IAENG Iteratoal Joural of Appled Mathematcs, 46:, IJAM_46 5 Lkelhood ad Bayesa Estmato Stress Stregth Model from Geeralzed Expoetal Dstrbuto Cotag Outlers Chupg L, Hubg Hao Abstract Ths paper studes the estmato of R = P(Y < X whe X ad Y are two depedet geeralzed expoetal dstrbutos cotag oe outler. The maxmum lkelhood estmator (MLE ad Bayesa estmator of R are obtaed uder exchageable ad detfable models, respectvely. Mote Carlo smulato s used to compare ad verfy the proposed model ad approaches. The smulato results show that the performace of MLE s more satsfactory tha Bayes estmator. Idex Terms geeralzed expoetal dstrbuto, outler, maxmum lkelhood estmator, bayes estmator. I I. INTRODUCTION N relablty cotexts, fereces about stress stregth model R=P(Y<X s a terest subect. For example, mechacal relablty of a system, f X s the stregth of a compoet whch s subect to the stress Y, the we kow that R s a measure of system performace. The system fals, f at ay tme the appled stress Y s greater tha ts stregth X. The problem of estmatg R=P(Y<X, X ad Y belog to a certa famly of probablty dstrbutos, has bee wdely studed the lterature, such as burr dstrbutos (Mokhls, 5, Webull dstrbuto (Kudu et al., 6, Gompertz dstrbuto (Saraçoglu et al., 7, expoetal dstrbuto (Jag et al., 8, the geeralzed gamma dstrbuto (Pham et al., 995 ad the geeralzed pareto dstrbuto (Rezae et al.,, et al.. Recetly, the stress stregth model lterature, Kudu ad Gupta (5 ad Raqab et al. (8 has cosdered the ordary samples from the geeralzed expoetal dstrbuto (GED, ad Baklz (8 has cosdered the record data from the GED. All these papers assume that the sample observatos are depedetly ad detcally dstrbuted. I fact, the sample data may cota outlers may cases, because outlers are Mauscrpt receved August, 8; revsed November 9, 5. Ths work was supported by the Humaty ad Socal Scece Youth Foudato of Mstry of Educato of Cha (No. 5YJCZH55, the Natoal Natural Scece Foudato of Cha (No.545, the Fudametal Research Fuds for the Cetral Uverstes ad the Graduate Trag Iovatve Proects Foudato of Jagsu Provce, Cha uder Grat No. CXLX_8, Youth Foudato of Hube Educatoal Commsso (No. Q63. C. P. L s wth the Departmet of Mathematcs, Hube Egeerg Uversty, Hube, 43, Cha. e-mal: lchupg35@63.com. H. B. Hao s the correspodg author wth the Departmet of Mathematcs, Hube Egeerg Uversty, Hube, 43, Cha. e-mal: haohubg979@63.com. usually caused by measuremet error or erroeous procedures (see Barett et al., 984. Km ad Chug (6 ad Jeevaad ad Nar (994 have cosdered the outler from the burr-x dstrbuto ad expoetal dstrbuto, but they oly cosdered the Bayes estmato of R. It s well kow that the pror fucto plays a mportat role Bayes method, thus the other estmato method should be cosdered. I ths paper, we focus o estmato of R=P(Y<X, X ad Y follow the GED(θ ad GED(β. The maxmum lkelhood estmator (MLE ad the Bayes estmato of R are obtaed from the samples cotag outlers, whch has ot bee studed before. The rest of the paper s orgazed as follows: I the ext Secto, the geeralzed expoetal dstrbuto ad the stress stregth model are troduced. I Secto 3, we troduce the ot dstrbuto of (X, X,, X wth oe outler. I Sectos 4 ad 5, the MLE ad the Bayes estmator of R uder exchageable ad detfable model are obtaed. I Secto 6, we preset some umercal results, ad compare the Bas ad the mea squares errors (MSE. Secto 7 cocludes the paper. II. GENERALIZED EXPONENTIAL DISTRIBUTION AND STRESS STRENGTH MODEL The geeralzed expoetal dstrbuto (GED has frstly bee troduced by Gupta ad Kudu (999. Due to the coveet structure of dstrbuto fucto, the GED ca be used to aalyze varous lfetme data. I recet years, may scholars have studed about ths dstrbuto, such as Raqab ad Ahsaullah(, Zheg(, Gupta ad Kudu (3. The probablty desty fucto wth oe parameter GED(θ s gve by x x f ( x e ( e, x, ( Ad the correspodg cumulatve dstrbuto fucto s x F ( x ( e, x, ( Suppose that X~GED(θ ad Y~GED(β are two depedet radom varables, the the relablty of stress stregth model ca be obtaed as R P( Y X P( Y X X x P( X x dx (3 (Advace ole publcato: 4 May 6
IAENG Iteratoal Joural of Appled Mathematcs, 46:, IJAM_46 5 III. JOINT DISTRIBUTION WITH OUTLIER Let X~GED(θ,X, X,, X be a radom sample from X, ad (- of them have the same probablty desty fucto as follow x x f ( x e ( e, x, (4 the remag oe s probablty desty fucto s gve as x x b g( x b e ( e, x, b, (5 Therefore, the ot probablty desty fucto of (X, X,, X ca be obtaed as (see Dxt ad Nooghab, (! f ( x, x,, x f ( x g( x! f ( x f(x ad g(x are gve (4 ad (5. The f ( x, x,, x b exp( x [ exp( x ] ( x ( x b exp( x exp [ T ( x b ( b T( x ] T ( x (6 T ( x log[ exp( x ], T ( x log[ exp( x ]. Smlarly, let Y~ GED(β ad Y, Y,, Y be a radom sample from Y, ad (- of them have the same probablty desty fucto as y y f ( y e ( e, y, the remag oe s probablty desty fucto s gve as y y c g( y c e ( e, y, c, The, we ca get the ot dstrbuto of Y, Y,, Y as f ( y, y,, y c exp( y exp [ T ( y ( c T( y ] T ( y (7 T ( y log[ exp( y ], T ( y log[ exp( y ] IV. MLE AND BAYES ESTIMATION OF R UNDER EXCHANGEABLE MODEL The exchageable model assumes that outlers are ot detfable ad ay observato sample s as lkely to be dscordat as ay other. I ths secto, we wll obta the MLE ad the Bayes estmato of R uder ths codto. A. The MLE of R Suppose that X=(X, X,, X follows GED(θ ad cotas oe outler, accordg to the expressos (6, the log lkelhood fucto s gve by l f l l l b x l{ exp[ ( T ( x ( b T ( x T ( x]} Dfferetatg lf wth respect to θ ad b, respectvely, we ca get l f exp[ ( T ( x ( b T( x T ( x][ T ( x ( b T( x ] (8 exp[ ( T( x ( b T( x T( x] l f b b exp[ ( T ( x ( b T( x T ( x][ T( x ] (9 exp[ ( T ( x ( b T( x T ( x] Equatg (8 ad (9 to zero, we have exp[ ( T ( x ( b T( x T ( x] ( exp[ ( T ( x ( b T( x T ( x][ T ( x ( b T( x ] exp[ ( T ( x ( b T( x T ( x] b exp[ ( T ( x ( b T( x T ( x][ T( x ] I the same way, we ca obta the MLE of β ad c as exp[ ( T ( y ( c T( y T ( y] exp[ ( T( y ( c T( y T( y][ T( y ( c T( y ] ( ( (Advace ole publcato: 4 May 6
IAENG Iteratoal Joural of Appled Mathematcs, 46:, IJAM_46 5 exp[ ( T ( y ( c T( y T ( y] c exp[ ( T ( y ( c T( y T ( y][ T( y ] (3 The, t s easy to obta the MLE of R uder exchageable model as R (4 B. The Bayes estmato of R I ths subsecto, we cosder the Bayes estmato of R uder the squared error loss fucto. Let X=(X, X,, X ad Y=(Y, Y,, Y be two depedet radom samples from GED wth parameters (θ, b ad (β, c, respectvely. The, the lkelhood fuctos are proporto to l( X, b b exp[ ( T ( x ( b T( x T ( x] ad l( Y, c c exp[ ( T ( y ( c T ( y T ( y] (5 I Bayesa framework, we assume that the parameters θ ad b are take to be depedetly dstrbuted to a gamma dstrbuto ad a o formatve pror dstrbuto. The the ot dstrbuto of (θ, b s proporto to p (, b exp( u I the smlar maer, the ot dstrbuto of (β, c s proporto to q (, c exp( v (6 p, q, u ad v are kow. Based o the above assumpto, we ca derve the ot probablty desty fucto of X ad Y as follows l( X, Y,, b, c l( X, b l( Y, c (, b (, c the posteror desty fucto of (θ, β, b, c s obtaed l( X, Y, b,, c (,, b, c X, Y l( X, Y, b,, c dd dbdc The (, X, Y (,, b, c X, Y dbdc Let r=θ/(θ + β ad ρ=θ + β, ρ>, <r<. We ca get ( r, X, Y ( r,( r, b, c X, Y dbdc Therefore, we obta the margal posteror desty fucto of R as ( r X, Y ( r, X, Y d m3 s3 ( ms C ( T( x T( y r ( r ( Ar [( Ar D ( ms4( Ar ( rt( x ( r T( y Q ( ms3( ms4( T( x T( y r( r] Q m p, s q, Q T ( x u, A, Q Q T ( y v ( m ( s, B( m, s, ( m s 4 C ( T( x T( y [( m T( x Q ] m Q [( s T( y Q ] B( m, s (7 Q Hece, uder the squared error loss fucto, the Bayes estmator of R s R E ( r date r ( r date dr C ( T( x T( y { Q H (,, ( ms4 T( x Q H(,, ( ms4 T( y Q H(,, ( ms3( ms 4 T( x T( y H(,,} (8 H( a, b, c F ( ms4 a, mb, msa3, A, B( mb, s c ( mb ( sc B( mb, sc, ( m s b c 3 b cb ( c t ( t F, ( a, b; c; z dt ( ( a b c b. ( tz V. MLE AND BAYES ESTIMATION OF R UNDER IDENTIFIABLE MODEL The detfable model assumes that outlers are detfable. We treat the largest observato the sample as a outler because the largest order statstcs the sample has the largest posteror probablty (see Kale ad Kale, 99. We wll obta the MLE ad Bayes estmatos of R uder ths model. A. The MLE of R Suppose that X=(X, X,, X follows GED(θ ad cotas oe outler, accordg to Kale ad Kale (99, we kow that the largest order statstcs the sample s the outler whe b>, ad we treat X( as outler. The we ca get the lkelhood fucto ad the log lkelhood fucto as L(, b X C f ( x f ( x ( ( ad C b exp( x exp[ ( T ( x ( b T( x T ( x] l L l C l l b x ( T ( x ( b T ( x T ( x ( (Advace ole publcato: 4 May 6
IAENG Iteratoal Joural of Appled Mathematcs, 46:, IJAM_46 5 C s a costat, T ( x log[ exp( x ], T ( x log[ exp( x ]. ( ( Dfferetatg ll wth respect to θ ad b, respectvely. We ca obta the followg solutos l L [ T ( x ( b T ( x( ] (9 l L [ T( x( ] b b ( From the above two equatos, we derve ( T ( x T ( x( Smlarly, suppose that Y=(Y,Y,,Y follows GED(β ad cotas oe outler, the log lkelhood fucto from the sample s gve by l L l C l l c y ( T ( y ( c T ( y T ( y T ( y log[ exp( y ], T ( y log[ exp( y ]. ( ( ( Smlar to (9 ad (, we ca get T ( y T ( y ( ( Therefore, from ( ad (, the MLE of R uder the exchageable model s R (3 B. The Bayes estmato of R I ths subsecto, we cosder the Bayes estmato of R uder the squared error loss fucto. Let X=(X, X,, X ad Y=(Y, Y,, Y be the two depedet radom samples from GED wth parameters (θ, b ad (β, c, respectvely. The, the lkelhood fuctos are proporto to ( l ( X, b b exp[ ( T ( x ( b T( x T ( x] ad ( l ( Y, c c exp[ ( T ( y ( c T( y T ( y] Assumg the ot pror dstrbuto of (θ, b ad (β, c are the same as gve (5, we ca get l( X, Y,, b, c l ( X, b l ( Y, c (, b (, c Usg smlar way, we ca get the margal posteror dstrbuto of R as follow ( r X, Y ( r, X, Y d 3 ( ( m3 s3 ( ms C ( T( x T( y r ( r ( Ar [( Ar D ( ms4( Ar ( rt( x ( r T( y Q ( ( ( ( ( m s 3( m s 4( T ( x T ( y r( r] m Q 3 [( ( ][( ( ] (, Q C m T x Q s T y Q B m s Hece, uder the mea squared error loss fucto, the Bayes estmator of R s R E ( r date r ( r date dr 3 ( C { Q H(,, ( m s 4 T( x Q H(,, ( m s 4 T ( y Q H (,, ( ( m s 3( m s 4 T( x T( y H(,,} (4 ( ( VI. MONTE CARLO SIMULATION STUDY I ths secto, Mote Carlo smulato s used to compare performace of the proposed models ad methods. Wthout loss of geeralty, let θ = 9, β = ad b = c =. We cosder sample sze to be (, m = (5, 5, (,, (5, 5, (,, (5, 5, (3, 3. For a gve geerated sample, compute the MLE ad Bayes estmators of R ad replcate the process 3 tmes. For the dfferet pror parameters: p = q = u= v =, p = q = u = v =, ad p = q = u = v =, we obta the Bayes estmators for R as Bayes-, Bayes-, ad Bayes-3, respectvely. The Bas ad MSE of the MLE estmator ad Bayes estmator are computed by 3 Bas( R R R 3, ad MSE( R ( R R 3 3 (5 Table I ad Table II show the smulato results for the Bas ad MSE of MLE ad Bayes estmators for R uder dfferet sample szes ad dfferet pror parameter values. From the smulato results, t s qute clear that the performaces of both the MLE ad the Bayes estmators uder the two models are qute satsfactory eve for very small sample szes. It s oted that the Bas ad MSE of the MLE are smaller tha the Bayes estmators uder the two models whe the sample cotas outlers. I addto, t s observed that the Bas ad MSE decrease for all the estmators uder the two models whe the sample sze creases. Meawhle, the Bases ad MSE of the Bayes estmators decrease whe the parameters of pror dstrbutos, p, q, u ad v crease. Moreover, from the smulato results, t s observed that the Bases ad MSE of the MLE, Bayes-, Bayes- ad Bayes-3 uder the detfable model are smaller tha the correspodg values uder the exchageable model. Based o all those kowledge, whe the sample cotas oe outler, we recommed use the MLE wheever the model s the exchageable or detfable. (Advace ole publcato: 4 May 6
IAENG Iteratoal Joural of Appled Mathematcs, 46:, IJAM_46 5 VII. CONCLUSION Ths paper deals wth the estmato of R=P(Y<X whe X ad Y are two depedet geeralzed expoetal dstrbuted radom varables. We assume that the sample from each populato cotas oe outler. The MLE ad Bayes estmator of R are obtaed uder the exchageable ad detfable models. A smulato study s preseted to TABLE I BIAS AND MSE UNDER THE EXCHANGEABLE MODEL (, m (5,5 (, (5, 5 (, (5, 5 (3, 3 MLE bas.43..7.4.5.9 mse.39..7.5.4.3 Bsyes- bas.79.636.573.58.487.449 mse.8.63.5.4.37.3 Bsyes- bas.6.57.458.43.396.368 mse.53.4.33.8.5. Bsyes-3 bas.56.454.49.4.379.355 mse.4.3.7.5.3. TABLE II BIAS AND MSE UNDER THE IDENTIFIABLE MODEL (, m (5,5 (, (5, 5 (, (5, 5 (3, 3 MLE bas.333.8.58.36..3 mse..7.5.4.3. Bsyes- bas.584.45.364.36.87.54 mse.5.8..5.3. Bsyes- bas.539.398.39.9.53.9 mse.43.43.3.8.33. Bsyes-3 bas.477.43.3.8.33. mse.35.5.6..9.6 compare the two estmato methods uder the dfferet model. Based o the smulato results, the performaces of the MLE are more satsfactory tha Bayes estmator eve for very small sample szes. Whe there s more tha oe outler, the problem becomes qute more complcated. The correspodg estmato methods eed to be explored the future. It may be metoed that although t has bee assumed that the samples are from geeralzed expoetal dstrbutos cotag oe outler, t may be exteded to some other dstrbutos also, for example, the Webull or gamma dstrbuto cotag outlers. Work s progress, ad t wll be reported later. [8] D. Kudu, R.D. Gupta, Estmato of P(Y<X for geeralzed expoetal dstrbuto, Metrka, vol.6, o.3, pp.9-38, 5. [9] M.Z. Raqab, M.T. Mad, D. Kudu, Estmato of P(Y<X for the 3-parameter geeralzed expoetal dstrbuto, Commu. Statst.Theor. Meth., vol.37, o.8, pp. 854-864, 8. [] A. Baklz, Lkelhood ad Bayesa estmato of P(X<Y usg lower record values from the geeralzed expoetal dstrbuto, Comput. Statst. Data Aal., vol.5, o.7, pp.3468-3473, 8. [] R. D. Gupta, D.Kudu, Geeralzed expoetal dstrbuto, Austral. N. Z. Statst., vol. 4, o., pp.73-88, 999. [] M. Z. Raqab, M. Ahsaullah, Estmato of locato ad scale parameters of geeralzed expoetal dstrbuto based o order statstcs, J. Statst. Computat. Smul., vol.69, o., pp.9-4,. [3] G. Zheg, O the Fsher formato matrx type-ii cesored data from the expoetated expoetal famly, Bometrcal J., vol.44, pp.353-357,. [4] R. D. Gupta, D. Kudu, Closeess of Gamma ad geeralzed expoetal dstrbuto, Commu. Statst. Theor. Meth., vol.3, o.4, pp.75-7, 3. [5] V. Barett, T. Lews, Outlers Statstcal Data, New York : Wley, 984. [6] C. Km, Y. Chug. Bayesa estmato of P(Y<X from Burr type X model cotag spurous observatos, Statst. Pap., vol.47, o.4, pp. 643-65, 6. [7] [7] E. S. Jeevaad, N. U. Nar, Estmatg P(Y<X from expoetal samples cotag spurous observatos, Commu. Statst. Theor. Meth., vol.3, o.9, pp.69-64, 994. [8] U.J. Dxt, M. J. Nooghab, Effcet estmato the Pareto dstrbuto wth the presece of outlers, Statst. Methodology, vol.8, o.8, pp.34-355,. [9] V.B. Kale, B.K. Kale, Outlers expoetal sample A Bayesa approach, Guarat Statstcal Revew, Khatr memoral, 99. [] M. G. Badar, A. M. Prest, Statstcal aspects of fber ad budle stregth hybrdcompostes, I: Hayash, T., Kawata, K., Umekawa, S., eds. Progress Scece ad Egeerg Compostes. Tokyo: ICCM-IV, pp.9-36, 98. [] H. Assareh ad K. Megerse, Bayesa estmato of the tme of a decrease rsk-adusted survval tme cotrol charts, IAENG Iteratoal Joural of Appled Mathematcs, vol.4, o.4, pp. 36-366,. [] J. La, L. Zhag, C.F. Duffeld, ad L. Aye, Egeerg relablty aalyss rsk maagemet framework: developmet ad applcato frastructure proect, IAENG Iteratoal Joural of Appled Mathematcs, vol.43, o.4, pp. 4-49, 3. REFERENCES [] A. Brol, Relablty Egeerg Theory ad Practce, Ffth edto. New York: Sprger-Verlag Berl Hedelberg, 7. [] N.A. Mokhls, Relablty of a stress stregth model wth Burr type III dstrbutos, Commu. Statst. Theor. Meth., vol.34, o.7, pp.643-657, 5. [3] D. Kudu, R.D. Gupta, Estmato of P(Y<X for Webull dstrbutos, IEEE Tras. Relab., vol.55, o., pp.7-8, 6. [4] B. Saraçoglu, M.F. Kaya, Maxmum lkelhood estmato ad cofdece tervals of system relablty for Gompertz dstrbuto stress stregth models, Selçuk J. Appl. Meth., vol.8, o., pp.5-36, 7. [5] L. Jag, A.C.M. Wog, A ote o ferece for P(X<Y for rght trucated expoetally dstrbuted data, Statst. Pap., vol.49, o.4, pp.637-65, 8. [6] T. Pham, J. Almhaa, The geeralzed gamma dstrbuto: Its hazard rate ad stress stregth model, IEEE Tras. Relab., vol.44, o.3, pp.39-397, 995. [7] S. Rezae, R. Tahmasb, M.Mahmood, Estmato of P(Y<X for geeralzed Pareto dstrbutos, J. Statst. Pla. Ifer., vol.4, o., pp.48-494,. (Advace ole publcato: 4 May 6