39, PP [21] Nasiri, P. ad Hosseii, S. (2012), Statistical Ifereces for Lomax Distributio Based o Record Values (Bayesia ad Classical), Joural of Moder Applied Statistical Methods, Vol.11, No.1, PP [22] Okasha, H. M. (2014), E-Bayesia Estimatio for the Lomax Distributio Based o Type-II Cesored Data, Joural of the Egyptia Mathematical Society, Productio ad hostig by Elsevier B.V. o behalf of Egyptia Mathematical Society, PP [23] Petropoulos, C. ad Kourouklis, S. (2004), Improved Estimatio of Extreme Quatiles i the Multivariate Lomax (Pareto II) Distributio, Metrika, Vol.60, PP [24] Sara, J. ad Pushkara, N. (1999), Momets of Order Statistics from a Doubly Trucated Lomax Distributio, Joural of Statistical Research, Vol.33, No.1, PP [25] Shojaee, O.; Azimi, R.; Babaezhad, M. (2012), Empirical Bayes Estimators of Parameter ad Reliability Fuctio for Compoud Rayleigh Distributio uder Record Data, America Joural of Theoretical ad Applied Statistics, Vol.1, No.1, PP [26] Shrestha, S. K. ad Kumar, V. (2014), Bayesia Aalysis of Exteded Lomax Distributio, Iteratioal Joural of Mathematics Treds ad Techology, Vol.7, No.1, PP [27] Sidhu, T. N. ; Aslam, M. ad Shafiq, A. (2013), Aalysis of the Left Cesored Data from the Pareto Type II Distributio, Caspia Joural of Applied Scieces Research, Vol.2, No.7, PP [28] Tadikamalla, P.R. (1980), A Look at the Burr ad Related Distributios, Iteratioal Statistical Review, Vol.48, No.3, PP [29] Vaseghi, S.V. (2000), Advaced Digital Sigal Processig ad Noise Reductio, Secod Editio, Joh Wiley & Sos Ltd. 23

8 Table (3): MSE Values for No-Bayes Estimators of λ with Differet Cases Case ML MO UMVU Best Estimator UMVU I UMVU UMVU UMVU II UMVU UMVU UMVU III UMVU UMVU UMVU IV UMVU UMVU UMVU V UMVU UMVU UMVU VI UMVU UMVU Table (4): MSE Values for Bayes Estimators of λ with Differet Cases Loss Fuctio Case Prior Squared LINEX Geeral Etropy Best Loss Error c= c= 0.8 p= p= 0.5 Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Best Prior I Levy I Levy Gamma I Levy Gamma Gamma LIN(0.8) I 30 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy Gamma I Levy Gamma Gamma LIN(0.8) II 30 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy Squared III Best Prior I Levy I Levy Gamma I Levy Gamma Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy Squared 24

9 IV Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy Squared Best Prior I Levy I Levy I Levy I Levy I Levy Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) V VI Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(-0.8) 25

10 Table (5): MSE Values for Empirical Bayes Estimators of λ with Differet Cases Case Prior Loss Fuctio Best Squared LINEX Geeral Etropy Loss Error c= c= 0.8 p= p= 0.5 Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) I 30 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) II 15 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) III 30 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) IV 30 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) V 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) 26

11 VI Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Gamma LIN(0.8) 10 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Best Prior Gamma I Levy I Levy I Levy Gamma LIN(0.8) 30 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Best Prior Gamma I Levy I Levy I Levy Gamma LIN(0.8) 50 Chi- Square LIN(0.8) Iverted Levy LIN(0.8) Best Prior Gamma I Levy I Levy I Levy Table (6): MSE Values for the Best No-Bayes Estimators of λ with Differet Cases I II III IV V VI 10 UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU UMVU Table (7): MSE Values for the Best Bayes Estimators of λ with Differet Cases I II III IV V VI 10 GLIN(0.8) GLIN(0.8) GLIN(0.8) GLIN(0.8) GLIN(0.8) GLIN(0.8) ILS ILS ILS GLIN(0.8) GLIN(0.8) GLIN(0.8) ILS ILS ILS GLIN(0.8) GLIN(0.8) GLIN(0.8) Table (8): MSE Values for the Best Empirical Bayes Estimators of λ with Differet Cases I II III IV V VI 10 ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8) ILLIN(0.8)

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