CHAPTER: 3 INVERSE EXPONENTIAL DISTRIBUTION: DIFFERENT METHOD OF ESTIMATIONS

CHAPTER: 3 INVERSE EXPONENTIAL DISTRIBUTION: DIFFERENT METHOD OF ESTIMATIONS 3. INTRODUCTION Th Ivrs Expoal dsrbuo was roducd by Kllr ad Kamah (98) ad has b sudd ad dscussd as a lfm modl. If a radom varabl X has a oal dsrbuo, h varabl = wll hav a Ivrs oal (IE) dsrbuo. A radom varabl X s sad o hav a Ivrs Expoal dsrbuo wh paramr f s pdf ad cdf ar gv rspcvly by; f ( x ) =, 0, > 0 x (3..) x x F ( x ) = ; x 0, > 0 (3..) x L, Rlably ad hazard fuco of IE dsrbuo ar gv by, R ( ; ) = F ( ) = ; 0, > 0 (3..3) Ad, Hazard ra, f () () h =, quvally w ca wr, R () ( ; ) h = ; 0, > 0 (3..4) Th oal dsrbuo s mos wdly usd dsrbuo for lfm daa aalyss, bcaus of s smplcy ad mahmacal fasbly. Howvr, ral world, w rarly com across h grg sysms whch hav cosa hazard ra hroughou hr lf durao. Thrfor, sms praccal o assum hazard ra as a fuco of m, whch ld o h dvlopm of alrav modl for lfm daa aalyss. A umbr of 4

lfm modls (lk Wbull, gamma, gralzd oal c.) hav b proposd o modl lf m daa ha hav mooocally crasg or dcrasg hazard ra fuco, hough, o-mooocy of h hazard ra has also b obsrvd may suao. For xampl, h cours of h sudy of moraly assocad wh som of h dsass, h hazard ra ally crass wh m ad rachs afr a pak afr som f prod of ms ad h dcl slowly, s, Sgh, al. (0). I vw of hs, Ivrd oal dsrbuo (IED) has b dscussd as a lf m modl by L, al. (989) dal. Thy obad Maxmum Lklhood smas (MLE), cofdc lms ad uformly mmum varac ubasd smaors for h paramrs ad rlably fuco of IED wh compl sampl. Dy (007) smad h paramr of IED by assumg h paramr volvd h modl as a radom varabl (r.v.). Prakash (009) dscussd h proprs of h Bays smaor, Shrkag smaor ad mmax smaor of h paramr. H also prsd h moms of h lowr rcord valu ad h smao of h paramr, basd o a srs of obsrvd rcord valus by h maxmum lklhood ad mom mhods. Rcly, Sgh, al. (0) proposd Bays smaors of h paramr ad rlably fuco for h sam udr h gral ropy loss fuco for compl, Typ-I ad Typ-II csord sampls. Th rmag scos go as follows. I Sco 3., w brfly dscuss h MLE's ad hr mplmaos. I Scos 3.3 o 3.6 w dscuss ohr mhods. Smulao rsuls ad dscussos ar provdd Sco 3.7. Comparsos of dffr mhods usg graph, ral lf applcao ad coclusos ar provdd scos 3.8, 3.9 ad 3.0. 3. MAXIMUM LIKELIHOOD ESTIMATORS I hs sco h maxmum lklhood smaors of IED () ar cosdrd, whr s ukow. If x, x... x s a radom sampl from IED () h h lklhood fuco, L(), s gv by, L = ( ; x, x,... x ) = f ( x, x,... x θ) = f ( x θ) θ (3..) 5

log L = L ( ) = = x = x = (3..) log L = L( ) = l( ) l x (3..3) x O dffrag (3..3) wh rspc o ad quag o zro, Th ormal quao bcoms: logl = = = 0 x (3..4) Afr solvg (3..4), w obad h sma of as, ˆ = (3..5) = x Dffra (3..4) wh rspc o, w hav, log L = Varac of ˆ ca b obad from, V ( ˆ ) = = (3..6) log L E Thrfor, V ( ˆ ) aas h Cramr-Rao lowr boud (= MLE ) O ca g Maxmum Lklhood Esma by usg xsv smulao chqu ad ad roo ma squar rror of Maxmum Lklhood Esma, usg (3..5) ad (3..6). 3.3 ESTIMATION OF RELIABILITY I sascs, rlably s a vry mpora cocp ha drms h prcso of masurms. Sascal rlably drms whhr or o h rm s rproducbl. Hr, R ( ) = F ( ) = (3.3.) 6

Dffra (3.3.) wh rspc o, w hav R () = Varac of sma of rlably fuco, R() ˆ ca b obad from, ( R () ˆ ) = R ( ˆ ) V ( ) V ( ˆ ) Now, Subsug h valus of R () ˆ hav, ad V ( ˆ ) (3.3.) (3.3.3), from (3..6) ad (3.3.) (3.3.3), w 4 ( () ˆ V R ) = 4 (3.3.4) O ca g sma of rlably ad roo ma squar rror of rlably by usg xsv smulao chqu, usg (3.3.) ad (3.3.4). 3.4 ESTIMATION OF HAZARD RATE Th hazard ra fuco h(), also kow as h forc of moraly or h falur ra, s dfd as h rao of h dsy fuco ad h survval fuco. Tha s, () () f Hazard ra, h () = (3.4.) R O pug h valu of () f ad R (), w hav 7

8 = h ) ( (3.4.) Dffra (3.4.) wh rspc o, w g () ( ) 3 0 + = h (3.4.3) () 3 3 4. + = h (3.4.4) Fally w g, () h as () 4 = h (3.4.5) Varac of sma of hazard ra, () h ˆ ca b obad from, () ( ) ( ) ( ) ( ) ˆ ˆ ' ˆ V h h V = (3.4.6) Now, Subsug h valus of () h ˆ ' ad ( ) ˆ V, from (3.4.5) ad (3..6) rspcvly (3.4.6), () ( ) h V s obad as, () ( ) h V 4 8. = (3.4.7)

O ca g sma of hazard ra ad roo ma squar rror of hazard ra by usg xsv smulao chqu usg (3.4.) ad (3.4.7). 3.5 ESTIMATORS BASED ON PERCENTILES If h daa coms from a dsrbuo fuco whch has a closd form, h s qu aural o sma h ukow paramrs by fg a sragh l o h horcal pos obad from h dsrbuo fuco ad h sampl prcl pos. Ths mhod was orgally lord by Kao (958, 959) ad has b usd qu succssfully for Wbull dsrbuo ad for h gralzd oal dsrbuo [s, Murhy al. (004) ad Gupa ad Kudu (00)]. I cas of a IE dsrbuo, s possbl o us h sam cocp o oba h smaor of basd o h prcls, bcaus of h srucur of s dsrbuo fuco. Now, h dsrbuo fuco of IE dsrbuo s gv as, F ( x ; ) =, x 0, > 0 x (3.5.) Takg Log of dsrbuo fuco of IE dsrbuo, w hav l F ( x ; ) =, x 0, > 0 x (3.5.) L X () dos h h ordr sasc,.., X () < X () <... < X (). If p dos som sma of F(x () ; ), h h sma of ca b obad by mmzg, h fuco l ( ) p + = x ( ) wh rspc o. (3.5.3) Dffrag (3.5.3) wh rspc o ad quag o zro, w hav o- lar quao for x () ad s gv by, l x ( p ) + = 0 = x ( ) ( ) (3.5.4) 9

W oc ha (3.5.4) s a o-lar quao of X (). So, s possbl o us som olar rgrsso chqus o sma. W call hs smaors as prcl smaors (PCE's). Svral smaors of p ca b usd hr [s, Murhy al. (004)]. I hs sco, w maly cosdr Th prcl smaor of, say, followg o-lar quao; p =, whch s h cd valu of F(X () ; ). + ˆ PCE ca b obad as h soluo of h l( p ) + = 0 = x ( ) = ( x ( ) ) (3.5.5) I cas of IE dsrbuo, f h shap paramr s ukow, h h PCE of, say, ˆ PCE of IED ca b obad by solvg (3.5.5). O ca us Nwo-Raphso mhod o solv o-lar quao (3.5.5). Now, o ca fd prcl sma ad roo ma squar rror of paramr by usg xsv smulao chqu. 3.6 LEAST SQUARES AND WEIGHTED LEAST SQUARES ESTIMATORS I hs sco, w provd h rgrsso basd mhod smaors of h ukow paramr, whch was orgally suggsd by Swa, al. (988) o sma h paramrs of Ba dsrbuos. Th mhod ca b dscrbd as follows: Suppos Y,Y,...,Y s a radom sampl of sz from a dsrbuo fuco G(. ) ad Y () < Y () <... < Y () dos h ordr sascs of h obsrvd sampl. I s wll kow ha, [ G ( y )] [ ( )] ( + ) E () = ad V G y () =, =,, 3.,. + ( + ) ( + ) [S, Johso al. (995)]. Usg h caos ad h varacs, wo procdurs of h las squars mhods ca b usd. 30

METHOD-3.6.: LEAST SQUARES ESTIMATORS Oba h smaors by mmzg, [ G ( y )] () = + (3.6.) wh rspc o h ukow paramr. Thrfor, cas of IE dsrbuo, h las squars smaor of, say, ˆ LSE ca b obad by mmzg, = x ( ) + wh rspc o. (3.6.) Dffrag (3.6.) wh rspc o ad quag o zro, w hav o- lar quao for x () ad s gv by, x () x () + x () x (). = = = 0 + (3.6.3) I cas of IE dsrbuo, f h shap paramr s ukow, h h LSE of, say, ˆ LSE of IED ca b obad by mmzg (3.6.). O ca us Nwo-Raphso mhod o solv o-lar quao (3.6.3). Now, o ca fd las squar sma ad roo ma squar rror sma of paramr by usg xsv smulao chqu. METHOD-3.6.: WEIGHTED LEAST SQUARES ESTIMATORS Th wghd las squars smaors ca b obad by mmzg, w [ G ( y )] () = + wh rspc o h ukow paramr, whr [ G( y )] () ( + ) ( + ) ( - + ) w = = ; = v,,3...,. (3.6.4) 3

Thrfor, cas of IE dsrbuo wghd las squar smaor of, say, ˆWLSE, b obad by mmzg ca w = x + () (3.6.5) wh rspc o, Dffrag (3.6.5) wh rspc o ad quag o zro, w hav o- lar quao for x () ad s gv by, x () x() + w w. = 0 = x() = x() + (3.6.6) I cas of IE dsrbuo, f h shap paramr s ukow, h h WLSE of, say, ˆ WLSE of IED ca b obad by mmzg (3.6.5). O ca us Nwo-Raphso mhod o solv o-lar quao (3.6.6). Now, o ca fd Wghd Las squar sma ad roo ma squar rror of paramr by usg xsv smulao chqu. 3.7 NUMERICAL EXAMPLES AND DISCUSSIONS I hs sco w prs rsuls of som umrcal rms o compar h prformac of h dffr smaors proposd h 3. o 3.6 scos. W prform xsv Mo Carlo smulaos o compar h prformac of h dffr smaors, maly wh rspc o hr bass ad roo ma squard rrors (RMSE's) for dffr sampl szs ad for dffr paramr valus. No ha, w cosdr = 0, 0, 30 ad =,, 3 all cass. W compu h bass ad RMSE's of smaors ovr 000 rplcaos. W cosdr h smao of, Wh s ukow. Th h sma ad roo ma squar rror of MLE, Rlably ad hazard ra ca b obad from (3..5), (3..6), (3.3.), (3.3.4), (3.4.) ad (3.4.7) rspcvly. Smlarly, h sma ad roo ma squar rror of prcls, las squars ad wghd las squars smaors of ca b 3

obad by mmzg (3.5.3), (3.6.) ad (3.6.5) or solvg o-lar quaos (3.5.4), (3.6.3) ad (3.6.6) rspcvly, wh rspc o. Th rsuls ar rpord Tabl: 3.7., 3.7., 3.7.3. I s obsrvd from Tabl: 3.7., 3.7., 3.7.3 ha mos of h smaors usually ovr sma, xcp PCE, whch udr sma all h m. Th RMSE's ad Bass ar also qu clos o h MLE ad LSE. I h cox of compuaoal complxs, MLE, Rlably ad hazard ra s ass o compu. I dos o volv ay o-lar quao, whr as h PCE, LSE ad WLSE volv o-lar quaos ad hy d o b calculad by som rav procsss. For, Ivrs oal dsrbuo, F ( x ) =, x hrfor, x =, whch s usful smulao procdur. log F ( x ) Tabl: 3.7. Smulad valus of bass ad RMSE s of smaors of, wh N=000, =, = 5. METHODS =0 MLE PCE LSE WLSE ˆ.094 0.583 0.976 6.5 Bas ( ˆ ) 0.094-0.46-0.03 5.54 RMSE ( ˆ ) 0.3658 0.595 0.9699 0.9588 R() ˆ 0.943 0.090 0.66 0.780 Bas R() ˆ 0.03-0.07-0.05 0.598 RMSE R() ˆ 0.0560 0.090 0.07 0.645 h() ˆ 0.790 0.96 0.86 0.0680 Bas h() ˆ -0.00 0.05 0.00-0. RMSE h() ˆ 0.0066 0.0099 0.048 0.68 33

= 0 MLE PCE LSE WLSE ˆ.044 0.6.09.8 Bas ( ˆ ) 0.044-0.387 0.09.84 RMSE ( ˆ ) 0.394 0.545.3589 6.75 R() ˆ 0.876 0.56 0.67 0.733 Bas R() ˆ 0.006-0.065-0.04 0.55 RMSE R() ˆ 0.038 0.094 0.7 0.598 h() ˆ 0.798 0.00 0.8 0.089 Bas h() ˆ -0.0008 0.039 0.0006-0.098 RMSE h() ˆ 0.0044 0.003 0.084 0.34 = 30 MLE PCE LSE WLSE ˆ.07 0.77.093 6.30 Bas ( ˆ ) 0.07-0.8 0.093 5.30 RMSE ( ˆ ) 0.908 0.5943.7005 9.79 R() ˆ 0.85 0.356 0.693 0.80 Bas R() ˆ 0.003-0.045-0.0 0.639 RMSE R() ˆ 0.03067 0.09 0.955 0.6795 h() ˆ 0.80 0.69 0.806 0.0583 Bas h() ˆ -0.0005 0.08 0-0. RMSE h() ˆ 0.0035 0.03 0.05 0.340 34

Tabl: 3.7. Smulad valus of bass ad RMSE's of smaors of, wh N =000, =, = 5. METHODS = 0 MLE PCE LSE WLSE ˆ.88.3.655 3.84 Bas ( ˆ ) 0.88-0.876-0.345.845 RMSE ( ˆ ) 0.73.09.090 6.988 R() ˆ 0.347 0.97 0.69 0.736 Bas R() ˆ 0.08-0.3-0.060 0.406 RMSE R() ˆ 0.0873 0.54 0.3 0.496 h() ˆ 0.59 0.78 0.69 0.075 Bas h() ˆ -0.00 0.05-0.006-0.087 RMSE h() ˆ 0.0 0.08 0.07 0.08 = 0 MLE PCE LSE WLSE ˆ.089.056.535.8 Bas ( ˆ ) 0.089-0.943-0.464 0.8 RMSE ( ˆ ) 0.478.007 0.756 4.997 R() ˆ 0.338 0.88 0.59 0.78 Bas R() ˆ 0.009-0.4-0.069 0.399 RMSE R() ˆ 0.060 0.5 0.04 0.490 35

h() ˆ 0.6 0.79 0.7 0.078 Bas h() ˆ -0.0008 0.07 0.008-0.083 RMSE h() ˆ 0.008 0.08 0.03 0.04 = 30 MLE PCE LSE WLSE ˆ.054.397.49 9.07 Bas ( ˆ ) 0.054-0.60-0.508 7.074 RMSE ( ˆ ) 0.386.0 0.70 0.98 R() ˆ 0.335 0.86 0.54 0.837 Bas R() ˆ 0.005-0.43-0.074 0.508 RMSE R() ˆ 0.049 0.5 0.098 0.453 h() ˆ 0.6 0.80 0.7 0.050 Bas h() ˆ -0.0008 0.08 0.009-0. RMSE h() ˆ 0.006 0.08 0.0 0. 36

Tabl: 3.7.3 Smulad valus of bass ad RMSE's of smaor of wh, N = 000, =3, = 5. METHODS =0 MLE PCE LSE WLSE ˆ 3.8.688.455.08 Bas ( ˆ ) 0.8 -.3-0.544 8.088 RMSE ( ˆ ).0975.53.58 4.53 R() ˆ 0.469 0.78 0.367 0.668 Bas R() ˆ 0.08-0.7-0.083 0.7 RMSE R() ˆ 0.03 0.0 0.6 0.334 h() ˆ 0.4 0.68 0.56 0.094 Bas h() ˆ -0.003 0.03 0.00-0.05 RMSE h() ˆ 0.06 0.06 0.03 0.079 = 0 MLE PCE LSE WLSE ˆ 3.33.585.303.90 Bas ( ˆ ) 0.33 -.84-0.696 9.900 RMSE ( ˆ ) 0.78.5.34 5.809 R() ˆ 0.460 0.67 0.360 0.78 Bas R() ˆ 0.009-0.83-0.090 0.77 RMSE R() ˆ 0.073 0.98 0.3 0.37 37

h() ˆ 0.44 0.70 0.57 0.080 Bas h() ˆ -0.00 0.04 0.0-0.065 RMSE h() ˆ 0.0 0.05 0.08 0.088 = 30 MLE PCE LSE WLSE ˆ 3.08.553.5 9.07 Bas ( ˆ ) 0.08 -.446-0.774 6.07 RMSE ( ˆ ) 0.574.55.057 0.98 R() ˆ 0.456 0.64 0.353 0.837 Bas R() ˆ 0.005-0.87-0.097 0.386 RMSE R() ˆ 0.060 0.97 0.4 0.453 h() ˆ 0.44 0.70 0.59 0.050 Bas h() ˆ -0.00 0.04 0.03-0.095 RMSE h() ˆ 0.008 0.05 0.07 0. REMARK: Hr, Rms s ad Bas ar obad from, 000 ˆ Bas = ˆ ad ' Rmss =. = 000 3.8 COMPARISONS OF DIFFERENT METHODS USING GRAPH For a quck udrsadg, h rlav bass ad h rlav RMSE s of h dffr smaors of h paramr s prsd fgur-3.8., 3.8., 3.8.3, 3.8.4, 3.8.5 ad 3.8.6 wh sampl szs 0, 0, 30. 38

FIGURE-3.8. ad 3.8. show h avrag rlav bass ad RMSE s of h dffr smaors of = wh dffr sampl szs 0, 0, 30. FIGURE-3.8.3 ad 3.8.4 show h avrag rlav bass ad Rms s of h dffr smaors of = wh sampl szs 0, 0, 30. FIGURE-3.8.5 ad 3.8.6 show h avrag rlav bass ad Rms s of h dffr smaors of = 3 wh sampl szs 0, 0, 30. 39

3.9 REAL LIFE APPLICATION I hs sco, a ral daa s usd as a xampl o fd h four smaors for h shap paramr of Ivrs oal dsrbuo. W apply h Kolmogorov-Smrov (K- S) sasc, ordr o vrfy whch smaor of mak h vrs oal dsrbuo fs br o hs daa. Th K-S s sasc s dscrbd dals D'Agoso ad Sphs (986). I gral, h smallr valu of K-S, h br f o h daa. Th MLE, PCE, LSE ad WLSE of ad h K-S s sascs for Ivrs oal dsrbuo wh dffr smaors of ar gv Tabl 4. Th followg daa s s prsd Aar s (987), rprss h lfms of 50 dvcs. 0., 0.,,,,,,, 3, 6, 7,,, 8, 8, 8, 8, 8,, 3, 36, 40, 45, 46, 47, 50, 55, 60, 63, 63, 67, 67, 67, 67,7, 75, 79, 8, 8, 83, 84, 84, 84, 85, 85, 85, 85, 85, 86, 86. W wll us h abov daa for fg h vrs oal dsrbuo. REMARK: Accordg o h K-S s sasc, h vrs oal dsrbuo fs o h gv daa bcaus hr calculad valu of K-S s (D ) = 0.60048 ad abulad valu of K-S s (D,α ) a 5% =.36. 40

Tabl 3.9.: Th valus of smaor of ˆ, () dsrbuo usg dffr mhod of smao, for h gv daa. Mhods ˆ () h ˆ ad R() ˆ for vrs oal R ˆ h() ˆ MLE.5954 0.3635 0.58079 PCE 0.538060 0.00 0.89438 LSE 3.480 0.99077 0.00877 WLSE 7.80003 0.97564 0.008405 REMARK: Accordg o Tabl-3.9., w ca s ha MLE ad PCE prform bs for ral daa xampl whl LSE ad WLSE do o prform wll. 3.0 CONCLUSIONS Comparg all h mhods, w coclud ha for ukow paramr, h MLE prforms h bs for mos dffr valus of ad cosdrd hr. Th WLSE do o prforms wll for mos dffr valus of ad cosdrd hr. For sampl sz 0, 0, 30, for WLSE, s xrmly ovr sma whl PCE ad LSE ar udr sma for all h m. Comparg h compuaoal complxs of h dffr smaors, s obsrvd ha wh h shap paramr s ukow, w d som rav chqus o compu PCE, LSE ad WLSE. Comparg h prformac of all h smaors, s obsrvd ha as far as bass or RMSE's ar cocrd, h MLE prforms bs mos cass cosdrd hr. Compuaoally, h MLE volv oly o dmsoal opmzao whras h rs of h smaors volv wo-dmsoal opmzao. As crass, h sma of dcrass, ad s basd crass for MLE, LSE ad PCE whl sma of dcrass, s basd crass for WLSE. From Tabl-3.7., 3.7. ad 3.7.3, w coclud ha for ukow paramr, as far as rlably s cocr, rlably crass for WLSE ad dcrass for MLE, LSE, ad 4

PCE for dffr sampl sz cosdrd. As far as hazard ra s cocr, hazard ra dcrass for MLE ad WLSE whl hazard ra crass for PCE ad LSE wh dffr sampl sz cosdrd. I hs chapr w hav also compard all h mhods usg graph. Hr o ca ak h comparso of dffr mhod by bass ad RMSE s. From comparsos of graph, w coclud ha MLE prform bs whl WLSE do o prforms wll for mos dffr valus of ad cosdrd hr. For a quck udrsadg, h rlav bass ad h rlav RMSE s of h dffr smaors of h paramrs s prsd fgur- 3.8., 3.8., 3.8.3, 3.8.4,3.8.5 ad 3.8.6 wh sampl szs 0, 0 ad 30 for MLE, PCE, LSE ad WLSE. Fally w ca say ha for vrs oal dsrbuo, MLE ad PCE prform bs whl LSE ad WLSE do o prform wll for ral daa xampl cosdrd hr. 4