Scale Parameter Estimation of the Laplace Model Using Different Asymmetric Loss Functions

Transcription

1 Scale Parameter Estimation of the Laplace Model Using Different Asymmetric Loss Functions Said Ali (Corresponding author) Department of Statistics, Quaid-i-Azam University, Islamaad 4530, Pakistan & Department of Decision Sciences, Bocconi University, Milan, Italy Muhammad Aslam Department of Statistics, Quaid-i-Azam University, Islamaad 4530, Pakistan Nasir Aas Department of Statistics, Government Post Graduate College, Jhang, Pakistan Syed Mohsin Ali Kazmi Sustainale Development Policy Institute Islamaad, 44000, Pakistan Received: January 9, 0 Accepted: Feruary 0, 0 Pulished: May, 0 doi:0.5539/isp.vnp05 Astract URL: In the last few decades, there has een an emergent interest in the construction of flexile parametric classes of proaility distriutions in Bayesian as compared to Classical approach. In present study Bayesian Analysis of Laplace model using Inverted Gamma, Inverted Chi-Squared informative, Levy and Gumel Type-II priors is discussed. The properties of posterior distriution, credile interval, highest posterior density region (HPDR) and Bayes Factor are discussed in current study. Bayes estimators are derived under squared error loss function (SELF), precautionary loss function, weighted squared error loss function and modified (quadratic) squared error loss function. Hyperparameters are determined through Empirical Bayes method. The estimates are also compared using the posterior risks (PRs) under the said loss functions. The priors and loss functions are compared using a real life data set. Keywords: Censored sampling, Squared error loss function (SELF), Precautionary loss function, Weighted squared error loss function (WSELF), Modified squared error loss function (MSELF), Credile interval (CI), Highest posterior density region (HPDR), Fixed test termination time, Informative prior, Posterior risk, Empirical Bayes. Introduction Very few real world events that we need to statistically study are symmetrical. Thus the popular normal model would not e a useful model for studying every phenomenon. The normal model at times is a poor description of oserved phenomena. Skewed models, which exhiit varying degrees of asymmetry, are a necessary component of the modeler s tool kit. A variety of different forms of the Laplace distriution have een introduced and applied in several areas of real world prolems. So Laplace distriution is getting popularity due to simplicity of its characteristic function as an alternative to Normal model. In particular, various forms of the Laplace distriution have een introduced and applied in several areas. In this article Laplace model is considered as a lifetime model using complete and censored data ecause in many experiment we cannot continue experiment up to last item failure due to cost or time constraints. So in these cases censoring is unavoidale. It needs to e mentioned here Laplace distriution has een considered efore in literature. Kappenman (975) otain Pulished y Canadian Center of Science and Education 05

2 the conditional confidence interval for the parameters of a doule exponential distriution y finding the conditional distriutions of the pivotal quantities for location and scale parameters. Kappenman (977) propose a procedure for otaining and lower proaility tolerance interval for a proportion of a population for the unknown two parameters of Laplace (doule exponential) distriution. Balakrishnan and Chandramouleeswaran (996) present an estimators for the reliaility function ased on the est linear uniased estimators (BLUEs) for the location and scale parameters of Laplace distriution ased on Type-II censored samples, and they show that otained estimator is almost uniased at varying level of reliaility, Childs and Balakrishnan (000) consider the progressively type-ii right censored sample for analysis of Laplace distriution. Childs and Balakrishnan (997) constrain the maximum likelihood estimators for the parameters of Laplace distriution under the general Type-II censored samples, which are simple linear functions of the order statistics. They also examine the asymptotic variance through the Fisher information matrix. Nadaraah (009) utilize the Laplace distriution random variales with application to price indices. Kozuowski and Nadaraah (00) motivated y the recent popularity of Laplace distriution, provide a comprehensive review of the known Laplace distriutions along with their properties and applications Nadaraah (00) otain two posterior distriutions for the mean of the Laplace distriution y deriving the distriutions of the product XY and the ratio X/Y when X and Y are Student s t and Laplace random variales distriuted independently of each other. The rest of the article is organized as follows. The Laplace model likelihood, posterior distriution using different informative priors and their properties are defined in Section for complete and censored data. Empirical Bayes method, predictive distriutions (prior and posterior) are provided in Section 3. Simulation study of different properties defined in section is conducted using simulated data and real data in Section 4. Section 5 contains discussion and derivation of Bayesian Interval Estimation using real data. Bayesian hypotheses testing discussed in Section 6, while Bayes Estimates and their respective Posterior Risks are evaluated using different loss functions in Section 7 for real and simulated data. A model comparison framework is presented in Section 8. Some concluding remarks and further research proposal are given in last Section 8.. Posterior Distriution and Likelihood Function The posterior distriution summarizes availale proailistic information on the parameters in the form of prior distriution and the sample information contained in the likelihood function. The likelihood principle suggests that the information on the parameter should depend only on its posterior distriution. Bayesian scientist s o is to assist the investigator to extract features of interest from the posterior distriution. In this section we will use the Laplace distriution as sampling distriution mingles with the informative and noninformative priors for the derivation of posterior distriution. A random variale x is said to possess a Laplace distriution if it has the following form: f (x) = ( exp x ), > 0, < x <, here is scale parameter and location parameter is zero. The likelihood function for a random sample x, x,... x n which is taken from Laplace distriution is: L(.) = n n n exp X i / () The likelihood function for censored data as descried Mendenhall and Hader (958) is: i= r L( x) f (x ) [ F(T)](n r) L(, x) = ( ) r e [ r= x ] +(n r)t Where T is time, r is for censored oservations and n-r are uncensored oservations.. Posterior Distriution Using Inverted Gamma Prior Informative priors are those that delierately insert information that researchers have at hand. This seems like a reasonale and reasoned approach since previous scientific knowledge should play a role in statistical inference. The author is delierately manipulating prior information to otain a desired posterior result. An informative prior provides more information than the non-informative priors, therefore the analysis using these prior more accurate and informative than classical approach. 06 ISSN E-ISSN

3 The Inverted Gamma prior of is defined as: f () = a Γ(a) ( ) (a+) exp ( /), a,, > 0 () The posterior distriution of parameter for the given data (x = x, x,..., x n ) using equation () and () is: p( x) = ( + ni= x i ) (a+n) Γ(a + n) (a+n+) exp n + x i /, > 0 which is the density similar to Inverted-Gamma distriution. So x G (, β) where G for Inverted Gamma distriution and = a + n, β = + n i= x i. For Censored data the posterior distriution of parameter ( r= ) (r+a) + (n r)t + ( ) (r+a+) p( x) = e [ r= x ] +(n r)t+, > 0 Γ(r + a) which is the density similar to Inverted-Gamma distriution. So x G ( a + r, r x ) = + (n r)t +. Posterior Distriution Using the Inverted Chi-squared (ICS) Prior Using ICS an informative prior with hyperparameters a and, this is defined y the following density: f () = (/) a a Γ (a/) i= ( ) ( a +) exp ( /), a,, > 0 (3) We otain the posterior distriution of for the given data (x = x, x,..., x n ) using equation () and (3) as: p( x) = ( (/) + ni= x i ) ((a/)+n) Γ ((a/) + n) ((a/)+n+) exp n (/) + x i /, > 0 Which is similar to inverted-gamma distriution, so x G (, β), where = a + n, β = + n i= x i. For Censored, the posterior distriution as follows p( x) = ( r= + (n r)t + Γ ( r + a ) ) (r+ a ) i= ( ) (r+ a +) e [ ] r= x +(n r)t+, > 0 Which is similar to inverted-gamma distriution, so x G (, β), where = a + r, β = r = + (n r)t +..3 Posterior Distriution Using Levy Prior The Levy prior of is defined as: f () = π ( ) (.5) exp ( /),, > 0 (4) The posterior distriution of parameter for the given data (x = x, x,..., x n ) using equation () and (4) is: ( p( x) = + n i= x i ) ( +n) Γ ( + n) ( 3 +n) exp n + x i /, > 0 which is the density similar to Inverted-Gamma distriution and we can write it as x G (, β), where = n, β = n i= x i and here G for Inverted Gamma distriution. For Censored data the posterior distriution of parameter p( x) = ( r= + (n r)t + Γ ( r + ) ) (r+ ) i= ( ) (r+ 3 ) e [ ] r= x +(n r)t+, > 0 Pulished y Canadian Center of Science and Education 07

4 Here also x G ( r, r = + (n r)t )..4 Posterior Distriution Using Gumel Type-II (GTII) Prior Using GTII an informative prior with hyperparameter, this is defined y the following density: f () = ( ) () exp ( /),, > 0 (5) We otain the posterior distriution of for the given data (x = x, x,..., x n ) using equation () and (5) as: ( + ni= x i ) (+n) p( x) = Γ ( + n) (+n) exp n + x i /, > 0 which is inverted-gamma distriution, x G (, β), where = + n, β = + n i= x i. For Censored, the posterior distriution as follows p( x) = ( r= + (n r)t + ) (r+) Γ (r + ) So x G (, β), where = + r, β = r = + (n r)t +..5 Properties of Posterior Distriution Assuming Different Priors i= ( ) (r+) e [ r= x ] +(n r)t+, > 0 Since our Posterior distriution is Inverted-Gamma distriution, here we give general forms of properties for Inverted- Gamma distriution. Mean = β/( ), >, Mode = β/( + ). Variance = β / ( ( ) ( ) ), >. Coefficient of Skewness = ( 4 ) / ( 3), > 3. Excess = (30 66) / (( 3) ( 4)), > 4, where and β are respective posterior distriution parameters. 3. Elicitation of Hyperparameters through Empirical Bayes Empirical Bayes procedures utilize past data as a means for ypassing the necessity of identifying a completely unknown and unspecified prior distriution having frequency interpretation. Graski and Sarhan (996), Sarhan (003) used empirical Bayes estimation in the case of exponential reliaility while Ahn et al. (006) consider this procedure for the hazard rate estimation of a mixture model with censored lifetimes. The empirical Bayes approach may e considered as a two-stage estimation procedure in which the hyperparameter is estimated from the marginal distriution and then the parameter is estimated using pseudo-prior where the hyperparameters are replaced y the estimation of first stage. Roins (955) sought a representation of the desired Bayes rule in terms of the marginal distriution of data (called prior predictive distriution) and then uses the data to estimate it rather than the prior distriution. For excellent discussion aout empirical Bayes see Kass and Steffey (989), Bansal (007). We use the Inverted Gamma, levy, Gumel type-ii and Inverted Chi-Squared informative priors for the cradle of prior predictive distriution which will e used for empirical Bayes procedure. Followings are the prior predictive equations used for Empirical Bayes procedure. Let Y e the random variale from Laplace distriution with unknown parameter. f (y ) = exp ( y ), > 0, < y < The prior predictive distriution is otained y using the following equation: p(y) = 3. Prior Predictive Distriution Using Inverted Gamma Prior The prior predictive distriution using Inverted Gamma prior is: 0 p() f (y, )d a a p(y) =, < y < (a+) ( y + ) We use aove equation of prior predictive distriution for the elicitation of hyperparameters a and. 08 ISSN E-ISSN

5 3. Prior Predictive Distriution Using Inverted-chi Prior The prior predictive distriution using Inverted Chi-Squared prior is: a (a/) p(y) =, < y < (a/)+ ((a/)+) ( y + (/)) We use aove equation of prior predictive distriution for the elicitation of hyperparameters a and. 3.3 Prior Predictive Distriution When Prior is Levy Distriution The prior predictive distriution using Levy prior is: p(y) =, < y < (5/) (3/) ( y + (/)) 3.4 Prior Predictive Distriution When Prior is Gumel Type-II Distriution The prior predictive distriution using Gumel Type-II prior is: 3.5 Predictive Distriution p(y) =, < y < () ( y + ) Often it is necessary to make predictions of future oservations, ased on our est inferences on parameters determined through oservations already made. Posterior predictive distriution is the product of the posterior distriution and (conditional) independence (given the parameters) of the new oservation from the learning sample. p(y x) = p( x)p(y ) d Where y = x n+ e the future oservation given the sample information x = x, x,...x n, from of the model with unknown parameter. Predictive Interval can e otained as: p(y x)dy = L p(y x)dy = = p(y x)dy 3.6 Predictive Distriution and Predictive Intervals Using Inverted-Gamma Prior The posterior predictive distriution for a future random variale y given that data x = x, x,...x n is: And Predictive interval is: = U p(y x) = (n + a) [ + n i= x i ] (a+n) ( ni= x i + y + ), < y < (6) (a+n+) [ + ni= x i ] (a+n) ( ni= x i + L + ) (a+n), = [ + ni= x i ] (a+n) ( ni= x i + U + ) (a+n) p(y x) = And Predictive interval is: [ r= ] (r+a) = + (n r)t + [ r= ] (r+a), + (n r)t + + L [ r= + (n r)t + ] (r+a) (r + a) [ r= + (n r)t + + y ] (r+a+), < y < = [ r= + (n r)t + ] (r+a) [ r= + (n r)t + + U ] (r+a) Pulished y Canadian Center of Science and Education 09

6 where a and are hyperparameters determined y empirical Bayes method. 3.7 Predictive Distriution and Predictive Intervals Using Inverted-Chi Prior The posterior predictive distriution for a random variale y given that data x = x, x,...x n is: Equations of Predictive Intervals are: = Equations of Predictive Intervals are: = ( ) [ n + a + n i= x i ] ( a +n) p(y x) = ( ni= x i + y + [ + n i= x i ] ( a +n) ( ni= x i + L + ) ( a +n), = ), < y < ( a +n+) [ + n i= x i ] ( a +n) ( ni= x i + U + ) ( a +n) ( ) r + a [ r= ] + (n r)t + (r+ a ) p(y x) = [ r= + (n r)t + + y ] (r+ a +), < y < ( r= ) + (n r)t + (r+ a ) ( r= + (n r)t + + L) (r+ a ), = Where a and are hyperparameters to e elicited y empirical Bayes procedure. 3.8 Predictive Distriution and Predictive Intervals Using Levy Prior ( r= ) + (n r)t + (r+ a ) ( r= + (n r)t + + U) (r+ a ) The posterior predictive distriution for a future random variale y given that data x = x, x,...x n is: p(y x) = ( ) [ n + + n i= x i ] ( +n) ( ), < y < ni= ( x i + y + 3 +n) And Predictive interval is: = [ + n i= x i ] ( +n) ( ni= x i + L + ) ( +n), = [ + n i= x i ] ( +n) ( ni= x i + U + ) ( +n) And Predictive interval is: <Tale > = p(y x) = [ r= ] + (n r)t + (r+ ) ( ) r + [ r= + (n r)t + + y ] (r+ 3 ), < y < [ r= ] + (n r)t + (r+ ) [ r= + (n r)t + + L] (r+ ), = 3.9 Predictive Distriution and Predictive Intervals Using Gumel Type-II Prior [ r= ] + (n r)t + (r+ ) [ r= + (n r)t + + U] (r+ ) The posterior predictive distriution for a random variale y given that data x = x, x,...x n is: p(y x) = (n + ) [ + n i= x i ] (n+) ( ni= x i + y + ) (n+), < y < 0 ISSN E-ISSN

7 Equations of Predictive Intervals are: = [ + ni= x i ] (n+) ( ni= x i + L + ) (n+), = [ + ni= x i ] (n+) ( ni= x i + U + ) (n+) p(y x) = (r + ) [ r= + (n r)t + ] (r+) [ r= + (n r)t + + y ] (r+), < y < Equations of Predictive Intervals are: = [ r= + (n r)t + ] (r+) [ r= + (n r)t + + L ] (r+), = [ r= + (n r)t + ] (r+) [ r= + (n r)t + + U ] (r+) 4. Simulation Study For conducting simulation study, data from Laplace model is generated using inverse transformation. Simulation study of different properties discussed in aove section are given in appendix. From Tales -8 in appendix, one can easily oserve that mode is more close to the true supposed parameter values than the mean and variance value increase as we increase the parameter value and especially for censored data it is clear that variance is more than complete data ecause in censored data we are using less information as compared to complete data set. Similar conclusion for skewness and kurtosis tales can e drawn. One thing is common in Tale - that with the increase of sample size our parameter values approach to true values of parameters and variances, skewness and kurtosis decreases. Also comparing the prior, one can easily oserve that GTII prior has etter results in terms of posterior risk, skewness and kurtosis than the other informative priors. 4. Real Data Findings Childs and Balakrishnan (996) consider the data of mean difference of 33 year in flood stage for two stations on the Fox River in Wisconsin. Following tale shows the different posterior distriution properties using real data set assuming IG and ICS informative priors. GT-II prior performance is etter than IG, ICHI and LP prior ased on variance, mode, skewness and kurtosis. Graphical presentation as follows: <Figure -4> Aove figures confirm the same ehaviors in terms of their shape ut Gumel Type-II prior is different from others and also from aove figure one can easily oserve that using censored data information lost ut due to application this is unavoidale. Figure (i) is for complete data set using informative priors and Figure (ii) is for complete and censored data using IG prior. Here in figure (ii) IG (c) denotes the censored data. One can easily oserve that using censored data we lost information. 5. Bayesian Interval Estimation In Bayesian analysis, credile interval ecomes the counterpart of the classical confidence interval when we put hyperparameter (s) values equal to zero. Also credile interval may e unique for all models. The difference etween credile interval and highest density region (HPDR) is that credile interval is comparale with classical interval ut HPDR is unique. 5. Credile Interval Assuming Informative Priors (B) Let A c =, B χ c = (B), where A, B are the parameters of respective posterior distriution. Thus a ( )00% credile (A), χ (A), interval is {A c B c }, (See Ahmed Au-Tale et al. (007) and Saleem and Aslam (009)). 5. Highest Posterior Density Region (HPDR) The HPD interval is defined on the posterior density such that the posterior density at every point inside the HPD interval is greater than the posterior density at every point outside the HPD interval. An interval (, ) would e a (- ) 00% Pulished y Canadian Center of Science and Education

8 HPD interval for if it satisfy the following two conditions simultaneously. p( x)d = p( x) = p( x) (7) 5.. HPDR for Inverted Gamma Prior Using aove two conditions given in equation (7) solving for IG prior we get (an+) ln( / ) (+ n i= x i )( ) = 0 and Γ(a + n, ( + n i= x i )/ ) Γ(a + n, ( + n i= x i )/ ) ( )Γ(a + n) = 0 where Γ(n, n i= x i / ) = n n exp( )d is incomplete gamma function. i= x i / For Censored data and Γ(a + r, ( + (n r)t + r i= x i )/ ) Γ(a + r, ( + (n r)t + r i= x i )/ ) ( )Γ(a + r) = HPDR for Inverted Chi-squared Prior HPDR equations for Inverted Chi-Squared prior as follows (0.5a + n + ) ln( / ) (0.5 + n i= x i )( Γ(n + 0.5a, (0.5 + n i= x i )/ ) Γ(n + 0.5a, (0.5 + n i= x i )/ ) ( )Γ(n + 0.5a) = 0. For censored data (0.5a + r + ) ln( / ) (0.5 + (n r)t + r i= x i )( ) = 0 and (43) 5..3 HPDR for Levy Prior ) = 0 and Aove two conditions given in equation (6), solving for Levy prior we get HPDR equations for Levy prior as follows: (n +.5) ln( / ) (0.5 + n i= x i )( ) = 0 and Γ(n + 0.5, (0.5 + n i= x i )/ ) Γ(n + 0.5, (0.5 + n i= x i )/ ) ( )Γ(n + 0.5) = 0. For censored data and Γ(r + 0.5, (0.5 + (n r)t + r i= x i )/ ) Γ(r + 0.5, (0.5 + (n r)t + r i= x i )/ ) ( )Γ(r + 0.5) = HPDR for Gumel Type-II Prior HPDR equations for Gumel Type-II prior as follows (n + ) ln( / ) ( + n i= x i )( ) = 0 and Γ(n +, ( + ni= x i )/ ) Γ(n +, ( + n i= x i )/ ) ( )Γ(n + ) = 0, where Γ(n, n i= x i / ) = n n exp( )d i= x i / is incomplete gamma function. For Censored data For real data set the HPD and Credile intervals for different level of significance evaluated in the following tale considering different informative priors. <Tale > The HPD has smaller and than credile interval. ICSP and GTII has wider CI and HPD than IGP and LP. 6. Bayesian Hypotheses Testing and Bayes Factor Bayesian hypothesis testing is less formal than non-bayesian varieties. In fact, Bayesian researchers typically summarize the posterior distriution without applying a rigid decision process. If one wanted to apply a formal process, Bayesian decision theory is the way to go ecause it is possile to get a proaility distriution over the parameter space and one can make expected utility calculations ased on the costs and enefits of different outcomes. Considerale energy has een given, however, in trying to map Bayesian statistical models into the null hypothesis testing framework, with mixed results at est. This section contains the Bayesian hypotheses for parameter under the posterior distriution, considering using noninformative priors. Posterior proailities are defined for the H : vs. H : < hypotheses: P(H ) = P( ) = p( x)d where p( x) is the posterior distriution of given x and P(H ) = P (H ). Bayes Factors are the dominant method of Bayesian model testing. These are Bayesian analogues of likelihood ratio tests. The asic intuition is that prior and posterior information are comined in a ratio that provides evidence in favor of one model specification verses another. By dividing posterior odds under null and alternative hypotheses we get Bayes Factor. Bayes Factors are very flexile, allowing multiple hypotheses to e compared simultaneously and nested models are not ISSN E-ISSN

9 required in order to make comparisons it goes without saying that compared models should oviously have the same dependent variale. The Bayes factor can e interpreted as the odds for H to H that is given y the data. While the Bayesian approach typically avoids aritrary decision thresholds, Jeffreys (96) gives the following typology for comparing H vs H. B H is supported 0 B minimal evidence against H, 0 B 0 sustantial evidence against H 0 B 0 strong evidence against H B < 0 decisive evidence against H. Here B is used for Bayes factor. <Tale 3> For the smaller parameter value than Bayes Estimator value we does not get strong support for H ut when we go further from it we get strong evidence support for H. For testing 6.0 we oserve that decisive evidence againsth for all priors, 7.5 we have strong evidence againsth for informative priors. Minimal evidence against H occur in case of 9.0 and in case of.0, H is strongly supported especially for Inverted Gamma prior. Similarly for censored data testing 5.0 we oserve that minimal evidence against H for all priors, 6.5 we have minimal evidence against H using informative priors. We have strong evidence against H occur in case of 8.0 and 9.5, H is strongly supported especially for Gumel prior. The values taken in complete data cannot e taken here ecause using that values we have decisive evidences due to involvement of time, we are using less information aout data. 7. Bayes Estimators Under Different Loss Functions This section spotlight on the derivation of the Bayes Estimator (BE) under different loss functions and their respective Posterior Risk (PR). The results are also compared their results for informative prior. Bayes decision is a decision d which minimizes risk function then d is the est decision. If the decision is choice of an estimator then the Bayes decision is a Bayes estimator. The Bayes estimator for different loss function is given elow: we use four loss functions name as squared error loss function (SELF), weighted squared error loss function (WSELF), precautionary loss function and modified (quadratic) squared error loss function (M/QSELF). The squared error loss function (SELF) was proposed y Legendre (805) and Gauss (80) to develop least square theory. Later, it was used in estimation prolems when uniased estimators of parameter were evaluated in terms of the risk function which ecomes nothing ut the variance of the estimators. It was also oserved that SELF is a convex loss function and, therefore, restricts the class of estimators y excluding randomized estimators. The difficulty with unounded loss function, like SELF is that Bayes estimates may change enormously when the oservation of the random variale changes infinitesimally. Therefore, the investigator has to e asolutely precise aout his proaility statements. Furthermore, in real life situations, it is usually eing impossile to lose an infinite amount of money. The extensive form of analysis provides Bayes estimate under SELF, as E( x). Also note that squared error loss function is not the only loss function for which posterior mean is the Bayes estimate. The natural exponential family f ( x) = a()(x) exp(x), the Bayes estimates under entropy loss function is the posterior mean. The Bayes estimate under the weighted SELF may not exist if the weight function w() i ncreases too fast to infinity. Norstrom (996) introduced an alternative asymmetric precautionary loss function, and also presented a general class of precautionary loss functions as a special case. These loss functions approach infinitely near the origin to prevent underestimation, thus giving conservative estimators, especially when underestimation may lead to serious consequence. Since in risk analysis, oth the potentiality of an undesired event and its consequences are investigated. This potentiality is usually measured y either a proaility or a failure rate. The Bayes approach is widely applied to estimate this failure rate. When dealing with disastrous consequences, it can e worse to underestimate the potentiality of an event than to overestimate it. This is important when risk-level is the asis of a risk-reducing initiative, either y reducing the potentiality or the consequences. An erroneously low estimated risk-level can lead to the asence of necessary initiative to reduce the risk level. It is unreasonale to use a loss function that allows one to estimate a failure proaility of zero. A positive loss function at the origin allows estimating zero, and in a risk analysis, estimating zero failure proaility simply means that no risk is anticipated. Hence, a precautionary loss function is used. Also optimal policy selection has traditionally een discussed in relation to symmetric and often quadratic loss functions. So y using non-symmetric loss functions one is ale to deal with cases where it is more damaging to miss the target on one side than the other. 7. Bayes Estimate and Posterior Risk Under Different Priors Following tales gives the comparison of Bayes estimates and Posterior risk under different priors. <Tale 4> 7. Simulation of Bayes Estimates (BE) and Posterior Risk (PR) Simulation is a flexile methodology, we can use it to analyze the ehavior of a pattern of proposed usiness activity, new Pulished y Canadian Center of Science and Education 3

10 product, manufacturing line or plant expansion, and so on (analysts call this the system under study). In simulation one generates a sample of random data in such a way that mimics a real prolem and summarizes that sample in the same way. It is one of the most widely used quantitative methods ecause it is so flexile and can yield so many useful results. There are different method such as Monte Carlo simulation and Bootstrap to simulate the data. Here we did simulation in Minita for non-informative prior. Following tales contains the Bayes estimates and their posterior risks. It is immediate from appendix Tale 7-4, as we increase sample size posterior risk comes down and also with the increase of parameters values posterior risk also increase. For censored data posterior risk is greater than the complete data ecause of time contriution. The choice of loss function as concerned, one can easily oserve that modified squared error loss function has smaller posterior risk than other three loss functions. 7.3 Real Data Bayes Estimates (BE) and Posterior Risk (PR) Following tales shows the Bayes Estimates (BE) and their Posterior Risk (PR) in rackets for real data <Tale 5> Modified squared error loss function has smaller posterior risk than other three loss functions using Gumel Type-II prior. 8. Model Comparison The comparison of model performances is proposed to e ased on the generated posterior predictive distriutions. The criterion used to compare them is ased on the use of the logarithmic score as a utility function in a statistical decision framework. This was proposed y Bernardo (979) and used, for example, y Walker and Gutiérrez-Peña (999) and Martín and Pérez (009) in a similar context. In situations where the uncertainty is contained in the value of a future oservation y = x n+, the logarithmic score log (p k (y x)) is used, where p k (y x) denotes the posterior predictive density under model M k. Then, the posterior predictive expected utility is given y: Ū k = log (p k (y x)) p k (y x)dy. The optimal solution to the decision prolem of choosing among the competing models M 0; M,...; M l is given y the model M k, such that: Ū k = max Ū k. From a practical viewpoint, Ū k can e estimated k {0,,...,l} as: ˆŪ k = mi= m log (p k (y i x)), where y ; y,..., y m are an independent and identically distriuted random sample from p k (y x). In order to illustrate this and the applicaility of the proposed approaches, a performance comparison etween posterior distriution using Inverted Gamma and Inverted Chi-Squared priors, a random sample of size 0 generated from laplace distriution with mean 0 and scale parameter equal to 4 using Minita v (0.9789, , , ,.09396, ,.77, , , 3.74). We get U IG =-.7349, U ICS = U Levy =-.756 and U GT II =-.74 so Gumel Type-II prior is est. 9. Conclusion and Suggestions We consider the Bayesian analysis of the Laplace Model using complete and censored data assuming informative prior. Based on different properties of posterior distriution, hypotheses testing, HPD and credile intervals, we conclude that Gumel Type II prior results are more precised than other prior. The choice of loss function as concerned, one can easily oserved ased on evidence (different properties as discussed aove) that modified squared error loss function has smaller posterior risk. Model comparison method also suggest that Gumel Type II prior is est. One thing is common as we increase sample size posterior risk comes down. Also note that we cannot compare the result of complete data with censored data ecause in censored data we are using less information than the complete data set. In future this work can e extended using truncated Laplace model and considering location parameter. References Ahn, S. E., Park, C. S., & Kim, H. M. (007). Hazard rate estimation of a mixture model with censored lifetimes. Stoch Environ Res Risk Assess,, Au-Tale, A. A., Smadi, M. M., & Alawneh, J. A. (007). Bayes Estimation of the Lifetime Parameters for the Exponential Distriution. Journal of Mathematics and Statistics, 3(3), Balakrishnan, N., & Chandramouleeswaran, P. M. (996). Reliaility estimation and tolerance limits for Laplace distriution ased on censored samples. Microelectron Relia., 36, N (3), Bansal, A. K. (007). Bayesian Parametric Inference. Narosa Pulishing House Pvt. Ltd. Bernardo, J. M. (979). Expected information as expected utility. Annals of Statistics, ISSN E-ISSN

11 Childs, A., & Balakrishnan, N. (000). Conditional inference procedures for the Laplace distriution when the oserved samples are progressively censored. Metrika, 5, N (3), Childs, A., & Balakrishnan, N. (997). Maximum likelihood estimation of place Laplace parameters ased on general Type-II censored samples. Statistical Papers, 38, N (3), Gauss, C. F. (80). Method des Moindres Carres Memoire sur la Comination des Oservations. Translated y J. Bertrand. (955). Mallet-Bachelier, City Paris. Graski, F., & Sarhan, A. (996). Empirical Bayes estimation in the case of exponential reliaility. Reliaility Engineering and System Safty, 53, Jeffreys, H. (96). Theory of Proaility (3rd Edt.). Oxford University Press. Kass, R. E., & Steffey, D. (989). Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models). Journal of the American Statistical Association, 84(407), Kappenman, Russell F. (975). Conditional confidence intervals for Doule Exponential distriution parameters. Technometrics, 7, N (), Kappenman, Russell F. (977). Tolerance intervals for the Doule Exponential distriution. Journal of the American Statistical Association (JASA), 7, N (360), Kozuowski, J. T., & Nadaraah, S. (00). Multitude of Laplace distriutions. Stat Papers, 5, Legendre, A. (805). Nouvelles Methodes Pour la Determination des Orites des Cometes, Courcier, City Paris. Martín, J., & Pérez, C. J. (009). Bayesian analysis of a generalized lognormal distriution. Computational Statistics and Data Analysis, 53, Mendenhall, W., & Hader, A. R. (958). Estimation of parameters of mixed exponentially distriuted failure time distriution from censored life test data. Biometrika, 45, 07-. Nadaraah, S. (009). Laplace random variales with application to price indices. AStA Adv Stat Anal, (009) 93, Nadaraah, S. (00). Some Posterior Distriutions for the Laplace Mean. Acta Mathematica Scientia, 30B, N(), Norstrom, J. G. (996). The use of precautionary loss functions in risk analysis. IEEE Trans. Relia., 45, N (), Sarhan, A. M. (003). Empirical ayes estimates in exponential reliaility model. Applied Mathematics and Computation, 35, Roins, H. (955). An Empirical Bayes Approach to Statistics. Proceedings of third Berkeley Symposium on Mathematical Statistics and Proaility,, Saleem, M., & Aslam, M. (009). On Bayesian analysis of the Rayleigh Survival Time assuming the Random censor Time. Pak. J. Statist., 5(), 7-8. Walker, S. G., Gutiérrez-Peña, E. (999). Roustifying Bayesian procedures. In: Bernardo, J. M., Berger, J. O., Dawid, A. P., Smith, A. F. M. (Eds.), Bayesian Statistics, 6. Oxford University Press, Tale. Properties of posterior distriution using real data set Prior Mean Mode Variance Skewness Excess IGP ICSP Levy GTII For Censored data, r = 6, T = 0 IGP ICSP Levy GTII Pulished y Canadian Center of Science and Education 5

12 Tale. HDR and CI using complete real data set; Parentheses contain credile intervals, Curly races HPDR and rackets classical intervals Prior 90% 95% 99% IGP (7.894,.7594) [7.787,.7754] {7.909,.7499} (6.840,3.56) [6.8304,3.5836] {6.84,3.569} ICSP (7.989,.7989) [7.787,.7754] {7.998,.7898} Levy (7.088,.559) [7.0845,.558] {7.0883,.55} GT (7.0005,.3509) [6.999,.3375] {7.00,.3305} IGP (.3633,5.938) [.48,6.865] {.3695,5.98} ICSP (.488,6.73) [.48,6.865] {.499,6.867} Levy (.37,5.90) [.33,5.855] {.330,5.800} GT (0.8656,4.376) [0.8558,4.3543] {0.8657,4.3559} (6.8500,3.607) [6.8304,3.5836] {6.850,3.6070} (6.7465,3.3468) [6.743,3.3400] {6.7469,3.3386} (6.665,3.88) [6.6580,3.046] {6.6668,3.038} (0.648,8.434) [0.663,8.846] {0.670,8.4336} (0.679,8.7708) [0.663,8.846] {0.67,8.7700} (0.4038,7.7074) [0.406,7.704] {0.4039,7.700} (0.63,6.668) [0.53,6.6390] {0.63,6.63} (6.58,5.338) [6.8,5.367] {6.99,5.37} (6.35,5.390) [6.8,5.367] {6.30,5.389} (6.403,5.0833) [6.37,5.0756] {6.409,5.073} (6.070,4.805) [6.0635,4.7944] {6.0730,4.790} (9.334,34.887) [9.3669,34.863] {9.3350,34.880} (9.3780, ) [0.3669,34.863] {9.3850,34.743} (9.544, ) [9.54,33.364] {9.590,33.368} (8.956,3.0030) [8.948,3.9744] {8.9559,3.9609} 6 ISSN E-ISSN

13 Tale 3. Posterior proailities under null and alternative hypotheses, Bayes factors using IG and ICS for real data Null Hypothesis Alt. Hypothesis Prior Posterior Proailities Bayes H H Distriution P (H ) P (H ) Factor B 6.0 > 6.0 IG ICS Gumel Levy > 7.5 IG ICS Gumel Levy > 9.0 IG ICS Gumel Levy >.0 IG ICS Gumel Levy > 5.0 IG ICS Gumel Levy > 6.5 IG ICS Gumel Levy > 8.0 IG ICS Gumel Levy > 9.5 IG ICS Gumel Levy Pulished y Canadian Center of Science and Education 7

14 Tale 4. Bayes estimators and respective posterior risk under different loss function for complete data set Squared Error Loss Function=L = L(, d) = ( d) Posterior Distriution Using Prior Bayes Estimator Posterior Risk d = E( x) E x L (, d) = Var( x) Inverse Gamma Inverse Chi-Squared Gumel Type-II Levy Inverse Gamma Inverse Chi-Squared Gumel Type-II Levy Inverse Gamma Inverse Chi-Squared Gumel Type-II Levy Inverse Gamma Inverse Chi-Squared Gumel Type-II Levy + n i= x i (a+n ) + n i= x i (+ n i= x i ) (a+n ) (a+n ) ni= x i ) ( a +n ) ( + ( a +n ) ( a +n ) + n i= x i n + n i= x i (n ) (+ n i= x i ) (n) (n ) ( + n i= x i ) (n ) (n 3 ) Modified /Quadratic Square Error Loss Function=L = L(, d) = ( d Bayes Estimator Posterior Risk d = E( x) E E( x) x L (, d) = x)] [E( E( x) + n i= x i a+n+ a+n+ + n i= x i ( a +n+) ( a +n+) + n i= x i (+n) (+n) + n i= x i ( 3 +n) ( 3 +n) Weighted Loss Function=L 3 = L(, d) = ( d) Bayes Estimator Posterior Risk d = E E( x) x L 3 (, d) = E( x) + n i= x i + n i= x i (a+n) (a+n)(a+n ) + n i= x i ( a +n) + n i= x i + n i= x i + n i= x i (n+) n(n+) + n i= x i ( +n) + n i= x i ( +n)(n ) ( a +n)( a +n ) E( x) Precautionary Loss Function=L 4 = L(, d) = ( d) d Bayes Estimator Posterior Risk d = E( x) E x L 4 (, d) = ( E( x) E( x) ) ( ) + n i= x i + n (a+n )(a+n ) i= x i (a+n )(a+n ) + n i= x i (a+n ) ( + n i= x i ( ( a +n )( a +n ) + n i= x i ) ( a +n )( a +n ) ( + n i= x i ) + ( n i= x i + ni= ) x (n)(n ) i (n)(n ) + n i= x i n ( ) ( + n i= x i ) + n i= x i (n (n ) + n i= x i (n )(n 3 ) Tale 5. BEs and PRs using IC and ICS priors under different LFs )(n 3 ) ( a +n ) ) ) Complete Data Censored Data Complete Data Censored Data LF IG ICS IG ICS Levy GTII Levy GTII L ( ) (3.006) (.9044) (.8844) ( ) ( ) (9.9866) (8.563) L ( ) (0.0939) ( ) ( ) ( ) (0.0857) ( ) ( ) L ( ) (0.993) ( ) (.09793) 9.54 (0.8355) ( ) (.03739) 5.53 ( ) L ( ) ( ) (.96677) (.46) (0.9957) (0.905) (.54473) (.0844) 8 ISSN E-ISSN

15 Tale 6. Posterior Distriution Properties via IGP for complete data n =0.5 = n =3.0 = Tale 7. Posterior distriution properties via ICSP for complete data n =0.5 = n =3.0 = Tale 8. Posterior distriution properties via IGP for censored data n =0.5 = n =3.0 = Pulished y Canadian Center of Science and Education 9

16 Tale 9. Posterior distriution properties via ICSP for censored data n =0.5 = n =3.0 = Tale 0. Posterior distriution properties via LP for complete data n =0.5 = n =3.0 = Tale. Posterior distriution properties via GTIIP for complete data n =0.5 = n =3.0 = ISSN E-ISSN

17 Tale. Posterior distriution properties via LP for censored data n =0.5 = n =3.0 = Tale 3. Posterior distriution properties via GTIIP for censored data n =0.5 = n =3.0 = Tale 4. Skewness and excess kurtosis of posterior distriution assuming different prior for complete data Prior IGP ICSP LP GTIIP n Skewness Excess Skewness Excess Skewness Excess Skewness Excess Pulished y Canadian Center of Science and Education

18 Tale 5. Skewness and excess kurtosis of posterior distriution assuming IG for censored data Prior IG ICSP n =0.5 =.0 =0.5 =.0 Skewness Excess Skewness Excess Skewness Excess Skewness Excess n =3.0 =4.0 =3.0 = Tale 6. Skewness and excess kurtosis of posterior distriution assuming LP for censored data Prior LP GTII n =0.5 =.0 =0.5 =.0 Skewness Excess Skewness Excess Skewness Excess Skewness Excess n =3.0 =4.0 =3.0 =4.0 =3.0 = ISSN E-ISSN

19 Tale 7. BEs and PRs using IGP under L Prior IGP ICSP n =0.5 = =3 =4 =0.5 = =3 = ( ) ( ) (0.4469) ( ) ( ) (0.0539) ( ) ( ) ( ) (0.0386) ( ) ( ) (0.0065) (0.0830) ( ) ( ) ( ) (0.0095) ( ) ( ) ( ) ( ) (0.0944) (0.6609) ( ) ( ) (0.0859) (0.0356) ( ) ( ) (0.0839) (0.0335) ( ) ( ) ( ) ( ) (0.0005) ( ) ( ) ( ) (0.0950) (0.0336) ( ) (0.4887) (0.067) (0.0954) (0.7060) ( ) ( ) ( ) (0.3667) (0.973) ( ) ( ) (0.389) (0.8949) ( ) ( ) (0.0670) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0564) (0.076) ( ) ( ) (0.0550) (0.0799) ( ) ( ) ( ) ( ) (0.0009) (0.0060) ( ) ( ) Tale 8. BEs and PRs using IGP under L Prior IGP ICSP n =0.5 = =3 =4 =0.5 = =3 = ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.0568) ( ) (0.0568) ( ) (0.067) ( ) (0.067) ( ) (0.0453) ( ) (0.0453) (0.0770) (0.0480) (0.0803) (0.0480) (0.0803) ( ) (0.0035) ( ) ( ) ( ) ( ) ( ) ( ) (0.0054) ( ) (0.0054) (0.0085) (0.0055) (0.0085) (0.0055) (0.0085) Pulished y Canadian Center of Science and Education 3