BOOTSTRAP PREDICTION INTERVALS FOR TIME SERIES MODELS WITH HETROSCEDASTIC ERRORS. Department of Statistics, Islamia College, Peshawar, KP, Pakistan 2

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1 Pak. J. Sais. 017 Vol. 33(1), 1-13 BOOTSTRAP PREDICTIO ITERVAS FOR TIME SERIES MODES WITH HETROSCEDASTIC ERRORS Amjad Ali 1, Sajjad Ahmad Khan, Alamgir 3 Umair Khalil and Dos Muhammad Khan 1 Deparmen of Saisics, Islamia College, Peshawar, KP, Pakisan Deparmen of Saisics, Abdul Wali Khan Universiy Mardan, KP, Pakisan 3 Deparmen of Saisics, Universiy of Peshawar, KP, Pakisan Corresponding auhor sajjadkhan@awkum.edu.pk ABSTRACT In his paper, we propose wo boosrap procedures o consruc predicion inervals for Auoregressive Fracionally Inegraed Moving Average wih Condiionally Heroscedasic errors (ARFIMA-GARCH) models. The firs mehod is based on he model based boosrap, in which he order of he model is assumed o be known. The second boosrap mehod is based on he idea of approximaing he ARFIMA par by an AR model. In modeling he ARFIMA-GARCH model, he firs sep is o deermine he order of ARFIMA par and deerminaion he order of ARFIMA model is a complicaed ask. To simplify he model building procedure, we approximae he ARFIMA par of he ARFIMA-GARCH model by an AR(p) model and fi an AR-GARCH model insead of ARFIMA-GARCH model. The mehodology has also been applied o ARMA-GARCH models. To check he performance of he proposed mehods, we perform simulaion sudy. KEYWORDS Boosrap; Time Series; Predicion inerval; Heroscedasic Errors. 1. ITRODUCTIO In ime series analysis a variey of models such as AR, ARMA, ARIMA are used o model he observed series and make predicions. For predicion based on hese models, i is ypically assumed ha he variance of he error erms is consan. The assumpion of consan variance of he error erm is no realisic for many financial and economic ime series. These series show burss of unusually high volailiy and he assumpion of a consan variance is no appropriae for such series. However, Engle (198) found ha he classical ARIMA model failed o achieve he desired effec of he fiing for UK inflaion rae. By carefully sudying he sequence of he residuals, hey found ha he series of he residuals faced he problem of heeroscdasiciy. Recenly, many researchers have shown ha various financial ime series exhibi heeroscdasiciy. They have found posiive relaionship beween he sandard deviaion and he level. Tha is, he sequence of flucuaion remains low wih low level of he sequence and he sequence of flucuaion becomes high wih increasing sequence of he level. For example, in financial ime series, 017 Pakisan Journal of Saisics 1

2 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors small reurns are followed by more small reurns (in case when marke crashes) and large reurns are followed by more large reurns in he growh period. In ime series analysis forecasing is an imporan objecive. In classical inerval forecasing, i is ypically assumed ha he innovaions of he model have some known disribuion. In mos of he applicaions, his assumpion is no saisfied and he predicion inervals based on i are no valid. To deal wih his problem, several boosrap procedures have been inroduced for he consrucion of predicion inervals in ime series analysis e.g. Thombs and Schucany (1990); Masaroo (1990); Grigoleo (1998) Cao e al. (1997); Alanso, A.M, e al. (00, 003); Pascual e al. (004), Clemens and Kim (007), Amjad e al. (015) among ohers. These boosrap predicion inervals have he assumpion of homescedasiciy for he innovaions of he model. In he conex of ime series models wih heroscedasic errors, Miguel and Olave (1999) proposed a boosrap mehod o consruc predicion inervals for ARMA-ARCH models. This mehod was furher improved by Pascual e al. (006) by incorporaing he parameer variabiliy and applied o consruc predicion inervals for ARMA-GARCH models. In he curren sudy, we propose wo boosrap procedures o consruc predicion inervals for Auoregressive Fracionally Inegraed Moving Average (ARFIMA) Models wih GARCH errors. The firs mehod is based on he parameric boosrap, in which he order of he model is assumed o be known. The second boosrap mehod is based on he idea of approximaing he ARFIMA par by an AR model. In modeling he ARFIMA-GARCH model, he firs sep is o deermine he order of ARFIMA par. Deerminaion of he order of ARFIMA model is a complicaed ask. To simplify he model building procedure, we approximae he ARFIMA par of he ARFIMA-GARCH model by an AR(p) model and fi an AR-GARCH model insead of ARFIMA-GARCH model.. THE MODE.1 The ARMA-GARCH Model The ARMA-GARCH model is he combinaion of he linear ARMA model wih GARCH errors. This is also called he condiional mean and condiional variance model. In mos of he applicaions he GARCH model is no direcly observed bu he innovaions of he linear ARMA process follows he GARCH model. The ime series X1, X,..., X n follows he ARMA(p,q)-GARCH(r,s) model if: X i i j j i1 j1 or ( B) X ( B) z r s w ii jj ii j1 z ~ i. i. d. (0,1) p q X. The ARMA(p,q) model for he condiional mean is assumed o be covariance saionary and inverible i.e. he roos of ( B) and ( B) lie ouside he uni circle.

3 Amjad Ali e al. 3 r For he condiional variance model o be saionary w 0, 0, 0 and s 1. ii i j1 j. The ARFIMA-GARCH Model The ARFIMA-GARCH model is obained by combing he ARFIMA model wih GARCH innovaions. The sochasic process X, R has ARFIMA(p,d,q)-GARCH(r,s) model if i saisfies wih d ( B) X ( B)(1 B), z r s w i i j j ii j1 z ~ i. i. d. (0,1), where ( B) and ( B) are polynomials of order p and q respecively and d is he long memory parameer. 3. BOOTSTRAP PREDICTIO ITERVAS FOR ARFIMA-GARCH MODES The consrucion of he predicion inervals hrough boosrap mehods is a nonparameric approach, which does no assume any parameric hypoheses on he error disribuion. In he curren sudy, we propose wo boosrap procedures o consruc predicion inervals for ARFIMA-GARCH models. One is model based boosrap in which we assume ha he model is known and he oher is a sieve ype boosrap procedure. These mehods are discussed as below. 3.1 The Model Based Boosrap Mehod o Consruc Predicion Inervals (A1) In he model based boosrap mehod also called he parameric boosrap mehod, we assume ha he order of he ARFIMA(p,d,q)-GARCH(r,s) is known. The seps o consruc model based boosrap predicion inervals are oulined as follows. 1) Esimae he fracional difference parameer d for he given series. A number of mehods are available o esimae d. Here he semi-parameric local While esimaor of d is used. ) Take he fracional difference of he given series by using d, esimaed in sep-1. The filered series hus obained will follow ARMA(p,q)-GARCH(r,s) model. 3) Esimae he ARMA(p,q)-GARCH(r,s) model for he filered series by quasimaximum likelihood. The vecor of esimaed parameers is given by ( ˆ,..., ˆ, d ˆ, ˆ,..., ˆ, wˆ, ˆ,..., ˆ, ˆ,..., ˆ ) 1 p 1 q 1 r 1 s where ˆd is he esimae of he fracional difference parameer calculaed in sep. j j

4 4 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors 4) Esimae he residuals ˆ from he fied model and calculae he sandardized residuals by ˆ zˆ for p 1,..., n. ˆ 5) Draw an i.i.d sample from ˆ G denoed by z, where Gˆ is he empirical disribuion funcion of he cenered residuals and is defined by ˆ 1 n G ( x ) I ( z x ), n 1 where z zˆ zˆ and zˆ ( n p) zˆ. n 1 p1 6) Generae he boosrap sample by recursion ˆ ˆ d ˆ( B)( X X ) ( B)(1 B) ˆ r s ˆ w ˆˆ i i j j ii j1 ˆ ˆ ˆ z ˆ ˆ ˆ. oe ha, we generae n 00 observaions by he above recursion in order o sabilize he series and discard he firs 00 values. 7) Esimae he model parameer ( ˆ,..., ˆ, ˆ, ˆ,..., ˆ,, ˆ,..., ˆ, ˆ,..., ˆ ) for 1 p d 1 q wˆ 1 r 1 s he boosrap sample ( X1, X,..., X n). The esimaion is done on he same lines as we did in seps (1-3) for he original series. 8) Calculae he h-seps ahead forecas values by using he recursion h1 h j ( h1 ) ˆ h1 j1 X X a X X r s ˆ h w ˆi ˆhi j h j ii j1 ˆ ˆ ˆ h zh ˆ h ˆ ˆ. The boosrap disribuion of he prediced values is obained by repeaing seps (5-8) B imes. 3. The Sieve Boosrap Approach o Consruc Predicion Inervals (A) In he model based boosrap, i was assumed ha order of he ARFIMA par of ARFIMA-GARCH model is known. In pracice, idenifying he order of ARFIMA model is no very simple and leads o wrong inferences if i is no correcly idenified. To deal wih his complexiy, we approximae i by an AR model as he idenificaion of an AR model is very simple compared o an ARFIMA model. Therefore, o consruc predicion

5 Amjad Ali e al. 5 inervals for ARFIMA-GARCH model, we fi an AR-GARCH model o he given series. The seps o consruc predicion inervals are oulined as follow. 1) Approximae he condiional mean equaion of he ARFIMA-GARCH model by an AR model and deermine he order p of he AR model. ) Esimae he parameers of he AR(p)-GARCH(r,s) model given by ( ˆ,..., ˆ, wˆ, ˆ,..., ˆ, ˆ,..., ˆ ). 1 p 1 r 1 s 3) From he model fied in sep, calculae he residuals ˆ and he sandardized ˆ residuals zˆ for p 1,..., n. ˆ 4) Define he empirical disribuion funcion ˆ residuals as where 1 G for he cenered sandardized ˆ 1 n G ( x ) I ( z x ). The cenered residuals are given z zˆ zˆ n zˆ ( n p) zˆ. n 1 p1 1 n 5) The boosrap sample ( X, X,..., X ) is generaed by he following recursion ˆ( B)( X X ) ˆ r s ˆ w ˆˆ i i j j ii j1 ˆ ˆ ˆ z ˆ ˆ ˆ, where ˆ ( B) (1 ˆ 1... ˆ p ) and X X for p. Using he above recursion, we generae (n+00) and remove he firs 00 values o minimize he effec of iniial values. 6) Fi he AR(p)-GARCH(r,s) model o he boosrap sample ( X1, X,..., X n ) generaed in sep 5 and esimae he model parameers given by ˆ ˆ ( ˆ,..., ˆ, wˆ, ˆ,..., ˆ,,..., ). 1 p 1 r 1 s 7) Calculae he h-seps ahead forecas values by using he recursion p h j ( h1 ) ˆ h1 j1 X X X X r s ˆ h w ˆi ˆhi j h j ii j1 ˆ ˆ ˆ h zh ˆ h ˆ ˆ. The boosrap disribuion of he prediced values is obained by repeaing seps 5-7, B imes.

6 6 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors 4. SIMUATIO STUDIES The finie sample performance of he model based boosrap and sieve boosrap mehods o consruc predicion inervals for ARFIMA-GARCH models has been invesigaed horough simulaion sudies. Here, we presen resuls for he following models. M1 : ARFIMA(0, d,0) GARCH(1, 1) M : ARFIMA(1, d,0) GARCH(1, 1) M 3 : ARFIMA(0, d,1) GARCH(1, 1) M 4 : ARFIMA(1, d,1) GARCH(1, 1) The value of he long memory parameer d is se o be 0.3 for all he models. For he ARFIMA par he values of he auoregressive parameer 1 and he moving average parameer 1 are fixed o be 0.5 and 0.3 respecively. We also apply he sieve boosrap approach o consruc predicion inervals for ARMA models wih condiional heroscedasic errors. The following models are considered in our simulaion sudy. M5: ARMA(0., 0.6) GARCH (1, 1) M 6 : ARMA(0.5, 0.3) GARCH (1, 1) The parameers of he GARCH(1,1) model are aken as w0.05, , and for he ARCH(1) model, hese are se o be w 1, We use wo sample sizes 00 and and hree differen error disribuions: he sandard normal, he -disribuion wih 5 degrees of freedom (i.e. lepokuric one) and he exponenial disribuion wih scale parameer equal o one (i.e. he asymmeric one). The exponenial and -disribuions are cenered and scaled o have zero mean and uni variance. We consruc h = 1, 3, 5, 10 seps ahead forecas inervals a he nominal coverage level of 90, 95 and 99 percen, bu here he resuls are given for 95 percen level of significance. To evaluae he performance of he predicion inervals, he empirical coverage and he lengh of he inervals are calculaed wih heir corresponding sandard errors. To check he performance of he model based boosrap and sieve boosrap predicion inervals, we calculae he empirical coverage and lengh of he inervals wih corresponding sandard errors. The number of simulaions is aken as S=100 and he number of boosrap resamples B is se o be For each combinaion of he model, parameers, sample size and innovaions disribuion, we perform he following seps. 1) Generae a realizaion of size n. Also generae R=1000 fuure values of T h X. These fuure values are generaed condiional on he pas n values of he generaed realizaion, he rue values of he parameers and he rue error disribuion.

7 Amjad Ali e al. 7 ) Calculae he boosrap forecas inerval B=1000 boosrap resamples, where Q Q, Q and Q 1 1 h and 1 h perceniles of he 1000 boosrap prediced values. based on are he 3) Using he rue R=1000 fuure values, we calculae he empirical coverage of he inerval. The empirical coverage ( ) is obained as he percenage of R fuure values ha lie in-beween inerval is calculaed as Q 1 and Q Q Q... The lengh of he We repea he above seps S=100 imes and obain he empirical mean lengh and he empirical mean coverage forecas inervals as follows. ( ) wih corresponding sandard errors for each of he S 1 i i1 S S 1 1/ i i1 SE( ) ( S ( S 1)) ( ) ) S 1 S i i1, S 1 1/ i i1 SE( ) ( S ( S 1)) ( ) ) The resuls for model 1 o 1 are presened in Tables 1 o 6. Boh mehods have reasonable coverage for n=00, bu increases wih increasing he sample size o n= as expeced. Since consrucing predicion inervals by boosrap are non-parameric mehods, herefore, differen error disribuions have no significan effec on he percenage coverage. This is rue for all models and boh sample sizes, bu in mos of he cases lengh of he predicion inerval for -disribuion is a lile bi wider han he normal and exponenial disribuion. While consrucing predicion inervals for ARFIMA- GARCH models our simulaion resuls reveal ha he performance of he Sieve Boosrap mehod becomes weaker as he long memory parameer d approaches 0.5 which is is limiing value. I is very naural as he performance of AR-approximaion deerioraes as he model becomes more persisen (Poski, 007). The same naure of performance has also been repored by Amjad e al. (015) while consrucing predicion inervals for ARFIMA models wih whie noise errors.

8 8 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors 5. COCUSIO This work is concerned wih forecasing of ime series models wih condiional heroscedasic errors hrough boosrap mehods. We considered he long memory ARFIMA and shor memory ARMA models wih condiional heroscedasic errors. In he curren sudy, wo boosrap mehods for he consrucion of predicion inervals have been proposed for ARFIMA-GARCH model; he model based boosrap and he sieve boosrap. Simulaion sudies showed ha boh he mehods have good coverage performance. The proposed sieve boosrap procedure showed good performance when applied o consruc predicion inervals for ARMA-GARCH models. REFERECES 1. Alanso, A.M., Pena, D. and Romo, J. (00). Forecasing ime series wih sieve boosrap. J. Sais. Planning Inference, 100, Alanso, A.M., Pena, D. and Romo, J. (003). On sieve boosrap predicion inervals. Saisics & Probabiliy eers, 65, Amjad, A., Alamgir, Khalil, U., Khan, S.A. and Khan, D.M. (015). A Sieve Boosrap Approach o Consrucing Predicion Inervals for ong Memory Time Series Models. Research Journal of Recen Sciences, 4(7), Clemens, M.P. and Kim, J.H. (007). Boosrap predicion inervals for auoregressive ime series. Compu. Sais. & Daa Analysis, 51, Cao, R., Febrero-Bande, M., Gonzalez-Maneiga, W., Prada-Sanchez, J.M. and Garcya-Jurado, I. (1997). Saving compuer ime in consrucing consisen boosrap predicion inervals for auoregressive processes. Comm. Sais. Simul Compu., 6, Engle, R.F. (198). Auoregressive condiional heeroscedasiciy wih esimaes of he variance of Unied Kingdom inflaion. Economerica, 50, Grigoleo, M. (1998). Boosrap predicion inervals for auoregressions: some alernaives. Inernaional Journal of Forecasing, 14, Masaroo, G. (1990). Boosrap predicion inervals for auoregression. Inerna. J. Forecasing, 6, Migue, J.A. and Olave, P. (1999). Boosrapping Forecas Inervals in ARCH Models. Tes, 8, Pascual,., Romo, J. and Ruiz, E. (1998). Boosrap predicive inference for ARIMA processes. Universidad Carlos III de Madrid, Madrid, W.P Pascual,., Romo, J. and Ruiz, E. (006). Boosrap Predicion for Reurns and Volailiies in GARCH Models. Compuaional Saisics and Daa Analysis, 50, Poski, D.S. (007). Properies of he Sieve Boosrap for Fracionally Inegraed and on-inverible Processes. Journal of Time Series Analysis, 9(), Thombs,.A. and Schucany, W.R. (1990). Boosrap predicion inervals for auoregression. J. Amer. Sais. Assoc., 95,

9 Amjad Ali e al. 9 Sepahead Sample size 1 00 Disr. Table 1 Simulaion Resuls for M1 A1 9.6(0.069) 9.8(0.064) 93.1(0.081) 94.3(0.060) 94.5(0.061) 93.9(0.073) 4.539(0.0087) 5.053(0.0101) 4.460(0.0130) 4.541(0.0055) 4.458(0.0077) 4.461(0.0087) A 91.9(0.070) 93.1(0.065) 9.9(0.099) 94.(0.074) 94.4(0.073) 93.8(0.093) 4.546(0.0079) 5.069(0.0099) 4.609(0.0086) 4.579(0.0066) 4.644(0.0074) 4.548(0.0066) (0.064) 4.575(0.0086) 9.1(0.060) 5.056(0.0078) 93.(0.067) 4.637(0.01) 93.1(0.060) 4.487(0.0095) 9.9(0.104) 4.494(0.0134) 93.0(0.109) 4.1(0.0106) 94.8(0.056) 4.561(0.0057) 94.6(0.057) 4.11(0.0063) 94.5(0.06) 94.8(0.075) 4.677(0.0071) 4.514(0.0081) 95.0(0.077) 94.7(0.066) 4.737(0.0065) 4.803(0.007) (0.085) 4.679(0.0083) 91.9(0.083) 4.83(0.0799) 9.4(0.094) 4.771(0.010) 9.3(0.08) 5.073(0.0093) 9.6(0.098) 4.573(0.013) 9.6(0.097) 4.981(0.0088) 94.7(0.055) 4.64(0.0073) 94.1(0.061) 4.646(0.0066) 94.3(0.051) 94.5(0.061) 4.78(0.0069) 4.995(0.0090) 94.5(0.059) 94.3(0.06) 5.089(0.0063) 4.746(0.0074) (0.098) 4.995(0.0083) 9.1(0.093) 5.8(0.0099) 9.1(0.096) 5.071(0.10) 9.0(0.080) 4.936(0.0103) 9.4(0.097) 4.597(0.0135) 9.1(0.074) 5.38(0.0110) 93.8(0.050) 4.51(0.0065) 94.0(0.064) 5.806(0.0050) 94.(0.05) 5.146(0.0070) 93.9(0.05) 5.809(0.0068) 94.6(0.056) 5.08(0.0081) 93.8(0.044) 4.987(0.0081)

10 10 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors Sepahead Sample size 1 00 Disr. Table Simulaion Resuls for M A1 93.0(0.071) 93.1(0.076) 9.9(0.103) 94.9(0.051) 94.6(0.057) 94.7(0.06) 4.686(0.0088) 5.17(0.0105) 4.55(0.0104) 4.355(0.0054) 4.470(0.0067) 4.501(0.0080) A 9.9(0.070) 93.3(0.068) 93.1(0.110) 94.0(0.051) 94.(0.044) 94.3(0.05) 4.756(0.0104) 4.995(0.0088) 4.671(0.0135) 4.446(0.0078) 4.668(0.0084) 4.474(0.0091) (0.053) 4.871(0.0080) 9.6(0.068) 4.865(0.0109) 9.8(0.055) 5.163(0.0110) 91.9(0.041) 4.990(0.0098) 9.4(0.090) 4.700(0.01) 91.9(0.077) 4.839(0.0117) 94.8(0.043) 4.348(0.0053) 94.6(0.051) 4.517(0.0081) 94.5(0.05) 94.7(0.071) 4.506(0.0067) 4.611(0.0090) 94.6(0.040) 94.8(0.066) 4.440(0.0085) 4.684(0.0095) (0.055) 4.996(0.008) 9.3(0.053) 5.03(0.0108) 9.6(0.084) 5.607(0.0117) 9.(0.050) 5.4(0.0095) 91.9(0.091) 4.901(0.017) 91.9(0.073) 4.976(0.0113) 94.9(0.043) 4.394(0.0057) 94.4(0.049) 4.459(0.0086) 94.5(0.05) 94.4(0.067) 4.736(0.0071) 4.706(0.0090) 94.1(0.041) 94.7(0.050) 4.68(0.0085) 4.845(0.0093) (0.061) 5.104(0.0083) 93.5(0.058) 5.150(0.0103) 93.7(0.067) 5.16(0.0105) 93.3(0.044) 5.10(0.0095) 9.9(0.087) 4.966(0.0136) 93.0(0.070) 5.053(0.0119) 94.7(0.037) 4.944(0.0058) 94.5(0.04) 4.957(0.0085) 95.0(0.050) 4.858(0.007) 94.4(0.03) 5.01(0.0085) 94.4(0.089) 4.848(0.0095) 94.9(0.06) 4.969(0.0096)

11 Amjad Ali e al. 11 Sepahead Sample size 1 00 Disr. Table 3 Simulaion Resuls for M3 A1 93.5(0.050) 93.1(0.051) 93.3(0.074) 94.(0.039) 94.1(0.040) 93.9(0.058) 4.331(0.0080) 4.616(0.0105) 5.058(0.0114) 4.983(0.0050) 5.095(0.0071) 4.994(0.0080) A 93.1(0.054) 93.(0.05) 93.0(0.073) 94.1(0.044) 94.0(0.04) 93.8(0.057) 4.345(0.0085) 4.78(0.0104) 4.544(0.0110) 4.971(0.0054) 5.13(0.0097) 5.061(0.0089) (0.055) 4.43(0.0081) 9.3(0.056) 4.575(0.0084) 9.3(0.057) 5.064(0.0110) 9.0(0.054) 4.896(0.0103) 93.6(0.077) 5.083(0.0115) 93.1(0.086) 5.5(0.0117) 94.7(0.040) 4.963(0.0053) 94.8(0.053) 4.961(0.0057) 95.0(0.046) 94.5(0.050) 4.934(0.0070) 5.015(0.0086) 94.7(0.039) 94.4(0.064) 4.960(0.0099) 5.041(0.010) (0.05) 4.968(0.0088) 93.0(0.051) 4.870(0.0083) 93.5(0.061) 4.850(0.010) 9.8(0.040) 4.987(0.0106) 9.9(0.060) 5.08(0.015) 93.(0.085) 4.456(0.0119) 94.4(0.031) 4.996(0.0051) 93.9(0.044) 5.106(0.0056) 94.0(0.031) 94.0(0.051) 5.394(0.0071) 5.076(0.0087) 94.3(0.031) 94.5(0.07) 4.199(0.0066) 5.097(0.0105) (0.043) 5.39(0.0084) 9.3(0.057) 5.303(0.0091) 9.6(0.048) 4.904(0.01) 9.(0.05) 5.105(0.0108) 93.0(0.057) 5.174(0.013) 9.6(0.090) 4.657(0.014) 94.5(0.05) 4.996(0.0053) 94.0(0.036) 5.365(0.0056) 94.6(0.04) 5.478(0.0073) 94.4(0.043) 5.0(0.0070) 94.7(0.056) 5.078(0.0084) 94.5(0.081) 5.03(0.0100)

12 1 Boosrap Predicion Inervals for Time Series Models wih Heroscedasic Errors Sepahead Sample Size 1 00 Disr. Table 4 Simulaion Resuls for M4 A1 93.1(0.056) 93.(0.063) 9.7(0.053) 94.8(0.04) 94.5(0.050) 94.1(0.037) 4.665(0.0090) 4.716(0.010) 4.505(0.013) 4.734(0.0056) 4.838(0.0080) 4.678(0.0094) A 9.9(0.058) 93.0(0.066) 9.9(0.056) 94.4(0.044) 93.9(0.051) 93.9(0.03) 4.70(0.009) 4.731(0.011) 4.494(0.0117) 4.834(0.0060) 4.916(0.0078) 6.747(0.0084) (0.05) 4.803(0.0093) 9.8(0.060) 4.800(0.0096) 93.1(0.057) 4.940(0.0110) 9.1(0.063) 4.913(0.010) 9.3(0.055) 4.635(0.0096) 9.1(0.058) 4.733(0.0103) 94.8(0.040) 4.839(0.0058) 94.8(0.040) 4.89(0.0058) 94.(0.053) 93.9(0.037) 4.859(0.0088) 4.965(0.0085) 94.1(0.05) 93.8(0.036) 4.98(0.0104) 4.97(0.009) (0.056) 4.98(0.0097) 9.6(0.056) 4.880(0.0097) 93.1(0.067) 4.944(0.0096) 93.0(0.06) 4.919(0.0093) 93.0(0.054) 5.165(0.0099) 93.1(0.055) 5.10(0.0110) 94.8(0.043) 4.866(0.0065) 93.8(0.043) 4.896(0.0065) 94.4(0.054) 94.3(0.040) 5.075(0.009) 5.33(0.0079) 93.0(0.051) 93.9(0.038) 5.33(0.0090) 5.164(0.0087) (0.05) 5.155(0.0098) 9.1(0.05) 4.955(0.0098) 9.7(0.066) 5.038(0.0100) 9.(0.043) 4.819(0.0081) 9.4(0.054) 5.174(0.0085) 9.8(0.045) 4.963(0.0088) 94.1(0.035) 5.355(0.0068) 94.1(0.035) 4.995(0.0068) 94.4(0.047) 5.45(0.0094) 93.0(0.031) 5.33(0.0090) 93.9(0.045) 5.377(0.0086) 93.8(0.040) 5.444(0.0093)

13 Amjad Ali e al. 13 Disr. 1 Table 5 Simulaion Resuls for M5 n=00 n= 93.5(0.051) 93.7(0.054) 9.9(0.063) 4.783(0.0106) 4.984(0.0111) 5.030(0.0107) 94.7(0.038) 94.9(0.039) 94.6(0.033) 4.968(0.0076) 5.097(0.0080) 5.130(0.0077) (0.055) 4.914(0.0109) 94.8(0.034) 5.194(0.0075) 93.7(0.058) 94.1(0.063) 5.19(0.0115) 5.045(0.0110) 94.5(0.040) 94.6(0.039) 5.43(0.008) 5.30(0.0079) 5 9.9(0.056) 4.983(0.0114) 94.4(0.038) 5.38(0.0079) 93.(0.064) 5.096(0.01) 95.1(0.046) 5.75(0.0084) 93.3(0.06) 4.361(0.0116) 94.6(0.04) 5.10(0.0081) (0.061) 5.17(0.0118) 94.4(0.041) 5.441(0.0083) 9.7(0.069) 5.88(0.0130) 94.7(0.046) 5.(0.0097) 9.(0.066) 5.86(0.0114) 94.6(0.043) 5.030(0.0087) Sepahead Sepahead Disr. 1 Table 6 Simulaion Resuls for M6 n=00 n= 93.5(0.06) 93.6(0.070) 93.4(0.061) 4.653(0.011) 4.718(0.014) 4.761(0.0116) 94.8(0.043) 94.7(0.05) 95.3(0.051) 4.95(0.0083) 5.073(0.0091) 4.961(0.0087) 3 9.8(0.066) 4.614(0.0114) 94.8(0.045) 4.990(0.0086) 9.9(0.071) 5.04(0.019) 95.0(0.057) 5.164(0.0101) 9.6(0.063) 5.030(0.0117) 94.9(0.05) 5.161(0.0096) (0.065) 5.199(0.0119) 94.4(0.050) 5.33(0.0083) 91.9(0.076) 5.531(0.0136) 94.8(0.058) 5.413(0.0104) EXP 9.(0.070) 5.46(0.019) 95.3(0.060) 5.361(0.0096) (0.073) 91.7(0.08) 5.514(0.017) 5.716(0.0145) 94.5(0.057) 94.1(0.060) 5.517(0.0091) 5.85(0.0100) 91.1(0.075) 5.45(0.0134) 94.3(0.06) 5.461(0.0098)