Appl. Math. If. Sci. 10, No. 1, 267-271 (2016 267 Applied Mathematics & Iformatio Scieces A Iteratioal Joural http://dx.doi.org/10.18576/amis/100127 Akaike Iformatio Criterio ad Fourth-Order Kerel Method for Lie Trasect Samplig (LTS Ali Algari 1, ad Ahmad Almutlg 2 1 Statistics Departmet, Faculty of Sciece, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia 2 Mathematics Departmet, Faculty of Sciece, Qassim Uiversity, Qassim, Saudi Arabia Received: 7 Mar. 2015, Revised: 20 May 2015, Accepted: 12 Aug. 2015 Published olie: 1 Ja. 2016 Abstract: Parametric ad oparametric approaches were used to fit lie trasect data. Differet parametric detectio fuctios are suggested to compute the smoothig parameter of the oparametric fourth-order kerel estimator. Amog the differet cadidate parametric detectio fuctios, the researcher suggests to use Akaike Iformatio Criterio (AIC to select the most appropriate oe of them to fit lie trasect data. More specifically, four differet parametric models are cosidered i this research. Where as two models were take to satisfy the shoulder coditio assumptio, the other two do ot. Oce the appropriate model is determied, it ca be used to select the smoothig parameter of the oparametric fourth-order kerel estimator. As the researcher expected, this techique leads to improve the performaces of the fourth-order kerel estimator. For a wide rage of target desities, a simulatio study is performed to study the properties of the proposed estimators which show the superiority of the resultig proposed fourth-order kerel estimator over the classical kerel estimator i most cosidered cases. Keywords: Akaike iformatio criterio, Maximum likelihood estimatio, Fourth-order kerel estimator, Expoetial model, Halformal model, Weighted Expoetial model, Reversed Logistic model. 1 Itroductio The smoothig parameter h of the fourth-order kerel estimator plays a vital role i its performace. Its performace becomes acceptable compared to the classical kerel estimator whe the method of determiig the smoothig parameter is chose correctly. I other words, there is a scope to improve the performaces of the fourth-order kerel estimator by cosiderig a suitable parametric method to determie h. mor survey give i Che [1, Clie ad Hart [2, Eidous [, Eidous ad bal-masri [4, Cowlig ad Hall [5, Eidous [6, Eidous ad Alshakhatreh [7, Gasser ad Muller [8, Gasser et al. [9, Eidous [10, Mack ad Quag [11, Wad ad Joes [12, Zhag ad Karuamui [1, Eidous [14. The four parametric models will be agai cosidered i this paper. However, choosig a suitable parametric model for the data is ot depedet o guesswork, istead the (AIC will be used for this purpose i this paper. To ivestigate the statistical properties of the resultat estimator, a simulatio study is performed lately i research Akaike Iformatio Criterio: Burham ad Aderso [16 illustrated that the (AIC is a quatitative method to select a suitable model for data. Iformatio theory ad a extesio of the maximum likelihood priciple usig AIC give i Akaike [17. They defied the AIC as: AIC= 2logL+2q, (1 where log L is the atural logarithm of the likelihood fuctio evaluated at the MLE of the model parameters ad q is the umber of parameters i the model. I this paper, the formulas for AIC will be computed for four parametric models. These parametric models are: Expoetial model f(x= 1 θ exp ( x θ, x>0, (2 Half-ormal model ( 1 x 2 f(x= exp 2Πσ 2 2σ 2, x>0, ( Correspodig author e-mail: ahalgari@kau.edu.sa Natural Scieces Publishig Cor.
268 A. Algari, A. Almutlg: Akaike iformatio criterio... Weighted Expoetial model f(x= 2θ exp( θx(2 exp( θx, x>0, (4 log Reversed Logistic model 2θexp( θx, x>0, (5 log(1+2exp( θx Also, we derived the smoothig parameter of the fourthorder kerel estimator based o each model separately. If X 1, X 2,..., X is a radom sample of size represetig the perpedicular distaces ad followig oe of the above pdf models f(x The correspodig smoothig parameters for each model are, h exp = 1.4408(θ1 ( 1, where( ˆθ 1 =x (6 h half = 1.0066(σ( 1 9,where ˆσ = x2 i h weight = 0.8689( 1 θ ( 1 7,where ˆθ = 7 6x h Rever 1 =.5400( (,where 1 ˆθ 4 = 1.078 (9 θ 4 x I the followig sectio, we derived the AIC that correspods to each model. 2 Akaike Iformatio Criterio for Some Parametric Models For local likelihood desity estimatio i lie trasect samplig see Barabesi [18 ad Barabesi [19 Let X 1, X 2,..., X be a radom sample of size perpedicular distaces followig the expoetial model. The likelihood fuctio is ( 1 1 L(θ 1 = exp( θ 1 θ 1 x i By takig the atural logarithm of both sides, we obtai logl(θ 1 = 1 θ 1 (7 (8 (10 x i logθ 1 (11 Multiply both sides of the last equatio by ( -2 we obtai 2logL(θ 1 = 2 θ 1 x i + 2logθ 1 (12 Sice there is oe parameter that eeds to be estimated, q=1. Therefore, the AIC for the expoetial model is Note that xis the ML estimator for θ 1. I the same way we ca derive the AIC for the other three models, which are give below: * For the Half-ormal model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are ˆσ L ( σ 2 ( ( 1 = exp 2Πσ 2 1 2σ 2 logl ( σ 2 = log 2Π logσ 1 2σ 2 This gives the AIC by x 2 i AIC half = 2log2+log [ 2π [ ˆσ 2 + +2 = 2log2+log 2π x2 i + +2.. (14 x 2 i (15 (16 where x2 i is the ML estimator of σ 2. * For the weighted expoetial model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are exp( L(θ = ( 2θ θ i (2 exp( θ x, x logl(θ =log( 2θ θ x i +log[ (2 exp( θ x, The AIC is AIC weight = 2log( 2 ˆθ + 2 ˆθ 2 log( 2 exp ( ˆθ x + 2. (17 (18 (19 where ˆθ is the ML estimator of θ, which does ot exist i closed form ad a umerical method such as the Newto- Raphso method is eeded to obtai the value of θ. * Fially, for the reversed logistic model, the likelihood fuctio ad the atural logarithm of the likelihood fuctio are L(θ 4 = ( 2θ4 log exp( θ 4 x i logl(θ 4 =log( 2θ4 log = log[ The AIC is (1+2exp( θ 4 x θ 4 x i (1+2exp( θ 4 x. (20 (21 AIC exp = 2ˆθ 1 x i+ 2log ˆθ 1 + 2 = 2(1+logx+2. (1 AIC Revr = 2log ( 2 ˆθ 4 + 2log(+ 2 ˆθ 4 x 2 log( 1+2exp ( ˆθ 4 x + 2. (22 Natural Scieces Publishig Cor.
Appl. Math. If. Sci. 10, No. 1, 267-271 (2016 / www.aturalspublishig.com/jourals.asp 269 where ˆθ 4 is the ML estimator of θ 4. Agai, a closed form for the value of ˆθ 4 does ot exist i closed form ad a umerical method is eeded to obtai it. Simulatio Study Desig we performed a simulatio study to ivestigate the properties of the proposed estimator. This estimator is the fourth-order kerel whose parameter h ca be computed by cosiderig ay reasoable parametric model. Oe of the four parametric models metioed i Sectio (.2 of this chapter is ow used to compute h. The AIC is a criterio for choosig the most appropriate model based o the data. The resultig estimator is deoted by ˆf(0. The proposed estimator ˆf(0 is computed as follows: 1.Simulate the perpedicular distaces X 1, X 2,..., X from oe of the target models metioed i Sectio (2.6. 2.For the simulated perpedicular distaces, compute the value of h that correspods to the expoetial, half-ormal, weighted expoetial, ad reversed logistic models. The values of these smoothig parameters are deoted by h exp, h hal f, h weight, ad h Rever, respectively..compute the AIC for the expoetial, half-ormal, weighted expoetial, ad reversed logistic models. 4.Select the model with the smallest AIC as a referece model to compute the value of h based o the selected model. The formula for computig h for each model are give i Sectio (.2. 5.Compute the value of the fourth-order kerel estimator based o the selected h of step (4 ad based o the perpedicular distaces X 1, X 2,..., X. The data are simulated from the 16 models that were give i Sectio (2.6 with the same values of, β, ad ϖ. For a simple compariso, the results of a classical kerel estimator ˆf k (0 are also preseted. The relative bias (RB, relative mea error (RMX, ad the efficiecy (EFF of f k (0 with respect to ˆf k (0 are give i Tables (.1-.4. Note that, EFF= MSE( ˆf k (0 MSE( f k (0 (2 as illustrated i Sectio (2. 4 Results Depedig o the simulatio results of Tables (.1-.4, several coclusios ca be draw by ispectig the results with regard to,, ad EFF. 1.It is obvious that the RBs that are associated with the proposed estimators fk (0 are smaller tha (i their magitude the correspodig RBs that are associated with the classical kerel estimator ˆf k (0. 2.The RMEs for the two estimators fk (0 ad ˆf k (0 decrease whe the sample size icreases. This is a strog idicatio of the cosistecy of these estimators..compared to fk (0, the performace of the classical kerel estimator seems to be reasoable for the EP model with β =2 at which the smoothig parameter of ˆf k (0 is computed uder this model (i.e., uder the half- ormal model. For the two cases, the HR model with (β,ω=(2.5,10 ad for the BE model with large value of β, the classical kerel estimator performs better tha fk (0. However, the efficiecies i these cases are aroud 1, which idicates that the performaces of the two estimators are similar. 4.With the exceptio of the three cases metioed i (, the performace of fk (0 is better tha that of ˆf k (0. I fact, out of the 16 target models, there are 12 cases i which fk (0 beats ˆf k (0. 5.I geeral, the performace of the fourth-order kerel estimator fk (0 is very good for almost all cosidered target models. I additio, for the cases i which ˆf k (0 seems to be better tha fk (0, the gai of efficiecy is ot sigificat compared with the large efficiecies of fk (0 over ˆf k (0 i certai cases. For example, the efficiecy of fk (0 for the model HR with (β,ω,=(1,5,200 is 4.086, which meas that the performace of ˆf k (0 becomes the same as that of fk (0 for a sample size of approximately 4.086 200 =800. 6.Fially, we ca say that Tables (.1-.4 demostrate clearly that there is a sigificat improvemet whe applyig the estimator fk (0 istead of the classical kerel estimator ˆf k (0. Refereces [1 Che, S. X. (1996. A kerel estimate for the desity of a biological populatio by usig lie trasect samplig. Applied Statistics, 45, 15-150. [2 Clie, D. B. H., & Hart, J. D. (1991. Kerel estimatio of desities of discotiuous derivatives. Statistics, 22, 69-84. [ Eidous, O. M., (2009. Kerel method startig with half-ormal detectio fuctio for lie trasect desity estimatio. Commu. i Statist.-Theory ad Methods. 8, 266-278. [4 Eidous, O. M.,& Al-Masri, A. (2010. Fourth-order kerel method usig lie trasect samplig. Advaces ad Applicatios i Statistics, 19, 65-79. [5 Cowlig, A., & Hall, P. (1996. O pseudodata method for removig boudary effects i kerel desity estimatio. Joural of the Royal Statistical Society, Ser. B, 58, 551-56. [6 Eidous, O. M., (2012. A ew kerel estimator for abudace usig lie trasect samplig without the shoulder coditio. Joural of the Koria Statistical Society, 41, 267-275. [7 Eidous, O. M., & Alshakhatreh, M. K. (2011. Asymptotic ubiased kerel estimator for lie trasect samplig. Natural Scieces Publishig Cor.
270 A. Algari, A. Almutlg: Akaike iformatio criterio... Table.1: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from expoetial power (EP detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = 1.0-0.16 0.1 1.000-0.292 0.0 1.000-0.267 0.27 1.000 fk (0 ω = 5.0-0.124 0.21 1.429-0.100 0.189 1.600-0.075 0.144 1.894 ˆf k (0 β = 1.5-0.14-0.069 1.000-0.121 0.155 1.000-0.105 0.124 1.000 fk (0 ω =.0-0.069 0.217 0.885-0.070 0.18 0.850-0.099 0.19 0.891 ˆf k (0 β = 2.0-0.07 0.15 1.000-0.066 0.12 1.000-0.042 0.091 1.000 fk (0 ω = 2.5-0.02 0.165 0.925-0.09 0.114 1.07-0.025 0.076 1.197 ˆf k (0 β = 2.5-0.050 0.151 1.000-0.07 0.115 1.000-0.01 0.088 1.000 fk (0 ω = 2.0-0.016 0.14 1.076-0.005 0.112 1.025-0.004 0.077 1.19 Table.2: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from hazard-rate (HR detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = 1.0-0.592 0.651 1.000-0.62-0.648 0.625 1.000 0.59 1.000 fk (0 ω = 5-0.414 0.45 1.496-0.57 0.72 1.679-0.292 0.0 1.954 ˆf k (0 β = 1.5-0.514 0.524 1.000-0.490 0.495 1.000-0.448 0.450 1.000 fk (0 ω = 25-0.154 0.21 2.258-0.105 0.168 2.96-0.058 0.110 4.086 ˆf k (0 β = 2.0-0.275 0.299 1.000-0.25 0.251 1.000-0.195 0.205 1.000 fk (0 ω = 14 0.007 0.18 1.6 0.02 0.145 1.72 0.01 0.128 1.605 ˆf k (0 β = 2.5-0.128 0.186 1.000-0.100 0.14 1.000-0.070 0.096 1.000 fk (0 ω = 10 0.045 0.229 0.814 0.020 0.152 0.879 0.016 0.1 0.726 Table.: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from the beta (BE family detectio fuctio. Parameter Estimator =50 =100 =200 ˆf k (0 β = 1.5-0.169 0.207 1.000-0.157 0.184 1.000-0.11 0.152 1.000 fk (0 ω = 1-0.140 0.196 1.055-0.19 0.166 1.106-0.119 0.19 1.094 ˆf k (0 β = 2-0.196 0.228 1.000-0.171 0.196 1.000-0.150 0.166 1.000 fk (0 ω = 1-0.172 0.227 1.005-0.152 0.198 0.991-0.145 0.160 1.02 ˆf k (0 β = 2.5-0.202 0.24 1.000-0.189 0.210 1.000-0.168 0.182 1.000 fk (0 ω = 1-0.14 0.24 0.961-0.168 0.218 0.965-0.179 0.189 0.962 ˆf k (0 β = -0.221 0.252 1.000-0.20 0.220 1.000-0.182 0.19 1.000 fk (0 ω = 1-0.154 0.258 0.977-0.161 0.21 0.95-0.179 0.209 0.925 Table.4: RB ad RME of the classical ad fourth-order kerel estimators whe the perpedicular distaces are simulated from the beta (BE family detectio fuctio. Parameter Estimator =50 =100 =200 β = 0.6 ˆf k (0 b=1-0.252 0.274 1.000-0.220 0.25 1.000-0.192 0.204 1.000 fk (0 ω = 1-0.10 0.25 1.166-0.094 0.209 1.128-0.05 0.148 1.72 ˆf k (0 b=5-0.147 0.19 1.000-0.114 0.152 1.000-0.100 0.120 1.000 fk (0 ω = 1-0.10 0.206 0.97-0.098 0.15 0.994-0.095 0.111 1.087 ˆf k (0 b=10-0.10 0.168 1. -0.084 0.12 1. -0.064 0.101 1. fk (0 ω = 1-0.060 0.17 0.970-0.061 0.115 1.14-0.045 0.085 1.19 ˆf k (0 b=0-0.049 0.149 1. -0.041 0.114 1. -0.09 0.099 1. fk (0 ω = 1-0.014 0.142 1.052-0.011 0.101 1.125-0.015 0.087 1.16 Natural Scieces Publishig Cor.
Appl. Math. If. Sci. 10, No. 1, 267-271 (2016 / www.aturalspublishig.com/jourals.asp 271 Commu. i Statist.-Theory ad Methods, 40(24, 45-46. [8 Gasser, T., & Muller, H. G. (1979. Kerel estimatio of regressio fuctios. I: Gasser, T., & Roseblatt, M. (Eds, Smoothig techiques for curve estimatio. Lecture otes i mathematics, Vol. 757, Sprig, Heidelberg, 2-68. [9 Gasser, T., Muller, H. G., & Mammitzsch, V. (1985. Kerels for oparametric curve estimatio. Joural of the Royal Statistical Society, Ser. B, 47, 28-256. [10 Eidous, O. M. (2005a. O improvig kerel estimator usig lie trasect samplig. Commu. I statistic.- Theory ad Methods, 4, 91-941. [11 Mack, Y. P. & Quag, P. X. (1998. Kerel methods i lie ad poit trasect samplig. Biometrics, 54, 606-619. [12 Wad, M. P. & Joes, M. C. (1995. Kerel smoothig. Lodo: Chapma & Hall. [1 Zhag, S., & Karuamui, R. J. (1998. O kerel desity estimatio ear edpoits. Joural of Statistical Plaig ad Iferece, 70, 01-16. [14 Eidous, O. M., (2011a. Variable locatio kerel method usig lie trasect samplig. Eviromets, 22, 41-440. [15 Ababeh, F. ad Eidous, O. M. (2012. A weighted expoetial detectio fuctio model for lie trasect data. Joural of Moder Applied Statistical Methods, 11(1, 144-151. [16 Burham, K. P., & Aderso, D. R. (1976. Mathematical models for oparametric ifereces from lie trasect data. Biometrics, 2, 25-6. [17 Akaike, H. (197. Iformatio theory ad a extesio of the maximum likelihood priciple. A iteratioal Symposium of Iformatio Theory, 2ded, Budapest, Hugary, 267-281. [18 Barabesi, L. (2000. Local likelihood desity estimatio i lie trasect samplig. Evirometrics, 11, 41-422. [19 Barabesi, L. (2001. Local parametric desity estimatio method i lie trasect samplig. Metro, LIX, 21-7. Ahmad Almutlg Takig his Diploma i Risk Maagemet from Europea Busiess (Learig Academy i the UK [October December 2010.I Dec.2014 takig the master i Statistics from Kig Abdulaziz Uiversity, Jeddah. Thesis Topic Akaike Iformatio Criterio ad Fourth-Order Kerel Method for Lie Trasect Samplig. Ali Algari has obtaied his PhD degree at 201 from School of Mathematics ad Applied Statistics, Wollogog Uiversity, Australia. He is a Ivestigator ad Data Aalyst at ceter of survey methodology ad data aalysis,wollogog. Australia. He is a member, Saudi Associatio of Mathematics ad Applied Statistics. He is presetly employed as assistat professor i Statistics Departmet, Faculty of Sciece, Kig Abdulaziz Uiversity, Jeddah, Saudi Arabia. Natural Scieces Publishig Cor.