Exponentiated Pareto Distribution: Different Method of Estimations

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1 It. J. Cotemp. Math. Sceces, Vol. 4, 009, o. 14, Expoetated Pareto Dstrbuto: Dfferet Method of Estmatos A. I. Shawky * ad Haaa H. Abu-Zadah ** Grls College of Educato Jeddah, Scetfc Secto, Kg Abdulazz Uversty, P. O. Box 3691, Jeddah 1438, Saud Araba * Permaet address: Fac. of Eg. at Shoubra, P.O. Box 106, El Maad 1178 Caro, Egypt (E-mal address: ** E-mal address: haaa_abuzadah@yahoo.com Abstract Recetly Gupta et al. (1998) troduced a ew dstrbuto, called the expoetated Pareto dstrbuto. I ths paper, we cosder the maxmum lkelhood estmato of the dfferet parameters of a expoetated Pareto dstrbuto. We also maly cosder fve other estmato procedures ad compare ther performaces through umercal smulatos. Mathematcs Subject Classfcato: 6G30; 6E15; 6G05; 6F10; 6F1; 6G0 Keywords: Expoetated Pareto dstrbuto; Maxmum lkelhood estmators; Bas; Fsher Iformato matrx; Asymptotc dstrbuto; Root mea squared errors; Ubased estmators; Order statstcs; Method of momet estmators; Least squares estmators; Weghted least squares estmators; Percetles estmators; L-momet estmators; Smulatos 1. INTRODUCTION A ew two-parameter dstrbuto, called the expoetated Pareto dstrbuto EP(θ,λ ) has bee troduced recetly by Gupta et al. (1998). The EP(θ, λ ) has the probablty desty fucto (pdf) ( 1) ( ;, ) [1 (1 ) ] 1 + f x θ λ = θ λ + x λ θ (1 + x), x > 0, λ > 0, θ > 0 (1.1) ad cumulatve dstrbuto fucto (cdf) F ( x; θ, = [1 (1 + x) λ ] θ, x > 0, λ > 0, θ > 0, (1.)

2 678 A. I. Shawky ad H. H. Abu-Zadah where θ ad λ are two shape parameters. Whe θ = 1, the above dstrbuto correspods to the stadard Pareto dstrbuto of the secod kd. Therefore, EP dstrbuto has a survval fucto S ( x; θ, = 1 [1 (1 + x) λ ] θ, x > 0, λ > 0, θ > 0 (1.3) ad a hazard fucto θ 1 ( λ + 1) θ λ [1 (1 + x) ] (1 + x) h( x; θ, =, x > 0, λ > 0, θ > 0. (1.4) θ 1 [1 (1 + x) ] Next, Gupta ad Kudu (1999) have troduced the geeralzed expoetal dstrbuto whch has bee studed qute extesvely. I (001), they dscussed also a dfferet method of estmatos of the parameters of a geeralzed expoetal dstrbuto. May authors have studed dfferet parameters estmators for certa dstrbutos; Balakrsha ad Basu (1995), Hossa ad Zmmer (003), Komor ad Hrose (004), Kotz ad Nadarajah (000), Kudu ad Raqab (005), Murthy et al. (004), Nguye (004), Surles ad Padgett (005), Wgo (1993), Wu ad L (003) ad Wu (003). Gupta et al. (1998) showed that the expoetated Pareto dstrbuto EP(θ,λ ) ca be used qute effectvely aalyzg may lfetme data. The EP(θ,λ ) ca have decreasg ad upsde-dow bathtub shaped falure rates depedg o the shape parameter θ. The ma am of ths paper s to study how the dfferet estmators of the ukow parameter/parameters of a EP dstrbuto ca behave for dfferet sample szes ad for dfferet parameter values. Here, we maly compare the maxmum lkelhood estmators ('s) wth the other estmators such as the method of momet estmators (MME's), estmators based o percetles (PCE's), least squares estmators (LSE's), weghted least squares estmators (WLSE's) ad the estmators based o the lear combatos of order statstcs (LME's), maly wth respect to ther bases ad root mea squared errors (RMSE's) usg extesve smulato techques. The remag sectos go as follows. I Secto, we brefly dscuss the 's ad ther mplemetatos. I Sectos 3 to 6 we dscuss other methods. Smulato results ad dscussos are provded Secto 7.. MAXIMUM LIKELIHOOD ESTIMATORS I ths secto the maxmum lkelhood estmators of EP(θ,λ ) are cosdered. We cosder two dfferet cases. Frst, cosder estmato of θ ad λ whe both are ukow. If x, 1 x,..., x s a radom sample from EP(θ,λ ), the the log-lkelhood fucto, L ( θ,, s

3 Expoetated Pareto dstrbuto 679 L ( θ, = l( θ ) + l( + ( θ 1) l[1 (1 + x ) ] ( λ + 1) l(1 + x ). = 1 = 1 (.1) The ormal equatos become: L = + l[1 (1 + ) x λ ] = 0, θ θ = 1 L (1 + x ) l(1 + x ) = + ( θ 1) l(1 + x ) = 0. λ λ λ = 1 [1 (1 + x ) ] = 1 (.) (.3) From (.), we obta the of θ as a fucto of λ, say ˆ θ (, where ˆ( θ =. (.4) l[1 (1 + ) x ] = 1 Puttg ˆ θ ( (.1), we obta g( = L( ˆ( θ, = l( ) l ( l[1 (1 + x ) ]) + l( = 1 (.5) l[1 (1 + x ) ] ( λ + 1) l(1 + x ). = 1 = 1 Therefore, of λ, say λˆ, ca be obtaed by maxmzg (.5) wth respect to λ. It s observed that g( λ ) s a umodal fucto, ad the λˆ whch maxmzes (.5) ca be obtaed from the fxed pot soluto of h ( λ ) = λ, (.6) where 1 (1 + x ) l(1 + x ) /[1 (1 + x ) ] 1 l(1 + x ) ( ) 1 h λ = = +. l[1 (1 ) ] = 1 [1 (1 + ) ] + x x = 1 (.7) A very smple teratve procedure ca be used to fd a soluto of (.6) ad t works very well. Oce we obta λˆ, the of θ say θˆ ca be obtaed from (.4) as ˆ θ = ˆ( θ ˆ λ ).

4 680 A. I. Shawky ad H. H. Abu-Zadah Now we state the asymptotc ormalty results to obta the asymptotc varaces of the dfferet parameters. It ca be stated as follows: [ ( ˆ θ θ ), ( ˆ λ )] N ( 0, I 1 ( θ, ), (.8) where I ( θ, s the Fsher formato matrx,.e., 1 ( / ) ( / ) 1 (, ) E L θ E L θ λ I I I θ λ = = ( / ) ( / ) I I E L λ θ E L λ 1 Usg the otatos 1 d ξ ( a, b) =, Ψ ( a) = l Γ( a) s the dgamma fucto ad Ψ (a) s b = 0 ( + a) da the dervatve of Ψ (a), we obta I I V ( ˆ θ ) ad V ( ˆ λ ) 11, I I I I I I where I = 11 θ θ I = I = { [ Ψ( θ ) Ψ(1)] [ Ψ( θ + 1) Ψ(1)]}, for θ λ θ 1 = ξ (,), for θ = 1 λ θ ( θ 1) I = {1 + ( Ψ (1) Ψ ( θ 1) + [ Ψ( θ 1) Ψ(1)] ) λ θ θ ( Ψ (1) Ψ ( θ ) + [ Ψ( θ ) Ψ(1)] )}, for θ = [1 + 4ξ (,3)], λ for θ =. Now cosder the of θ, whe the shape parameter λ s kow. If λ s kow the of θ s ˆ θ =. (.9) l[1 (1 + ) x ] = 1 The dstrbuto of θˆ s the same as the dstrbuto of (θ /Y), where Y follows Gamma(,1). Therefore, for > ˆ E ( θ ) = θ, 1

5 Expoetated Pareto dstrbuto 681 ˆ V ( θ ) = θ, ( 1) ( ) ˆ + MSE ( θ ) = θ. ( 1)( ) Clearly θˆ s ot a ubased estmator of θ, although asymptotcally t s ubased. From the expresso of the expected value, we cosder the followg ubased estmator of θ, say θˆ, UBE ˆ 1 ˆ 1 θ = θ =, (.10) UBE l[1 (1 + ) x ] = 1 where ˆ ˆ θ V ( θ ) = MSE( θ ) =. (.11) UBE UBE Therefore, V ( ˆ θ ) s closer to the Cramer-Rao lower boud ( = θ / ) UBE compared to the. Now cosder the of λ whe the shape parameter θ s kow. For kow θ the of λ say λˆ ca be obtaed by maxmzg u ( λ ) = l( + ( θ 1) l[1 (1 + x ) ] ( λ + 1) l(1 + x ), (.1) = 1 = 1 wth respect to λ. It ca be easly show that u ( s a umodal fucto of λ ad λˆ whch maxmzes u ( ca be obtaed as the fxed pot soluto of v ( λ ) = λ, (.13) where 1 [1 (1 ) ]l(1 ) 1 θ + x + x v ( =. λ 1 [1 (1 + ) ] = x From the asymptotc propertes of the, t follows that E( ˆ λ ) λ ad V ( ˆ λ ) I 1.

6 68 A. I. Shawky ad H. H. Abu-Zadah 3. METHOD OF MOMENT ESTIMATORS I ths secto we provde the method of momet estmators (MME's) of the parameters of a EP dstrbuto. Frst, we cosder the case whe both of the parameters are ukow. If X follows EP(θ,λ ), the μ = E ( X ) = θ Β( θ,1 1/ 1; λ > 1, (3.1) σ = V ( X ) = θ Β( θ,1 / [ θ Β( θ,1 1/ ] ; λ >. (3.) Here Β (.,. ) deotes the beta fucto. Smlarly, the mea ad varace of the radom sample x, 1 x,..., x from EP(θ,λ ) are x = x / ad = S ( x x) /( 1). = 1 = 1 Therefore, equatg the mea ad varace of the sample wth the mea ad varace of the populato, we obta x = θ Β( θ,1 1/ 1; λ > 1, (3.3) S = θ Β( θ,1 / [ θ Β( θ,1 1/ ] ; λ >. (3.4) The, the MME's of θ ad λ, say θˆ ad λˆ, respectvely, ca be MME MME obtaed by solvg the two equatos (3.3) ad (3.4). It s ot possble to obta the exact varaces of θˆ or λˆ. MME MME If the shape parameter λ s kow, the the MME of θ ca be obtaed by solvg the o-lear equato (3.3) wth respect to θ. I the same way, f the shape parameter θ s kow, the the MME of λ ca be obtaed by solvg the o-lear equato (3.3) wth respect to λ. 4. ESTIMATORS BASED ON PERCENTILES If the data comes from a dstrbuto fucto whch has a closed form, the t s qute atural to estmate the ukow parameters by fttg a straght le to the theoretcal pots obtaed from the dstrbuto fucto ad the sample percetle pots. Ths method was orgally explored by Kao (1958, 1959) ad t has bee used qute successfully for Webull dstrbuto ad for the geeralzed expoetal dstrbuto [see, Murthy et al. (004) ad Gupta ad Kudu (001)].

7 Expoetated Pareto dstrbuto 683 I case of a EP dstrbuto, also, t s possble to use the same cocept to obta the estmators of θ ad λ based o the percetles, because of the structure of ts dstrbuto fucto. Frst let us cosder the case, whe both of the parameters are ukow. Sce F ( x; θ, = [1 (1 + x) λ ] θ, therefore, l[ F ( x; θ, ] = θ l[1 (1 + x) λ ]. (4.1) Let X () deotes the -th order statstc,.e., X ( 1) < X () <... < X ( ). If p deotes some estmate of F ( x ; θ, ) ( ) λ, the the estmate of θ ad λ ca be obtaed by mmzg (l( ) θ l[1 (1 ) λ ]) p + x, (4.) ( ) = 1 wth respect to θ ad λ. We otce that (4.) s a o-lear fucto of θ ad λ. It s possble to use some o-lear regresso techques to estmate θ ad λ smultaeously. We call these estmators as percetle estmators (PCE's). Several estmators of p ca be used here [see, Murthy et al. (004)]. I ths paper, we maly cosder p =, whch s the expected value of F ( X ( ) ). +1 Now let us cosder the case whe oe parameter s kow. If the shape parameter θ s kow, the the estmator of λ ca be obtaed by mmzg (4.) wth respect to λ oly. The percetle estmator of λ for kow θ, say λˆ, ca be obtaed as the soluto of the followg o-lear equato; PCE (1 + x ) l(1 + x ) ( ) ( ) (l( ) l[1 (1 + ) λ p θ x ]) ( ) = 1 [1 (1 + x ) ] ( ) = 0. (4.3) If the shape parameter λ s kow, the the estmator of θ ca be obtaed by mmzg (4.) wth respect to θ oly. The percetle estmator of θ for kow λ, say θˆ, becomes PCE l( p ) l[1 (1 + x ) ] ( ) ˆ θ = = 1. (4.4) PCE (l[1 (1 + x ) ]) ( ) = 1 Iterestgly, θˆ s also a closed form lke θˆ whe λ s kow. PCE

8 684 A. I. Shawky ad H. H. Abu-Zadah 5. LEAST SQUARES AND WEIGHTED LEAST SQUARES ESTIMATORS I ths secto, we provde the regresso based method estmators of the ukow parameters, whch was orgally suggested by Swa et al. (1988) to estmate the parameters of Beta dstrbutos. The method ca be descrbed as follows: Suppose Y 1, Y,..., Y s a radom sample of sze from a dstrbuto fucto G (. ) ad Y ( 1) < Y() <... < Y( ) deotes the order statstcs of the observed sample. It s well kow that ( + 1) E ( G( Y )) = ad V ( G( Y )) =, ( ) + 1 ( ) ( + 1) ( + ) [see, Johso et al. (1995)]. Usg the expectatos ad the varaces, two varats of the least squares methods ca be used. Method 1 (Least Squares Estmators): Obta the estmators by mmzg ) ) ( G( Y, (5.1) ( ) = wth respect to the ukow parameters. Therefore, case of EP dstrbuto the least squares estmators of θ ad λ, say θˆ ad λˆ, respectvely, ca be LSE LSE obtaed by mmzg ([1 (1 ) ] ) + x λ θ, (5.) ( ) = wth respect to θ ad λ. Method (Weghted Least Squares Estmators): The weghted least squares estmators ca be obtaed by mmzg ) ) w ( G( Y, (5.3) ( ) = wth respect to the ukow parameters, where w = (1/( V ( G( Y )))) = ( + 1) ( + ) / ( + 1). Therefore, case of EP ( ) dstrbuto the weghted least squares of θ ad λ, say θˆ ad λˆ, WLSE WLSE respectvely, ca be obtaed by mmzg ([1 (1 ) ] ) w + x λ θ, (5.4) ( ) = wth respect to θ ad λ oly.

9 Expoetated Pareto dstrbuto 685 I case of EP dstrbuto, f the shape parameter λ s kow, the the LSE ad WLSE of θ ca be obtaed by mmzg (5.) ad (5.4), respectvely, wth respect to θ. I the same way, f the shape parameter θ s kow, the the LSE ad WLSE of λ ca be obtaed by mmzg (5.) ad (5.4), respectvely, wth respect to λ. 6. L-MOMENT ESTIMATORS I ths secto we propose a method for estmatg the ukow parameters of a EP dstrbuto based o the lear combatos of order statstcs, see, for example Davd (1981), Hoskg (1990), or Davd ad Nagaraja (003). The estmators obtaed by ths method are popularly kow as L-momet estmators (LME's). The LME's are aalogous to the covetoal momet estmators but ca be estmated by lear combatos of order statstcs,.e., by L-statstcs. The LME's have theoretcal advatages over covetoal momets of beg more robust to the presece of outlers the data. It s observed that LME's are less subject to bas estmato ad sometmes more accurate small samples tha eve the 's. Frst, we dscuss the case how to obta the LME's whe both the parameters of a EP dstrbuto are ukow. If x ( 1) < x() <... < x( ) deotes the ordered sample, the usg the same otato as Hoskg (1990), we obta the frst ad secod sample L-momets as 1 l = x, l = ( 1) x l (6.1) 1 ( ) ( ) 1 = 1 ( 1) = 1 ad the frst two populato L-momets are λ = θ Β( θ, 1 1/ 1, λ = θ Β(θ, 1 1/ θ Β( θ,1 1/ ; λ > 1, (6.) 1 respectvely. Note that (6.) follows from the dstrbuto fucto of the -th order statstc of a EP radom varable. Now to obta the LME's of the ukow parameters θ ad λ, we eed to equate the sample L-momets wth the populato L-momets. Therefore, the LME's ca be obtaed from l = θ Β( θ, 1 1/ 1; λ > 1, (6.3) 1 l = θ Β(θ, 1 1/ θ Β( θ,1 1/ ; λ > 1. (6.4) The, the LME's of θ ad λ, say θˆ ad λˆ, respectvely, ca be LME LME obtaed by solvg the two equatos (6.3) ad (6.4). It s ot possble to obta the exact varaces of θˆ or λˆ. LME LME If the shape parameter λ or θ s kow, the the LME of θ or λ ca be obtaed by solvg the o-lear equato (6.3) wth respect to θ or λ,

10 686 A. I. Shawky ad H. H. Abu-Zadah respectvely. It s terestg to ote that f λ or θ s kow, the the LME of θ or λ s the same as the correspodg momet estmator. 7. NUMERICAL EXPERIMENTS AND DISCUSSIONS I ths secto we preset results of some umercal expermets to compare the performace of the dfferet estmators proposed the prevous sectos. We perform extesve Mote Carlo smulatos to compare the performace of the dfferet estmators, maly wth respect to ther bases ad root mea squared errors (RMSE's) for dfferet sample szes ad for dfferet parameter values. Note that, we take λ =3 all cases cosdered. We cosder θ = 0.5, 1.0, 3.0 ad = 10, 30 ad 50. We compute the bases ad RMSE's of estmators over 1000 replcatos for dfferet cases. Frst cosder the estmato of θ whe λ s kow. Whe λ s kow the, ubased estmator (UBE) ad PCE of θ ca be obtaed from (.9), (.10) ad (4.4), respectvely. The methods of momet estmator of θ ca be obtaed by solvg the o-lear equato (3.3). Smlarly, the least squares ad weghted least squares estmators of θ ca be obtaed by mmzg (5.) ad (5.4), respectvely, wth respect to θ oly. The results are reported Table 1. It s observed Table 1 that most of the estmators usually overestmate θ, except PCE, whch uderestmates all the tmes. As far as bases are cocered, the UBE's are more or less ubased as expected ad cosderg the mmum RMSE's for most dfferet values of θ ad cosdered here. The RMSE's of the UBE's are also qute close to the 's ad PCE's. I the cotext of computatoal complextes, UBE, ad PCE are easest to compute. They do ot volve ay o-lear equato solvg, whereas the MME, LSE ad WLSE volve solvg o-lear equatos ad they eed to be calculated by some teratve processes. Comparg all the methods, we coclude that for kow shape parameter λ, the UBE should be used for estmatg θ. Now cosder the estmato of λ whe θ s kow. I ths case the of λ ca be obtaed by maxmzg (.1) wth respect to λ. The MME ad PCE ca be obtaed by solvg the o-lear equatos (3.3) ad (4.3), respectvely. Fally the LSE ad the WLSE ca be obtaed by mmzg (5.) ad (5.4) wth respect to λ oly for fxed θ. The results are reported Table. I ths case t s observed that the estmators usually overestmate λ. Comparg the bases of all the estmators, t s observed that the WLSE performs the best for most dfferet values of θ ad cosdered here. The performace of the 's ad LSE's are qute close to the WLSE whe θ 1. As far as RMSE's are cocered, PCE ad LSE outperforms others whe θ 1 ad θ > 1,

11 Expoetated Pareto dstrbuto 687 respectvely, ad the performs the best for most dfferet values of θ ad cosdered here. Comparg the computatoal complextes of the dfferet estmators, t s observed that whe the shape parameter θ s kow, we eed some teratve techques to compute, MME, PCE, LSE ad WLSE. Summg up, we recommed to use the for estmatg λ whe the shape parameter θ s kow. Now we cosder the estmato of θ ad λ whe both of them are ukow. We cosder sx dfferet estmators, amely, MME, PCE, LSE, WLSE, ad LME. The results for θ ad λ are reported Tables 3 ad 4, respectvely. Comparg the performace of all the estmators, t s observed that as far as bases or RMSE's are cocered, the performs best most cases cosdered here. Iterestgly, whle estmatg λ, the bases ad RMSE's of the LME are lower tha the other estmators most of the tmes. Computatoally, the volve oly oe dmesoal optmzato whereas the rest of the estmators volve two dmesoal optmzato. Eve though, the estmators ca be obtaed by performg oe or two dmesoal optmzato, we recommed to use the for estmatg θ ad λ whe both are ukow.

12 688 A. I. Shawky ad H. H. Abu-Zadah Table 1: Smulated values of bases ad RMSE's of estmators of θ whe λ s kow Method θ =.5 θ =1 θ = MME PCE LSE WLSE UBE MME PCE LSE WLSE UBE MME PCE LSE WLSE UBE Note: The frst etry s the smulated bas. The secod etry s the smulated RMSE.

13 Expoetated Pareto dstrbuto 689 Table : Smulated values of bases ad RMSE's of estmators of λ whe θ s kow Method θ =.5 θ =1 θ = MME PCE LSE WLSE MME PCE LSE WLSE MME PCE LSE WLSE Note: The frst etry s the smulated bas. The secod etry s the smulated RMSE.

14 690 A. I. Shawky ad H. H. Abu-Zadah Table 3: Smulated values of bases ad RMSE's of estmators of λ whe θ s ukow Method θ =.5 θ =1 θ = MME PCE LSE WLSE LME MME PCE LSE WLSE LME MME PCE LSE WLSE LME Note: The frst etry s the smulated bas. The secod etry s the smulated RMSE.

15 Expoetated Pareto dstrbuto 691 Table 4: Smulated values of bases ad RMSE's of estmators of θ whe λ s ukow Method θ =.5 θ =1 θ = MME PCE LSE WLSE LME MME PCE LSE WLSE LME MME PCE LSE WLSE LME Note: The frst etry s the smulated bas. The secod etry s the smulated RMSE.

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