Sri Laka Joural of Applied Statistics, Vol (5-3) Modelig ad Estimatio of a Bivariate Pareto Distributio usig the Priciple of Maximum Etropy Jagathath Krisha K.M. * Ecoomics Research Divisio, CSIR-Cetral Leather Research Istitute, Adyar, Cheai 600 00, Idia * Correspodig Author: jagath.krisha@gmail.com Received: 4, November 03 / Revised:, July 04 / Accepted: 9, October 04 IAppStat-SL 03 ABSTRACT I this paper we modeled a bivariate Pareto I distributio usig the method of priciple of maximum etropy probability distributio. Properties of the model are discussed. Further the estimatio of the parameters ivolved i the model is doe i two stages usig two differet methods amely, priciple of maximum etropy estimatio (POME) ad maximum likelihood estimatio. From the simulatio study coducted to compare the performace of the estimates obtaied by the above two methods, we coclude that POME method is performig better tha MLE ad the two methods are comparable. Keywords: Bivariate Pareto I, Maximum Etropy probability Distributio, POME, MLE, Bivariate Failure Rate.. Itroductio Every probability distributio has some ucertaity associated with it. Accordigly, for some give partial iformatio about some characteristics of the distributio or the radom variate, we wish to derive a model that best approximate the distributio which is cosistet with the give iformatio. A approach to produce a model for the data geeratig distributio is the well-kow maximum etropy method. Let X be a o-egative radom variable represetig the lifetime of a compoet with distributio fuctio F( x) P[ X x], survival fuctio F( x) F( x) ad probability desity fuctio f( x ). The the etropy fuctio (Shao (948)) which provides a quatitative measure for the ucertaity of the radom variable X is give by IASSL ISBN-39-4987 7
Jagathath Krisha K.M. H( f ) f ( x)log f ( x) dx. (.) 0 The maximum etropy distributio is the distributio whose probability desity fuctio f (.) maximizes H( f ) i a set of distributios with give costraits. I maximum etropy priciple, we begi with the fact that the distributio fuctio is ukow. It is well kow that amog the set of distributio, the most ucertai distributio is the uiform. Thus i maximum etropy procedure, we have access to some iformatio regardig the characteristic properties of the life time of the compoet or system. The additioal iformatio or costraits help us to obtai a appropriate model that maximizes the etropy fuctio H( f ) (see Kapur (989)). I reliability studies, the problem of iterest is the residual life time of a compoet which has survived beyod a age t. Ebrahimi (996) has defied the residual etropy fuctio applicable to such situatios. The maximum etropy distributio for the uivariate residual life time distributios has bee discussed i Ebrahimi (000) ad Asadi et. al. (004). Now let ( X, X) represets the life time of a two compoet parallel system with survival fuctio F x x PX x X x (, ),, the bivariate desity fuctio f ( x, x ) ad total failure rate ( t ) Cox s (97). The the bivariate residual etropy fuctio (Asha et. al (009)) H( f, t, t, t ) is give by where H( f, t, t, t ) H( f, t), H ( f, t, t ), H ( f, t, t ) (.) X H( f X, t) ( x) FX( x)log ( x) dx; t 0 F () t (.3) X t F( x, u) (,, ) ( ) log ( ) ; F( t, u) u t ut H f t t x t x t dx t t u ut (.4) F( u, x) (,, ) ( ) log ( ) ; F( u, t) u t ut H f t t x t x t dx t t u ut (.5) 7 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME The paper is orgaized ito seve sectios. I sectio, we derived a bivariate Pareto model ad discussed its properties. I sectio 3, we focused o the estimatio of oly the scale parameter by usig the priciple of maximum etropy estimatio (POME) ad maximum likelihood estimatio (MLE) from the desity of miimum. The estimatio of the rest of the parameters are obtaied by substitutig the estimate of the scale parameter obtaied from the uivariate miimum desity i the MLE for the bivariate Pareto I distributio. Also a asymptotic property of the model is also discussed i sectio 4. I sectio 5, simulatio studies for the estimates of the parameters by usig the two methods are doe ad root mea square errors of the estimates have also bee discussed. I sectio 6, a discussio based o the simulatio is give ad fially cocludig remarks is give i sectio 7.. Maximum Etropy Probability Distributio I this sectio, we formalized a bivariate maximum etropy distributio. The model is obtaied by maximizig the bivariate residual etropy fuctio H( f, t, t, t ) give i (.) subject to a series of costraits ad is discussed as follows. Result : Maximize H( f, t, t, t) f ( x, x ) with support of (, ) (, ) satisfyig (i) f ( x, x) 0 for all x, x (ii) f x x dxdx overall probability desity fuctio (, ) ad pi P Xi X j, i, j,, i j. d() t ( t t) ( t t) dt t t ( t) ( t t) ( t t) (iii),, k( t), k ( t t ), k ( t t ) with ( ) k, ( t ) k ad ( t ) k is the bivariate Pareto I distributio. Proof: To obtai the maximum etropy fuctio differetiate (.3), (.4) ad (.5) with respect to ttad, t ad equatig to zero. Thus we have IASSL ISBN-39-4987 73
Jagathath Krisha K.M. d H ( f, t X ) ( t ) log ( t ) H ( f X, t ) 0 dt H( f, t) log ( t) (.) X H ( f, t, t) ( t t) log ( t t) H ( f, t, t) 0 t t H ( f, t, t ) log ( t t ) (.) H ( f, t, t ) ( t t ) log ( t t ) H ( f, t, t ) 0 H ( f, t, t ) log ( t t ) (.3) Now we have to obtai the optimum values for ( t), ( t t) ad ( t t) that maximizes (.), (.) ad (.3). Hece we will maximize the etropy fuctios (.), (.) ad (.3) subject to the costraits (i), (ii) ad (iii). From (iii), we have, d() t dt k (.4) () t ( t t) t k (.5) ( t t) ( t t) t k (.6) ( t t) Itegratig (.4) with respect to t, we get () t kt Usig the coditio ( ) i costrai (iii), the iequality becomes k () t kt Itegratig (.5) with respect to t, we get ( t t) kt 74 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME Usig the coditio ( t ) k i costrai (iii), the iequality becomes ( t t) kt Itegratig (.6) with respect to t, we get ( t t) kt Usig the coditio ( t ), i costrai (iii) the iequality becomes k Hece ( t ),, kt k t k t ( t t) kt ( t) p ( t), ( t) p ( t). ad 0 0 The Cox s TFR uiquely determie the distributio through the expressio. x p ku ku kx x kx x f ( x, x) exp du du p k k k k f ( x, x ) x x ; x x Similarly for x x. kk Thus k p f ( x, x ) k k k x x ; x x kk k p k k k ; kk x x x x (.7) Writig,,, p p ad k k k k k we get, ( ) x x ; x x ( ) x x ; x x f ( x, x ) (.8) The model (.8) has a depedecy similar to the Freud s (96) bivariate expoetial distributio. This model is differet from all the models IASSL ISBN-39-4987 75
Jagathath Krisha K.M. discussed i literature i the sese that, it do ot have Pareto margial but have mixture Pareto margial. However, this distributio ejoys the bivariate extesio of several properties of the uivariate Pareto distributio. This model is applicable to a two compoet parallel system. Whe oe of the compoets fails, the other bear a extra load ad works with a reewed parameter. The survival fuctio of the model (.8) is give by F( x, x) ( ) x x x ( ) ( ( ) ) ( ) ( ) ( ) x ( ) x x x x ( ) ( ( ) ) ( ) ( ) ( ) x ; x ; (.9) Properties of the model:. Let X ( X, X ) be a radom vector i the support of (, ) (, ) with survival fuctio F( x, x ) specified i (.9), the X follows bivariate Pareto I distributio (.8), if ad oly if X is totaly dull at the poit t ( t, t) (Asha ad Jagathath (008)). A distributio of X ( X, X ) is called dull at the poit t ( t, t) wheever P[ X s t, X s t X t, X t] P[ X s, X s ] (.0) for all s, s.. A bivariate radom variable X has desity fuctio specified by (.8) if ad oly if ( x) is reciprocal liear give by ( ) ( x) x, x, x with 0 ( ) x ad 0 x ( ) x (Asha ad Jagathath (008)). x The importat task after modelig is the estimatio of the parameters ivolved i the model. Now we look ito the problem of estimatio. 3. Estimatio of Parameters from the Desity of the Miimum I reliability ad survival aalysis, the distributio of miimum of a set of compoets is of iterest to may of the researchers. I this sectio we cosidered two methods of estimatig parameters from the desity of the 76 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME miimum. The methods are Priciple of Maximum Etropy estimatio (POME) ad Maximum Likelihood Estimatio (MLE). The probability desity fuctio correspodig to the miimum is give by ( ) x f X ( x), x, (3.) 3.. Priciple of Maximum Etropy Estimatio (POME) I this method, usig the give costraits (iformatio) we maximize the etropy fuctio. To obtai a estimate of the parameters, we proceed as follows. The etropy fuctio for the desity fuctio f ( x) give i (3.) is obtaied by isertig (3.) i the defiitio of etropy fuctio give i (.). That is x H( f ) log f ( x) dx log f ( x) dx From (3.), the costraits are obtaied as ad X (3.) f ( x) dx (3.3) x X log f ( x) dx E log (3.4) The first costrait specifies the total probability ad the secod costrait represets the geometric mea. These costraits are uique ad sufficiet to explai the model. Usig the geeral expressio for the probability desity fuctio give i Kapur (989), we have f( x) exp 0 log x (3.5) where 0 ad are lagragia multipliers. By applyig coditios (3.3) to (3.5), we obtai 0 log (3.6) IASSL ISBN-39-4987 77
Jagathath Krisha K.M. Substitutig (3.6) i (3.5) yields, x f( x) (3.7) Comparig (3.7) with (3.), we get. Takig logarithm of equatio (3.7), log f( x) log log log x (3.8) The egative expected value of equatio (3.8) gives the etropy fuctio H( f ) log log E log X (3.9) Now the lagragia multipliers are obtaied by takig the partial derivative of (3.9) with respect to ad equatig to zero. Thus we have the equatio, X E log Differetiatig (3.6) with respect to ad usig (3.0), we get (3.0) From Tribus (969), we have X 0 E log 0 X V log (3.) (3.) Hece the parametric estimatio of POME cosists of two equatios (3.) ad (3.). Isertig, (3.) ad (3.) becomes, E log X V log X (3.3) (3.4) The estimates of the parameters ad ca be obtaied by solvig the equatios (3.3) ad (3.4) umerically. 78 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME 3.. Maximum Likelihood Estimatio (MLE) To obtai the maximum likelihood estimate for the parameters ad, cosiderig the log-likelihood fuctio. log L log log ( ) log xi Accordig to Arold (983) for a fixed, the likelihood is maximized whe is set equal to Mi( X, X ). Differetiatig with respect to ca the be used to obtai the maximum value of. Thus we have, i ˆ Mi( X, X ) ad ˆ i X i log ˆ 4. Maximum Likelihood Estimatio for the Bivariate Pareto I Model. Estimatio of the parameters,,,, by the method of MLE is a tedious job. By cosiderig ˆ Mi( X, X ) for some fixed ', s the problem of estimatio becomes easier. I this sectio we obtai the estimates for the shape parameters by cosiderig the estimate of obtaied i sectio 3. Cosider a radom sample of size from a populatio havig bivariate desity fuctio give i (.8). Let ad be the sample sizes correspodig to x x ad x x, where X ad X are life times of the compoets. Now the likelihood fuctio of the sample is give by ( ) ( ) ( ) i i i L x x Cosiderig the log-likelihood, i x x i i log L log log( ) log log( ) ( )log log x i log xi i i log xi log x i i i (4.) (4.) IASSL ISBN-39-4987 79
Jagathath Krisha K.M. The estimates are obtaied by solvig the five simultaeous ormal equatios. Ufortuately, the solutios shows covergece problem. Hece we have to estimate the parameters either by cosiderig some of the parameters to be kow or to be estimated from some other methods. The estimate of the scale parameter estimated from the desity of the miimum by the method of POME ad MLE discussed i sectio 3 are used to estimate the rest of the parameters. Thus the estimates of the shape parameters are give by ˆ (4.3) log x log x log ˆ ˆ i i i i ˆ ˆ i i i i log x log x log ˆ log x log x i i i i log x log x i i i i (4.4) (4.5) (4.6) Where ˆ is the estimate of which is obtaied by two methods discussed i sectio 3. 5. Simulatio Study Bivariate Pareto I radom samples are geerated from Freud (96) bivariate expoetial distributio by usig the trasformatio give i the followig theorem. Xi rt Theorem 5.. Let X be a radom vector with e i for, 0. r The X is distributed as i (.8) if ad oly if ( T, T ) is distributed as Freud (96) bivariate expoetial distributio with i, i, i r i,. i r For each populatio, 0000 radom samples of sizes 5, 50 ad 00 were geerated ad the parameters are estimated i two phases as discussed i sectio 3 ad sectio 4. Several combiatios of parameters were cosidered for the simulatio study. From the simulatio study, it is observed that whe 80 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME the parameters are havig higher values, i.e. for greater tha 3 ad rest of the parameters above 5 is givig comparatively large bias or root mea square error values compared to that of parameters with smaller values. But still the estimates obtaied usig POME is performig better tha that of MLE. For illustratio, root mea square error ad variace of the estimates for two differet sets of parameters are obtaied ad are give i Table 6. ad Table 6.. 6. Results ad Discussio The simulatio study has bee coducted for differet set of parameters. Simulatio studies for two sets of parameters are give i Table 6. ad table 6.. Table 6.: Compariso of Estimatio Methods - Simulatio Study Sample Size Parameters Measures POME MLE =5 =50 =00 (.8) (.73) (.04) RMSE 0.034 0.53 Variace 0.007 0.064 RMSE 0.008 0.44 Variace 0.0403 0.946 RMSE 0.887 0.3756 Variace 0.0356 0.4 RMSE.085.085 (.38) Variace.78.78 RMSE 0.3484 0.3484 (0.69) Variace 0.4 0.4 (.8) (.73) (.04) RMSE 0.0743 0.53 Variace 0.0055 0.0508 RMSE 0.35 0.307 Variace 0.09 0.090 RMSE 0.085 0.567 Variace 0.08 0.0659 RMSE 0.639 0.639 (.38) Variace 0.3893 0.3893 RMSE 0.59 0.59 (0.69) Variace 0.050 0.050 (.8) (.73) (.04) RMSE 0.0535 0.006 Variace 0.009 0.040 RMSE 0.067 0.089 Variace 0.0045 0.0436 RMSE 0.0647 0.774 Variace 0.004 0.035 RMSE 0.403 0.403 (.38) Variace 0.66 0.66 RMSE 0.53 0.53 (0.69) Variace 0.035 0.035 IASSL ISBN-39-4987 8
Jagathath Krisha K.M. Table 6.: Compariso of Estimatio Methods - Simulatio Study Sample Size Parameters Measures POME MLE =5 =50 =00 () (4) (3.3) RMSE 0.340 0.3364 Variace 0.0548 0.3 RMSE 0.484 0.399 Variace 0.00 0.088 RMSE 0.40 0.89 Variace 0.096 0.080 RMSE 0.7488 0.7488 (4.6) Variace 0.5608 0.5608 RMSE 0.464 0.464 (.7) Variace 0.0607 0.0607 () (4) (3.3) RMSE 0.0 0.940 Variace 0.049 0.0865 RMSE 0.083 0.80 Variace 0.0069 0.0475 RMSE 0.0784 0.80 Variace 0.006 0.033 RMSE 0.444 0.444 (4.6) Variace 0.97 0.97 RMSE 0.6 0.6 (.7) Variace 0.060 0.060 () (4) (3.3) RMSE 0.0800 0.47 Variace 0.0064 0.06 RMSE 0.0486 0.50 Variace 0.004 0.08 RMSE 0.0466 0.69 Variace 0.00 0.06 RMSE 0.95 0.95 (4.6) Variace 0.0855 0.0855 RMSE 0.04 0.04 (.7) Variace 0.0 0.0 From the Table 6. ad Table 6., it is evidet that irrespective of the values of the parameters ad the sample sizes, the estimates obtaied by method of POME for the miimum probability desity applied i MLE for a bivariate Pareto distributio performs better tha the method of MLE for all the parameter. The RMSE seems to be comparatively smaller for POME tha MLE. 8 ISBN-39-4987 IASSL
Modelig ad Estimatio of a Bivariate Pareto Distributio usig the POME From Table 6. it ca be observed that, for smaller parametric values, as the sample size icreases the estimates obtaied for scale parameters usig POME as a plug i estimator is givig least RMSE, whereas the RMSE for the estimates obtaied from MLE is higher. Simulatio study is also coducted for larger parameter values. From Table 6., we ca observe that the estimates obtaied for all the parameters are givig least RMSE values as the sample size icreases for both the method of estimatio. This shows that whatever be the choice of parameter values, higher the sample values the least is the RMSE. From the simulatio study carried out, we recommed POME method of estimatio for estimatig parameters to MLE. 7. Coclusio I this paper, a bivariate Pareto I distributio has bee modeled usig the priciple of maximum etropy distributio method. The parameters ivolved i the model are estimated usig two phases, the first phase correspodig to the desity of the miimum ad the secod phase by MLE correspodig to bivariate radom variable. A simulatio study has bee coducted to study the performace of these methods. From the simulatio study, it ca be see that POME performs better tha that of MLE. Here i this study, we cosidered two method of estimatio of the parameters amely, POME ad MLE. POME method is applied oly for estimatig ad is beig used as a plug i estimate i MLE. Istead of that if a bivariate coditioal POME method is derived, it may give more efficiet estimates tha those obtaied from the methods discussed i this article. Ackowledgemets The author is thakful to the CSIR-Cetral Leather research Istitute, Govermet of Idia, for the support redered to carry out the work ad ackowledge the CSIR-CLRI Commuicatio No.08. The author is idebted to the ukow referee for the critical reviews ad suggestios which helped i improvig the article. Refereces. Arold B.C. (983). Pareto distributios, Iteratioal Co-operative Publishig House, Fairlad, MD.. Asadi M., Ebrahimi N., Hamedai G.G. ad Soofi E.S. (004). Maximum dyamic etropy models. Joural of Applied Probability, 4, 379-390. DOI:0.39/jap/08999073. IASSL ISBN-39-4987 83
Jagathath Krisha K.M. 3. Asha G. ad Jagathath Krisha K.M. (008). Modelig ad characterizatios of a bivariate Pareto distributio. Joural of Statistical Theory ad Applicatios, 7(4), 435-45. 4. Asha G., Jagathath Krisha K.M. ad Muraleedhara Nair K.R. (009). Characterizatio of bivariate distributios usig residual measure of ucertaity. Joural of Probability ad Statistical Sciece, 7(), 39-5. 5. Cox D.R. (97). Regressio models ad life tables. Joural of Royal Statistical Society. B, 34, 87-0. 6. Ebrahimi N. (996). How to measure ucertaity i residual life time distributios. Sakhya A, 58, 48-56. 7. Ebrahimi N. (000). The maximum etropy method for lifetime distributios. Sakhya A, 6, 36-43. 8. Freud J. (96). A bivariate extesio of expoetial distributio. Joural of America Statistical Associatio, 56, 97-977. DOI:0.080/06459.96.04838 9. Kapur J.N. (989). Maximum etropy models i sciece ad egieerig. Wiley Easter Limited, New Delhi. 0. Shao C.E. (948). A mathematical theory of commuicatio. Bell System Techical Joural, 7, 379 43. DOI:0.00/j.538-7305.948.tb0338.x, DOI:0.00/j.538-7305.948.tb0097.x. Tribus M. (969). Ratioal Descriptios, Decisios ad Desigs. Pergamo, New York, USA. 84 ISBN-39-4987 IASSL