Objective Priors for Estimation of Extended Exponential Geometric Distribution

Transcription

1 Journal of Modern Appled Statstcal Methods Volume 13 Issue Artcle Objectve Prors for Estmaton of Extended Exponental Geometrc Dstrbuton Pedro L. Ramos Unversdade de São Paulo, São Paulo, Brazl, pedrolramos@hotmal.com Fernando A. Moala Unversdade Estadual Paulsta, São Paulo, Brazl, femoala@fct.unesp.br Jorge A. Achcar Unversdade de São Paulo, São Paulo, Brazl, achcar@fmrp.usp.br Follow ths and addtonal works at: Part of the Appled Statstcs Commons, Socal and Behavoral Scences Commons, and the Statstcal Theory Commons Recommended Ctaton Ramos, Pedro L.; Moala, Fernando A.; and Achcar, Jorge A. (014) "Objectve Prors for Estmaton of Extended Exponental Geometrc Dstrbuton," Journal of Modern Appled Statstcal Methods: Vol. 13 : Iss., Artcle 1. DOI: 10.37/jmasm/ Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCommons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCommons@WayneState.

2 Journal of Modern Appled Statstcal Methods November 014, Vol. 13, No. 1, Copyrght 014 JMASM, Inc. ISSN Objectve Prors for Estmaton of Extended Exponental Geometrc Dstrbuton Pedro L. Ramos Unversdade de São Paulo São Paulo, Brazl Fernando A. Moala Unversdade Estadual Paulsta São Paulo, Brazl Jorge A. Achcar Unversdade de São Paulo São Paulo, Brazl A Bayesan analyss was developed wth dfferent nonnformatve pror dstrbutons such as Jeffreys, Maxmal Data Informaton, and Reference. The am was to nvestgate the effects of each pror dstrbuton on the posteror estmates of the parameters of the extended exponental geometrc dstrbuton, based on smulated data and a real applcaton. Keywords: Extended exponental geometrc dstrbuton, Jeffreys, MDIP, Reference, Bayesan, nonnformatve, pror Introducton Adamds & Loukas (005) ntroduced an extenson of the exponental geometrc dstrbuton (Adamds & Loukas, 1998), namng t as an extended exponental geometrc (EEG) dstrbuton, to analyze lfetme data. Ths dstrbuton provdes ncreasng or decreasng hazard functons, dependng on the values of ts parameters. In ths way, EEG gves a great flexblty of ft for the data. If T s a random varable denotng the lfetme of a component wth an extended exponental geometrc (EEG) dstrbuton, then the probablty densty s gven by: f t, e t t 11 e, (1) wth t > 0 and parameters γ > 0 and λ > 0. Let us denote ths dstrbuton as EEG( γ, λ ). Dr. Ramos s n the Faculty of Medcne of Rberão Preto. Emal hm at: pedrolramos@hotmal.com. Dr. Moala s a Professor n the Faculty of Scence and Technology. Emal hm at: femoala@fct.unesp.br. Dr. Achcar s a Professor n the Insttute of Mathematcs and Computer Scence. Emal hm at: achcar@fmrp.usp.br. 6

3 RAMOS ET AL. The survval and hazard functons of EEG( γ, λ ) dstrbuton, for a fxed tme t, s gven by t e S t;, and ht;, t 1 1 e 1 1 e t, () respectvely. The mean and varance of the EEG dstrbuton are gven, respectvely, by ET 1,1,1 and var 1,,1 1,1,1 T (3) where Ψ( z, s, a ) s known as Lerch transcendental functon (Erdely et al., 1953), gven by s1 au k 1 u e z z, s, a du ; z 1; a, s 0 s. u s 1 ze 0 k0 a k Adamds et al. (005) and Ktdamrongsuk (010) gave addtonal propertes of the EEG dstrbuton. Fgures 1 and present dfferent forms for the densty, survval and hazard functons for the EEG dstrbuton consderng dfferent values of γ and λ. The motvaton here s to present a Bayesan analyss when there s lttle pror knowledge avalable or that reflects manly the nformaton from the sample.. In ths stuaton, t s mportant to use nonnformatve prors, however, t can be dffcult to choose a pror dstrbuton that represent one of ths stuatons. Thus, the man am of ths paper s to choose a nonnformatve pror dstrbuton s for the parameters parameters λ and γ of the EEG dstrbuton and to study the effects of these dfferent prors n the resultng posteror dstrbutons, especally n stuatons of small sample szes, a common stuaton n applcatons. 7

4 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION Fgure 1. Probablty densty functons for the EEG dstrbuton wth values for the scale and shape parameters gven respectvely, by λ = 0.5, 1.0, 1.5 and γ = 0.5, 1.5,.0,.5, 3.0, 3.5. Fgure. Survval functons and hazard functons for the EEG dstrbuton wth values for the scale and shape parameters gven respectvely, by λ = 0.5, 1.0 and γ = 0.5, 1, 3, 5. Commonly used nonnformatve pror dstrbutons are derved, such as unform (Bayes, 1763; Laplace, 1774), Jeffreys (1967), reference prors (Bernardo, 1979; Berger & Bernardo, 199), and the uncommon MDIP pror (Zellner, 1977, 1984). A smulaton study s conducted comparng ther performance n terms of ther summares and coverage rates of credble ntervals. 8

5 RAMOS ET AL. Numercal ntegraton based on stochastc smulaton methods as the Markov Chan Monte Carlo (MCMC) wll be used to smulate samples of the margnal posteror dstrbuton of nterest. In partcular, we wll be usng the Metropols- Hastngs algorthm to obtan the posteror summares of nterest (see Gelfand & Smth, 1990 or Chb & Greenberg, 1995). Methodology Maxmum Lkelhood Estmaton Let X 1,, X n be a random sample from EEG( γ, λ ) dstrbuton wth densty (1). The lkelhood functon for the parameters γ and λ s gven by n Lx;, exp x 1 1 e 1 1 n n x, (4) where γ > 0 and λ > 0. The logarthm of the lkelhood functon (4) s gven by n n x. (5) l x;, nlog x log 1 1 e 1 1 By settng l(x; γ, λ ) / γ = 0 and l(x; γ, λ ) / λ = 0 and after some algebrac manpulatons, we obtan the lkelhood equatons n n x n xe x 1 0 and x 1 1 e n x n e 0 x 1 1 e (6) whose solutons provde the maxmum lkelhood estmators of the parameters γ and λ. Note that the solutons of the lkelhood equatons (6) cannot be obtaned analytcally and hence numercal approaches need to be used. Adamds et al. (005) propose to use the EM algorthm (Dempster et al., 1977) to solve the nonlnear equatons (6) and fnd the MLE of γ and λ. The EM teratons are gven by 9

6 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION ˆ j1 j ˆ ˆ j j j n 1 1 exp x n ˆ exp ˆ x j j n x 1 1 ˆ exp ˆ 1 x j n. j j 1 ˆ ˆ ˆ x exp 1 (7) Ktdamrongsuk (010) shows n detal the computatons of the expected Fsher nformaton matrx Ι(γ,λ) of the EEG dstrbuton, gven by I,, n 1 log I11, 3 1 n 1 log n (8) wth where I 11, (9) n 31 1 L 1, n 1 L 1 1 log, p Lr p 1, 0 p 1 r s a polygarthmca functon (Erdely et al., 1953). The maxmum lkelhood estmates for γ and λ are based for small sample problems. In the case of large samples they become unbased and asymptotcally effcent. Such estmates are asymptotcally Normal dstrbuted wth jont dstrbuton gven by 30

7 RAMOS ET AL. Bayesan Analyss ˆ 1 ˆ, ~ N,, I,, when n. (10) In ths secton we consder the Bayesan estmaton of the unknown parameters λ and γ. Frst, a pror dstrbuton whch expresses lttle nformaton on the parameters γ and λ can be obtaned from unform denstes, whch do not favor any partcular value of λ and γ. In ths case, the jont pror dstrbuton for λ and γ s gven by, constant. (11) U Another wdely-used method to specfy pror nformaton s through the product of ndependent gamma dstrbutons for each parameter λ and γ, snce γ > 0 and λ > 0, that s, γ ~ Gamma(α 1, β 1 ) and λ ~ Gamma(α, β ), where Gamma(a,b) denotes a gamma dstrbuton wth mean a/b and varance α/b ; and α 1, α, β 1 and β are known hyperparameters. Thus, the jont pror dstrbuton for λ and γ s gven by 1 1 1, exp. (1) G 1 Assume α 1 = α = β 1 = β = 0.01, that s, a non-nformatve pror gven by (1). An another well-known exstng non-nformatve pror, whch represents a stuaton wth lttle a pror nformaton on the parameters was ntroduced by Jeffreys (1967), also known as the Jeffreys rule. The Jeffreys pror has been wdely used due to the nvarance property for one to one transformatons of the parameters. The Jeffreys pror s defned as, det I,. (13) J where Ι(γ, λ) s the Fsher nformaton matrx defned n (8) and (9). From the equaton (13), we get 31

8 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION J, 3 L log 1 log 3 J, 1 ; 0 1, log 3 log L (14) 1 ; 1. It s nterestng to observe t was found n (14) ndependent prors for the parameters λ and γ, but ths jont pror has a dependence structure. Zellner (1977, 1984) proposed a non-nformatve pror based on the Shannon's entropy (1948). The dea s to maxmze the nformaton from the data n relaton to the pror nformaton on the parameters. Ths non-nformatve pror dstrbuton known as "Maxmal Data Informaton Pror" s obtaned from the soluton of the equaton Z, exp log f x, f x, dx. (15) 0 For the EEG dstrbuton gven n (1) the resultng non-nformatve pror s gven by log Z, exp 1,1,1, 1 (16) where Ψ(z,s,a) s defned n (5). The proposed Zellner pror dstrbuton (15) has lmted nvarance propertes, where nvarance s only verfed under lnear transformatons of the vector (γ, λ) and not for all dfferentable one by one transformatons. Bernardo (1979) and Berger & Bernardo (199) use the Kullback-Lebler dstance between the posteror dstrbuton p (θ x) the pror dstrbuton π (θ) to maxmze the nformaton from the data n relaton to the known pror nformaton for the parameters to fnd a nonnformatve pror. Addtonal nformaton about the reference pror can be found n Bernardo (005). 3

9 RAMOS ET AL. An mportant feature n ths approach s the dfferent treatment for nterest and nusance parameters when θ s a vector of parameters. In the presence of nusance parameters, a typcal case n ths paper, one must establsh an ordered parameterzaton wth the parameter of nterest sngled out and then follow the procedure below. The algorthm of Berger and Bernardo (199) to derve the reference pror can be descrbed n four steps, as follows. We wll present here the two-parameters case n detals. Let θ = (θ 1, θ ) be the two parameters vector; θ 1 wll be consdered the parameter of nterest and θ s the nusance parameter. The algorthm used to obtan the reference pror s gven by Step 1: gven by Fnd the condtonal reference pror π (θ θ 1 ), assumng that θ 1 s I,, (17) 1 1 where I (θ 1, θ ) s the term of order (,) of the nformaton Fsher matrx. Step : Normalze π (θ θ 1 ). If π (θ θ 1 ) s mproper, choose a sequence of sets Ω 1 Ω Ω, where π (θ θ 1 ) s proper. Fnd p m c m 1 1 (18) d m 1 c 1. (19) 1 m 1 1 m Step 3: Fnd the reference pror for θ 1. The result s gven by the soluton of 1 det I 1, m1 exp pm 1 log d. I 1, m (0) 33

10 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION Step 4: parameter Fnd the pror dstrbuton for (θ 1, θ ), when θ s the nusance c m 1 m 1 1, lm 1, m * * cm1 m1 (1) * where 1 s any fxed pont wthn the postve densty for all π m. For the EEG dstrbuton gven n (1), the reference pror when λ s the parameter of nterest s gven by, 1 L log L () The reference pror when γ s the parameter of nterest s gven by, for 0 1 and, for 1 3 L 1 1 log 1 log L 3log 3 log (3) Fnally, derve the pror dstrbutons for the parameters the resultng jont posteror dstrbutons for γ and λ s proportonal to the product of the lkelhood functon (4) and the pror dstrbutons π (γ, λ) gven n (11), (1), (14), (16), () and (3), that s n n n x,, exp 1 1 p x x e. (4)

11 RAMOS ET AL. By usng any pror dstrbuton proposed s not possble to derve the margnal posteror dstrbutons n an analytcal form for the parameters γ and λ. Thus, to obtan the posteror nformaton on the parameters of nterest as the pont estmator and Bayes credblty ntervals, we use MCMC algorthms to smulate samples of the values of γ and λ from the jont posteror dstrbutons. Results Two applcatons of the theoretcal results dscussed n the prevous sectons are presented. The frst nvolves a comparson of the estmaton methods based on smulated data; the second shows an applcaton of the EEG dstrbuton to real data. Analyss va numercal smulaton In ths example, some smulatons are performed va the Monte Carlo method. The goal s to study the effect of dfferent non-nformatve pror dstrbutons on the posteror summares and also to compare these results wth the obtaned results usng classcal nference analyss. Posteror summares of nterest are evaluated usng Monte Carlo Markov Chan (MCMC) methods. The nfluence of sample sze on the accuracy of the obtaned estmators s also examned. The followng procedure was adopted: 1. Determne the values of γ and λ.. Specfy the sample sze n. 3. Generate values of a dstrbuton EEG(γ,λ) wth sze n. 4. Usng the data obtaned n Step 3, calculate the estmates for the parameters γ and λ usng MCMC n the Bayesan approach and MLE n the classcal approach. 5. Repeat the steps 3 and 4 N tmes. Consder two set of the true values for the parameter (γ, λ) gven by (γ, λ) = (0.5, ) and (γ, λ) = (, 4) representng decreasng and ncreasng hazard functons, respectvely. The smulated data are generate from EEG dstrbuton wth the parameter values above for dfferent sample szes, as n = 10, 5 and 50. Tables 1 and show the posteror mean and medan, respectvely, by consderng the non-nformatve prors proposed n ths paper for the parameters. The maxmum lkelhood estmates (MLE) are also avalable. 35

12 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION Table 1. Posteror medans and MLE for λ = and γ = ½ for 1000 samples of szes 10, 5 and 50. λ = Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n=10.98(1.6).80(0.84).96(0.87).88(0.81) 3.61(1.54) 1.15(1.06) 3.44(.61) n=5.6(1.10) 3.00(1.05).93(0.88).9(0.83).95(1.00).01(1.06).54(1.6) n=50.17(0.76).5(0.70).58(0.67).71(0.67).56(0.7).07(0.77).6(0.80) γ = ½ Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= (0.43) 1.31(0.77) 1.1(0.17) 1.13(0.14) 1.50(0.77) 0.35(0.49) 1.51(.0) n=5 0.87(0.39) 0.9(0.4) 0.95(0.3) 0.99(0.1) 0.97(0.46) 0.56(0.41) 0.81(0.68) n= (0.3) 0.71(0.7) 0.81(0.7) 0.89(0.6) 0.76(0.3) 0.56(0.9) 0.64(0.36) Table. Posteror medans and MLE for λ = 4 and γ = for 1000 samples of szes 10, 5 and 50. λ = 4 Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= (1.1) 4.6(1.50) 5.31(1.55) 4.73(0.99) 5.08(1.5) 5.08(1.5) 4.86(.03) n=5 4.56(0.81) 4.0(0.77) 3.8(0.71) 3.90(0.73) 4.77(0.87) 3.50(0.84) 4.41(1.4) n= (0.57) 3.98(0.61) 3.51(0.41) 3.56(0.40) 3.40(0.64) 3.67(0.65) 4.(0.94) γ = Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= (0.50) 1.81(0.40) 3.5(0.4) 3.33(0.40) 3.6(0.4) 3.6(0.4) 3.65(3.3) n=5.70(0.49) 1.76(0.34) 1.6(0.5) 1.74(0.7).94(0.53) 1.5(0.56).87(.06) n= (0.45) 1.90(0.41) 1.53(0.9) 1.60(0.3) 1.95(0.45) 1.70(0.5).43(1.16) From Tables 1 and, t s observed that when the hazard functon s decreasng (0 < γ < 1) the pror dstrbuton gven by product of ndependent gamma dstrbutons gves the best estmaton for the parameters whle for the ncreasng hazard functon (γ > 1) the MDIP pror dstrbuton provdes the best one for all sample szes consdered. A crteron for comparson of the pror dstrbutons conssts on checkng the frequentst coverage probabltes of the posteror ntervals. We therefore compare the frequency at whch the true values of γ and λ are ncluded n ther 95% posteror ntervals. Ths frequency should be close to 95% for large numbers of repeated experments. 36

13 RAMOS ET AL. Table 3. Coverage probabltes for λ = and γ = ½ for 1000 samples of szes 10, 5 and 50. λ = Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= % 99.10% 95.70% 96.70% 90.60% 96.50% 95.0% n=5 95.0% 91.60% 89.70% 90.00% 93.40% 97.0% 95.00% n= % 94.30% 95.00% 91.30% 9.0% 96.60% 94.70% γ = ½ Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= % 98.00% 95.70% 94.80% 90.60% 97.10% 94.60% n= % 97.80% 97.90% 97.70% 95.00% 97.80% 95.50% n= % 97.90% 96.00% 95.50% 93.70% 97.70% 95.40% Table 4. Coverage probabltes for λ = 4 and γ = for 1000 samples of szes 10, 5 and 50. λ = 4 Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= % 96.60% 88.10% 98.00% 91.0% 91.0% 96.60% n= % 98.0% 96.80% 97.00% 94.00% 95.60% 94.30% n= % 98.40% 96.00% 97.60% 98.50% 96.60% 95.00% γ = Jeffreys MDIP Ref. λ Ref. γ Unform Gamma MLE n= % 99.50% 93.80% 97.00% 96.60% 96.60% 9.50% n= % 99.10% 99.50% 99.90% 98.50% 97.10% 9.00% n= % 98.90% 97.00% 99.50% 98.80% 96.50% 93.90% An example wth lterature data Now consder a lfetme dataset related to an electrcal nsulator subjected to constant stress and stran, ntroduced by Lawless (198). The dataset does not have censored values and represents the lfetme (n mnutes) to falure: 0.96, 4.15, 0.19, 0.78, 8.01, 31.75, 7.35, 6.50, 8.7, 33.91, 3.5, 16.03, 4.85,.78, 4.67, 1.31, 1.06, and Assume that the EEG dstrbuton s approprated to analyze ths dataset, and then t wll be compared wth other lfetme dstrbutons such as Webull, Gamma, and Lognormal. As shown, the effcency of the dfferent non-nformatve pror dstrbutons changes wth the shapes of the hazard functons, therefore, to get good nferences on parameters of nterest t s necessary to have some pror nformaton on how the hazard functon behaves for the Lawless data set. In ths way, Barlow & Campo (1975) proposed a smple graphcal technque that has been wdely used to verfy the behavor of the rsk functon called TTT plot (total tme for testng). The graph s constructed wth the plot of the consecutve quanttes [r/n, G (r/n)], where G (r/n) s a functon gven by 37

14 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION r 1 t n r t r G r / n, t where r = 1,..., n and t (), = 1,, n are the order statstcs n the sample. Usng the TTT curve n an emprcal scale, one can determne the shape of the hazard functon for the lfetme data. A dagonal lne ndcates that the data have a constant hazard functon; f the curve s convex the rsk s decreasng; f t s concave, there s an ndcaton that the rsk s ncreasng; f frst s convex and after ths s concave then there s an ndcaton that there s a bathtub shape for the hazard functon; f t s frst concave and after ths convex, there s an ndcaton of nverse form of the bath for the hazard functon. The Fgure 3 shows how to verfy the behavor of the hazard functon. n 1 Fgure 3. TTT plots for dfferent dstrbutons ndcatng the shape of the hazard functon. Some TTT transformatons can be studed to solve other problems. Nar et al. (008) show some of these transformatons appled n survval analyss. Fgure 4 shows the TTT plot for the Lawless data set. 38

15 RAMOS ET AL. Fgure 4. TTT plot for the dataset lfetme related to an electrcal nsulator subjected to constant stress and stran (Lawless data). It s observed n Fgure 4 that the TTT plot s convex; then t can be concluded that the rsk s a decreasng functon. When the hazard functon s decreasng, t was observed from the results of secton 4.1, that non-nformatve prors obtaned through the product of ndependent gamma dstrbutons s the best pror wth lttle pror nformaton about the parameters of nterest. The jont posteror dstrbuton of λ and γ (4) s obtaned by replacng π(γ, λ) by (1). It s necessary to use numercal methods to extract nformaton from the margnal posteror dstrbutons λ and γ. MCMC methods are used to smulate samples for the jont posteror dstrbuton; that s, also for the margnal posteror dstrbutons of nterest. It was generated 110,000 teratons wth a burn-n of 10,000 values and jumps of sze 10; so we get chans of the margnal posteror dstrbutons for λ and γ of sze 10,000 obtaned usng MCMC methods. To verfy the convergence of the chans, we have used Geweke (199) dagnoss, whch ndcated the convergence of the two chans. The convergence and autocorrelatons s also observed n the trace-plots of the smulated seres gven n Fgure 5. To verfy the performance of other lfetme dstrbutons we also consder as non-nformatve pror, the product of ndependent gamma dstrbutons γ ~ Gamma (0.01, 0.01), λ ~ Gamma (0.01, 0.01) assumng the followng lfetme dstrbutons: EGE, Webull, Gamma and Lognormal dstrbuton. The results are compled n Tables 5 and 6, respectvely. 39

16 OBJECTIVE PRIORS FOR ESTIMATION OF GEOMETRIC DISTRIBUTION Fgure 5. Trace-plots and autocorrelaton graphs for the generated values of λ and γ. Table 5. Posteror estmates (means) for γ and λ consderng dfferent probablty dstrbutons for the Lawless data set. Parameters EGE Webull Gamma Log-normal γ (0.0161) 0.769(0.1356) 0.675(0.1870) (0.3969) λ (0.1649) 7.850(4.50) (0.0183).6410(1.000) Table 6. Obtaned results for the DIC, BIC and AIC crtera for the dfferent probablty dstrbutons for the Lawless data set. Crtera EGE Webull Gamma Log-normal DIC BIC AIC Based on any of the crtera used by the table t can be concluded that EGE was the best ft to the offered data. Concluson The use of extended exponental geometrc (EEG) dstrbutons showed a good flexblty of ft for lfetme data applcatons and could be an alternatve 40

17 RAMOS ET AL. dstrbuton to other usual dstrbutons n Survval analyss. The great number of exstng non-nformatve pror dstrbutons can cause dffcultes n the choce of an adequate pror wth lttle nformaton a pror, manly when these pror dstrbutons do not produce smlar posteror summares. In ths way, the development of a general theory for the constructon of non-nformatve pror dstrbutons s an mportant topc to be nvestgated by researchers n the Bayesan nference. The results showed the effects of dfferent non-nformatve pror dstrbutons related to the changes n the rsk functon usng extended exponental geometrc (EEG) dstrbutons. Therefore, we recommend the product of gamma dstrbutons Gamma (0.01, 0.01) when the hazard functon s decreasng and the nonnformatve MDIP pror dstrbuton when the hazard functon s ncreasng. Wth these choces of pror we surely get better nferences for the parameters. References Achcar, J. A, Moala, F. A, & Boleta, J. (March, 010). Generalzed Exponental Dstrbuton: A Bayesan approach usng MCMC methods. Poster presented at the 10th Bayesan Statstcs Brazlan Meetng, Ro de Janero, Brazl. Adamds, K, Dmtrakopoulou, T, & Loukas, S. (005). On an Extenson of the Exponental Geometrc Dstrbuton. Statstcs and Probablty Letters, 73, Adamds, K, & Loukas, S. (1998). A Lfetme Dstrbuton wth Decreasng Falure Rate. Statstcs and Probablty Letters, 39, Barlow, R. E., & Campo, R. A. (1975). Total Tme on Test processes and applcatons to falure data analyss. In R. E. Barlow, J. B. Fussel, & N. D. Sngpurwalla (Eds.), Relablty and Fault Tree Analyss: Theoretcal and Appled Aspects of System Relablty and Safety Assessment ( ). Phladelpha, PA: Socety for Industral and Appled Mathematcs. Bayes, T. R. (1958). Essay towards solvng a problem n the doctrne of changes. Reprnted n Bometrka, 45, , Berger, J. O, & Bernardo, J. M. (199). On the Development of the Reference Pror Method. Fourth Valenca Internatonal Meetng on Bayesan Statstcs, Span. 41

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