IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 Estimation o f the Two Parameters for Generalized Rayleigh Distribution Function Using Simulation Technique A. A .AL- Naqeeb, A. M .Hame d College of Health, of Medical Technology, Baghdad Mathematical Department College of Education, Ibn Al-Haitham, Unive rsity of Baghdad Abstract In this p aper, suggested formula as well a conventional method for estimating the two- p arameters (shap e and scale) of the Generalized Rayleigh Dist ribution was p rop osed. For different sample sizes (small, med ium, and lar ge) and assumed several contrast s for the two p arameters a percentile estimator was been used. M ean Square Error was imp lemented as an indicator of p erformance and comp arisons of the p erformance have been carried out through data analysis and computer simulation between the suggested formulas versus the st udied formula accordin g to the app lied indicator. It was observed from the results that the suggested method which was p erformed for the first time (as far as we know), had highly advantage than the st udied method, since the whole suggested outcomes of statistics in the suggested method are registered. Introduction During the last century , vast activities have b een observ ed in generalizing the distributions. These distributions were formulated by st atisticians, mathematicians, and engineers t o mathematically model or represent certain behavior. R ecently Surles and Pad gett (2001) introduced two-p arameter Burr Typ e X distribution, which can also be d escribed as Generalized Rayleigh Dist ribution and It was observed that this p articular skewed distribution can be used qu ite effectively in analy zing lif etime d ata [1]. Raqab and Kundu; (2003) considered this distribution and discussed its different p rop erties and emp loy ed different methods of estimators. It was concluded that the two-p arameter Generalized Rayleigh Dist ribution is a p articular member of the generalized Weibull Dist ribution, originally p rop osed by Mudholkar and Srivastava [2]. Rayleigh distribution, which is a sp ecial case of Weibull distribution, is widely used to model events that occur in different fields such as medicine, social and natural sciences [3]. The Generalized Rayleigh with two p arameters (shap e and scale) (GR) distribution has the dist ribution function as following:-  ( ) 2 ( ; , ) 1 tF t e       t, α and λ > 0. ---------------------- (1) Here α and λ ¸ are the shap e and scale p arameters resp ectively, and the two-p arameter GR distribution will be denoted by GR (α; λ). Therefore, GR distribution has the density function 2 2 2 ( ) ( )1( ) 2 (1 ) t t tf t e e      ------------------------ (2) IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 And t he surv ival funct ion 2( ) ( ; , ) 1 (1 ) t S t e       Where the hazard function is ( ; , ) ( ; , ) 1 ( ; , ) f t h t F t         2 2 2 2 ( ) ( ) 1 ( ) 2 (1 ) 1 (1 ) t t t e e e               It was observed that for α < 1/2, the p robability density function (p.d.f) of a GR distribution is a decreasing function and it is a right skewed unimodal fun ction for α > 1 /2. And the hazard function of a GR distribution can be either batht ub type or an increas ing function, depending on the shap e parameter α. for α < 1/2, the hazard functions of GR (α, λ) are batht ub type and for α > 1/2, it has an increasing h azard function [2]. The main aim of this p ap er is to study how the estimator of the different unknown p arameters behaves for different sample sizes and for different p arameter values. We mainly comp are p ercentiles estimator between the suggested formulas versus the st udied methods accordin g to the ap p lied indicator by using extensive simulation techniques. The rest of the p ap er is organized as follows, a briefly descrip tion of the p ercentile estimator methods (PCE) and their imp lementations including the conventional and the suggested methods. Followed by an emp irical work, as well as the simulation results and conclusion. Estimators Based on Percentiles If the data come from a distribution function which has a closed form, then it is quite natural to est imate the unknown p arameters by fitting st raight lin e to t he theoretical percentile p oints obtained from the distribution function and the samp le p ercentile p oints. This method was originally p rop osed by Kao (1958, 1959) and it has been used quite successfully for Weibull distribution and for the generalized exp onential distribution. In this p aper, we app ly the same technique for the GR distribution [2]. An est imator is statist ic that specifies how t o use the samp le d ata to estimate an unknown p arameter of the p op ulation [4]. In the following sections three estimation p rocedures are considered, the p ercentiles estimators, and compare their p erformances through numer ical simulation for different sample sizes and for different p arameters values. In p ercentile methods t he Generalized Rayleigh Dist ribution has the exp licit distribution function, therefore in this case the unknown p arameters α and λ ¸ can be estimated by equating the sample p ercentile p oints with the p op ulation p ercentile p oints and it is known as the p ercentile method [5]. Among the most easily obtained estimators of the p arameters of the Weibull distribution are the gr aphical app roximation to the best linear unbiased estimators. It can be obtained by fitt ing a st raight line to the theoretical p oints obtained from the distribution function and the sample p ercentile p oints. In case of a GR distribution also it is p ossible to use the same concept to obtain the estimators of α and λ based on the percentiles, because of the st ructure of its distribution function, when both the parameters are unknown [6]. Since   2( ) ( ; , ) 1 ; 0, x F x e x         IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 Therefore   1 2 2 1 ln 1 ( ; , )X F x            -------------------------------- (3) If Pi denotes some estimate of F ( ( )ix ; α, λ), then the estimate of α and λ can be obtained by minimizing (2) as followin g:- 2 2 2 (1/ ) ( ) 1 ln(1 ) n i i i x p       --------------------------- (4) 2 2 (1/ ) 3 (1/ ) ( ) 1 2 ln(1 ) 2 ln(1 ) n i i i i x p p                  Not e that (4) is a non-linear function and it has to be min imized usin g some non- linear optimization technique. We call the corresp onding estimators as the p ercentile estimators PCE's. Several estimators of Pi can be used here and in this p aper, we mainly consider Pi = i/n+1, which is the exp ected value of F (T (i)). Wher e 1 i i P n   represent t he studied formula And ( ( )) 1 i i E F t n   the exp ected value Which 1( ) ( ) 1 i i E t F n    F(t) represents cumulative distribution function (c.d.f)for distribution, and ( )iE t named (inverse p robability of the cumulative samp ling d istribution) [7]. Then the suggested formula Pi wi ll be [8] as follows:- 0.5 0.5 i i P n    -Algorithms of the Suggested Methods The cumulative distribution function of the Generalized Rayleigh d istribution can be written in the form:- 2 ( ) 1( ) ( ) t F t e       ------- --------------- (5) Since the model (5) involves α and λ in a nonlin ear way as shown in (4) so it can be transformed and taking its logarithms t o the base e as follows:- 2 1( ) 1 ( ) t e F t        2 1( ) 1 ( ) t e F t        1 2( ) ln(1 ( ) )t F t       Therefore 11 ln 1( ( ) )t F t        ---------------------------------------- (6) Using un iform d istribution and generating U where IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 1 [0,1] 0 otherwise t U     Since U = 1 - U in-case of generating continu es uniform r andom v ariable, t hen 1 1 ( ) 1 ( )F t F t      --------------------- - (7) Takin g the logarithm, for the both side of equation [5], then the followin g equation will p roduce: 1 1 ln( ) l n( ) ln ln ( ) 2 t F t           ----------------------------- (8) Thus t he equation (8) is intrinsically lin ear for m, in whi ch ln( )iiy t   , 1 ln ln ( ) 2 ix F t       and 0 1 ln( )     In equation (8), the slop e is a constant and equals to 1, which indicates that Δ x = Δ y, using si mple lin ear r egression equation, then 0 1 + + i i iY x e  Where 1 1  Emp loying the initial valu e of α & λ in the right sid e of (8) with substitut ion of the generating un iform v alues in ( ) iF t u   , to obtain the left side exp(ln( ))t t    And if error is added to t his model, then ii it t e    Since that E (e) = 0, where e ~ exp (1), so the errors are independent and uncorrelated [9].In order to m ake a comp arison b etween the two methods (suggested and st udied methods) for PCE the same p rocedure in finding equation (8) wi ll b e repeated twice time and the equation of the st raight lin e will p roduce. Throu gh out solving them the value of   &   will be founded. Hence ln( ) ln( )i i i it e t y     1 & ln[ ln( )] 2 i i x u  And t he est imator 0 1 ln          represents the suggested method which is obtained for the first time (as far as we know). -Percentile Esti mator (studied) Accordin g to equation (8) we have 1 1 ln ln ln ln 2 1 i t n                  1 ln ln ln ln 2 2 1 i t n                  1 2 2 ln 2 ln ln ln 1 i t t n                   IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 ln ln ln 2 ln ( ) 1 21 i t t n                   1 2 1 2( ) exp ln ln 2 ln ( ) 1 i t t n                       ----------------------- (9) Wher e: 11 1t t e    22 2t t e    And t he same method is used to estimate ( )   and obtain: 1 2 1 1 ( ) exp ln ln ln ( ) 2 1 2 i t t n                    -------------- (1 0) -Percentile Esti mator (suggested) ( )   & ( )   in su ggested formu la of Pi will be as follows:- 1 0.5 ln ln ln ln 1 2 0.5 i t n                    1 0.5 ln ln ln ln 2 2 0.5 i t n                    1 2 0.5 2 ln 2 ln ln ln 0.5 i t t n                    0.5 ln ln ln 2 ln ( ) 1 20.5 i t t n                    1 2 1 2 0.5 ( ) exp ln ln 2 ln ( ) 0.5 i t t n                        ------------------ (11) Where: 11 1t t e    And 22 2t t e    1 2 1 0.5 1 ( ) exp ln ln ln ( ) 2 0.5 2 i t t n                      ------------------- (12) Therefore, it can be con clud ed that the same procedure can b e ap p lied for obtainin g the estimators (α & λ) of GR distribution to other est imation methods such as (Ordinary Least Square (OLS), M aximum Likelihood (M LH)…) as in example in case o f OLS the obta inin g equations will b e flowi n g: Accordin g to equation (8) we have IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 1 1 1 ln ln ln ln 2 1 i t n                   The same method is used to est imate ( )   & ( )   , hence 1 2 1 2( ) exp ln ln 1 2 ln ( ) 1 i t t n                        1 2 1 1 ( ) exp ln ln 1 ln ( ) 2 1 2 i t t n                     And ( )   & ( )   in suggested formula of Pi wi ll b e 1 2 1 2 0.5 ( ) exp ln ln 1 2 ln ( ) 0 .5 i t t n                         1 2 1 0.5 1 ( ) exp ln ln 1 ln ( ) 2 0.5 2 i t t n                       Em pirical work One of the most imp ortant app lications of comp uter science is Computer simulation. It is an att empt to model a real-life on a comp uter so t hat it can be studied to see how t he sy stem works. By changing variables, p redictions may be made about the behavior of the sy stem Computer simulation has beco me a usefu l p art of modeling many systems in economics, finance, and several app lications [10]. Simulation app roaches offer great opp ortunities for working out p robabilities, confiden ce intervals and similar concepts [11].T his analysis may be done, sometimes, through analytical or numerical methods, but the model may be too comp lex to be dealt with. Essentially , simulation p rocess consists of building a comp uter model that describes the behavior of a sy st em and exp erimenting with this comp uter model to reach conclusions that supp ort decisions [12]. Sometimes, it is not feasible or p ossible, to build a prototype, yet we may obtain a mathematical mod el d escribing, throu gh equations and constraints, the essential behavior of the system. In such extreme cases, we may use simulation to replicate real world st udies that cannot be done, simulation exercises may encounter st atist ical p itfalls that degrade their p erformance, or fail to take advantage of the opp ortunities st atistics can provide for controlling si mulation error and p roducing st atist ically reliab le results [12].In order to make comp arison of the two methods of PCE for the p arameters of the Generalized Rayleigh Dist ribution which were st udied in the p revious chap ters to reach into the best estimated method of the shap e p arameter and scale p arameter we make a simulation p rototype and p rovide assump tion of many cases which it can b e exist ed in real world and use the basic step p rocess in any simulation exp eriment once we have estimated the corresp onding simulation model. Algorithms steps First ste p: - sp ecified the assumed values by choosing differ ent sample size of Generalized Rayleigh distribution, such as small sample size (n=20) and medium sample size (n=50) and lar ge sa mple size (n=100).And then choosing the v alues of assump tion parameters (α, λ) in each sev eral contrast s and choosing for the initial values of the two p arameters (shap e and scale) parameters as shown in. IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 Assume d contrast paramete rs [Introduced by the researche r]. Se cond ste p:- Generation of data which include :- -Generated the random data which was taken from the uniform distribution in the interval (0, 1) using Excel, and SP SS, soft ware computer package. - The gener ation of errors for all d ata and in methods t he random errors hav e been gener ated using the standard exp onential distribution instead of normal distribution which has been used in conventional methods introduced by Gupta and Kundu. Thi rd ste p:- This step contains the following:- -Using the same valu e of t  for three methods and apply ing the equation 11 ln(1 )t U       as mentioned in [5]. -findin g the time event (t) by using the equation ii it t e    Where i=1, ------, n -The values of   &   of the generalized Rayleigh distribution can be determined accordin g to the estimator of each method usin g the equations (9), (10), (11), and (12) resp ectively. Fourth ste p:- smoothing the obtained values -In this step the iteration of data will be rep eated 100 times to generate a new different error, so we obtain 100 value of  , and 100 value of   for each contrast . Then the mean of each case will b e calculated to find the estimated α and λ. Fifth ste p:- In this step the followin g comparison indicator will b e emp loyed to make a comparison between different methods by M ean Square Error (MSE). Results and Conclusion As a consequence for p ractical work and taking the mean square error as the indicator of p reference between the different est imator methods, the following r esults are obtained:- 1-For the conventional methods and for different sample sizes the following results are obtained :- (i)S mall sample size (n=20) For the assumed contrast p arameters (2, 1), (1, 2), (2, 2) and (1, 1) the PCE estimator method was given the best results. (ii)Medium sample size (n=50). IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 α λ 1 1 1 2 2 1 2 2 For the assumed contrast p arameters (2, 2), (2, 1), (1, 2), and (1, 1) the PCE estimator method was given the best result. (iii)Large sample size (n=100). For the assumed contrast p arameters (1, 1), (2, 2), (1, 2), and (2, 1) the PCE estimator method was given the best result. 2-For t he suggested method t he following results are obtained:- (i)S mall sample size (n=20) For the assumed contrast p arameters (1, 1), (2, 2), (1, 2), and (2, 1), the PCE estimator method was given the best result. (ii)Medium sample size (n=50) For the assumed contrast p arameters (2, 1), (1, 1), (1, 2) and (2, 2) the PCE estimator method was given the best results. (iii)Large sample size.(n=100) For t he assumed contrast p arameters (2, 1), (2, 2), (1, 1), and (1, 2) the PCE estimator method was given the best results. 3-The comparison between the studied and suggested methods are summarized as follows :- (i) S mall sample size (n=20) For the assumed contrast p arameters (1, 1) and (2, 2) the PCE (suggested) estimator method was given the best results in studied. For the assumed contrast p arameters (2, 1), and (1, 2) the PCE (st udied) estimator method was given the best results in suggested method. (iii)Medium sample size (n=50) For t he assumed contrast p arameters (2, 2), (1, 1), and (2, 1) the PCE (st udied) estimator method was given the best results in suggested method. For the assumed contrast p arameters (1, 2), the PCE (suggested) estimator method was given the best results in studied method. (iv)Large sample size.(n=100) For the assumed contrast p arameters (2, 2), (1, 1), (1, 2), and (2, 1) the (suggested) PCE estimator method was given the best results in studied methods. 4-It can be mentioned that when the samp le size increased the mean square error decr eased. 5-It can be noticed that when the assumed values are equal, the values of   and   are equal also, and that consist ent with the gener al mathematical concept. 6-In this work t he white nose error is generated in Rayleigh Dist ribution and it was followed the dist ribution and gave best results. The results of simulation (estimation of scale and shap e p arameters) of the generalized Rayleigh d istribution for studied and suggested methods for different sample size (n=20, 50, and100) are list ed in the table (1) Re ferences 1. Surles, J. G. and Pad gett W. J., (2005), "Some p rop erties of a scaled Burr typ e X distribution", Journal of Statist ical Plannin g and Infer ence, 128: 271-280. 2. D.Kundu, D. and Raqab M ., (2005)," Generalized R ayleigh Dist ribution: Different M ethods of Estimations", Comp utational Statist ics & Data Analysis, 49: 18. 3. M ahdi1, S. and Cenac M ., (2006), "Estimatin g and Assessing the Parameters of the Logist ic and Rayleigh Dist ributions from Three M ethods of Estimation”, Journal of M athematical Comp uter Science, 13 : 25-34. IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 4. M cclave, T., 1990,"Probability and Statistics for Engineers", 3rd Edition, Pws-Kent p ublishing comp any, Bost on. 5. Gupt a, R. D. and Kundu D., (2007), " Generalized Exp onential Dist ribution: Existing results and some recent developments ", Journal of Statistical Plannin g and Inference, 137(11): 3537-3547. 6. Gupt a, R. D. and Kundu D., (2001), " Generalized Exp onential Dist ribution: Different methods of estimation ", Journal of Statist ical Computations and Simulations, 69(4): 315-338. 7. Al –Naqeeb, A., (1990)," Est imation of the Residual Variance in Linear M odels within Cases Normal Dist . And Non-Normal dist.", Colle ge of Administ ration & Economics- University of Al- M ustansiriy ah. 8. Al –Naqeeb, A., (2007)," Estimation some non-Linear Typ e by Using M in Ch square method for Binary Data by simulation method", Resear ch Journal of Al epp o University , Basic Science Series. 9. Al –Naqeeb, A., 1993,"M easures The Coincidence Power of the Two Random Variables (Dependant And Residu al) In The Linear Regr ession M odels By Using The Probability Plot M ethod" ,5 th international conference of st atist ical scien ces, Baghd ad, Iraq. 10. "Simu lation ", (2009), From Wik ipedia, the free encyclop edia, Wikimedia Foundation, Inc., All text is availab le under the terms of t he GNU Free Documentation License . 11. M ichael Wood, (2005),"The Role of Simulation App roaches in Statist ics", University of Port smouth, U.K., Journal of Statist ics Education, 13(3). 12. Insua, D. R. and etal, (2005), "Simulation in Indust rial Statist ics ", Statist ical and App lied M athematical Sciences Inst itute, Technical Report, Research Triangle Park, NC 27709-4006, PO Box 14006, available at www.samsi.info. IBN AL- HAITHAM J . FO R PURE & APPL. SC I. VO L.22 (4) 2009 Assumed Parameter Estimators Indicator Samp le M ethod α λ     M SE 1 1 0.130645 0.130645 3.10325 1 2 0.205795 0.346934 2.446409 2 1 0.260983 0.181641 1.614332 PCE st udied 2 2 0.125566 0.177578 2.594462 1 1 0.13282 0.13282 1.456683 1 2 0.184391 0.359877 2.504632 2 1 0.221384 0.158 2.702558 20 PCE suggested 2 2 0.127188 0.179871 2.051962 1 1 0.168913 0.168913 2.667632 1 2 0.141036 0.283179 2.082187 2 1 0.209621 0.146927 1.617342 PCE st udied 2 2 0.149932 0.212036 1.308038 1 1 0.1721 0.1721 2.473099 1 2 0.144298 0.431658 2.478395 2 1 0.210311 0.149445 1.817146 50 PCE suggested 2 2 0.153072 0.2164759 3.208128 1 1 0.194146 0.194146 1.593123 1 2 0.118901 0.237803 2.388022 2 1 0.217849 0.152913 2.73559 PCE st udied 2 2 0.1536 0.2172234 2.011512 1 1 0.196815 0.196815 1.753823 1 2 0.119837 0.240805 2.22064 2 1 0.217955 0.154118 1.214269 100 PCE suggested 2 2 0.155098 0.2193424 1.385011 Tables (1) Esti mation of S cale and Shape Paramete rs of GED For suggested Table (4-8) The Results of Si mulation methods 2009) 4 (22 المجلد مجلة ابن الھیثم للعلوم الصرفة والتطبیقیة تقدیر معلمتي توزیع رالي العام بأستخدام تقنیة المحاكاة األء ماجد حمد ، عبد الخالق النقیب بغداد الكلیة التقنیة الطبیة ، ، ابن الهیثم ، جامعة بغدادقسم الریاضیات ، كلیة التربیة لخالصةا ة التقلیدیة لغرض تقدیر معلمتي الشكل و القیاس لتوزیع رالي العام صیغ الً فضال عنجدیدة صیغة حتاقترفي هذا البحث ) .PCE( افتراضیة للمعلمتین بأستخدام طریقة عدیدةمع تولیفات) كبیرة ، متوسطة ، صغیرة ( عینات مختلفة ووألحجام تحلیل القیم ة التقلیدیة والمقترحة من خالل صیغ ألفضل اداء ومقارنة بین الًام مؤشر معدل مربعات الخطأ مؤشراستخد و م وت( ة المقترحةصیغلقد تمت المالحظة من خالل النتائج التي تم الحصول علیها بأفضلیة ال. والمحاكاة الحاسوبیة .ة المدروسة التقلیدیةصیغعلى ال) استخدامها ألول مرة على حد علمنا