مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 نموذج االنحدار الخطي أ االختبار األولي المعدلة لمعالم ومقدرات ثومبسن ذ البسیط عابدةنوره اسام د،كلیة االدارة واالقتصاد جامعة بغدا 2011 تموز 13: ، قبل البحث في 2011 ایار 23:استلم البحث في الخالصة ي المقلصه ذف بمقُلتي تعرا یتعلق موضوع هذا البحث بمقدرات ثومبسن المرحلة الواحدة مع يدرات االختبار االول م . () عن طریق عامل تقلص موزون ةبعض التعدیالت على صیغته العام اقترح هذا النوع من المقدرات لتقدیر المعال () نموذج االنحدار الخطي البسیط عند توافر تقدیرات مسبقه حول هذه المعالم بشكل ال0 . ان قیمة0شار لها في ُ ی .ة حول المعلم(point guess)االدبیات االحصائیه على شكل تقدیر نقطي ة أُعطیت النتائج العددیة الخاص. ة النسبیة للمقدرات المقترحةیمتوسط مربعات الخطأ والكفاواشتقت معادالت التحیز، .  ةمقدرات االختبار االولي بمستوى معنویوات، عندما تكون هذه المقدرة بالمقدرات المقترحةبالمعادالت اعاله والمتعلق ة ی من حیث الكفاة لبیان افضلیة المقدرات المقترحة والمشابهة مع المقدرات االعتیادیةأجریت مقارنات بین المقدرات المقترح .النسبیه ومتوسط مربعات الخطأ مقدر التقلص، مقدر االختیار االولي، االنحدار الخطي البسیط، طریقة المربعات الصغرى، : الكلمات المفتاحیة .ة النسبیةیالتقدیر االولي، التحیز، متوسط مربعات الخطأ والكفا مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Modified Thompson –Type Testimators for the Parameters of Simple Linear Regression Model N. U. Abde Economic and Administrative College, University of Baghdad Received in:23May 2011, Accepted in:13July2011 Abstract This paper is concerned with Thompson-Type estimators, which is known as preliminary test shrinkage estimator with some modification on its form via shrinkage weight factor (). This type of estimators have been considered for estimating the parameters  of simple linear regression model, when a prior estimate of the parameter value () is available, say 0. This 0 has been referred in statistical literatures as guess point about the parameter . The expressions for Bias, Mean Squared Error (MSE) and Relative Efficiency of the proposed estimators are obtained. Numerical results are provided when the proposed estimators are testimators of level of significance . Comparisons with the usual (L.S.M) and existing estimators were made to show the usefulness of the proposed estimators in the sense of Relative Efficiency and Mean Squared Error. Key Words: Simple linear regression, least square method, Shrinkage estimator, preliminary test est imat or, prior estimator, Bias, Mean Square Error and Relat ive Efficiency. Introduction Some time we may have a prior estimate value (point guess) of the parameter to be estimated. If this value is in the vicinity of the true value, the shrinkage technique is useful to get an improved estimator. Thompson in [14], Mehta and Srinivasan in [8], Singh at el in [12] and others suggested shrunken estimators for different distributions when a prior estimate or guess point is available. They showed that these estimators perform better in the term of Mean Square Error when a guess value 0 close to the true value . Consider the following simple linear regression model: yi =  + (xi – x ) + ei, i = 1,2,…,n (1) where xi is the independent variable and yi is the response variable,  =  +  x , x is the mean of xi, ei is the random error which is distribute as normal distribution with zero Mean and Variance 2 and Cov(ei,ej) = 0 and yi  N[ + (xi – x ),2( i x (x x)1 n SS   )], see [5], [6]. Thompson-Type estimator in [14] is considered for estimating the parameter  ( may refer to  or ) of previous model when a guess point 0 is available about  due the past experience or similar cases. From the empirical studies it has been established that the shrunken estimators performs better than the usual estimator when our guess point be very close to the true value of the parameter. Therefore to make sure whether  is closed to 0 or not, we may test H0: = 0 against H1:   0, so we denote by R to the critical region for above test. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Thompson suggested shrinking the least square estimator ̂ of  towards the prior guess point 0 and proposed the estimator ˆ ˆ ˆ    % , where ˆ  represents the experimenters belief in the guess point 0. He was found the estimator % is more efficient than ̂ if the true value  is close to 0 (H0 accepted) but may be less efficient otherwise, therefore to resolve the uncertainty that a guess point value is approximately the true value or not, a preliminary test of significance may be employed. So he take the least square estimator ̂ when  is far a way from 0 (H0 rejected) after he made the preliminary test. Thus, the preliminary test shruken estimator has the following form 0 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ, if R            % (2) where R is the preliminary test region for acceptance the null hypothesis H0 as we mentioned above, ̂ is the least square estimator of , ˆ( )  is a shrinkage weight factor such that 0  ˆ( )   1 which may be a function of ̂ or may be a constant (ad hoc basis). Several authors had studied a preliminary test shrunken estimator which is defined in (2) for special population by choosing different weight factors ˆ( )  . See for example [1], [2], [3], [4], [7], [10], [11] and [13]. The aim of this paper is to modify the preliminary test shrunken estimator which is defined in (2) for estimating the parameters () of the proposed simple linear regression model (1). Therefore, the form of the proposed preliminary test shrunken estimator is as below:- 1 1 PT 2 2 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ ˆ ˆ( ) (1 ( )) , if R                % (3) where i ˆ( )  , i = 1,2 is a shrinkage weight factor such that 0  i ˆ( )   1. The expressions for Bias, Mean Square Error and Relative Efficiency of the estimator PT% above are derived. Numerical results of these expressions were made to show the validity and the usefulness of the proposed estimator when it is compared with the least square and existing estimators. Preliminary Test Single Stage Shrunken Estimator PT % In this section recall the estimator which is defined in (3) for estimating the parameter  of assuming model as below مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 1 1 0 1 PT 2 2 0 1 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ ˆ ˆ( ) (1 ( )) , if R                   % (4) where 0 is a prior guess value of , ̂ is a unbiased estimator (L.S.M.) of  and R1 is a preliminary test region of acceptance of size  for testing the hypothesis H0: = 0 against the hypothesis H1 :   0. i.e. 2 2 1 0 0 ,n 2 ,n 2 x x2 2 R t , t SS SS                , see [5] …(5) where n i i i 1 n i i 1 (x x)(y y) ˆ (x x)          2 x ˆ, ) SS     and n 2 x i i 1 SS (x x)   , see [6] .. (6) while t/2, n – 2 is the 100(/2) percentile of t-distribution with (n – 2) degree of freedom. Now, put forward 1 ˆ( ) 0   and 2 ˆ( )  = k = e – 10 / n . The expressions for Bias and Mean Square Error (MSE) of % are respectively given by PT 1 PTBias( ,R ) E( )    % %  1 0 1 x [1 k kJ (a*,b*)] kJ (a*,b*) SS       …(7) where Jℓ(a*,b*)= 2 b* t / 2 a * 1 t e dt, 0,1,2 2     l l …(8) and x x 0 1 1 / 2,n 2 1 / 2,n 2 ˆSS ( ) SS ( ) t , ,a* t , b* t                  …(9) we denote to the Bias ratio of PT % as PTB( )% and defined as below PT 1 PT x Bias( , R ) B( ) / SS      % % …(10) and مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012   2 PT 1 PT 2 2 2 2 2 2 1 2 1 1 0 1 x 1 1 1 0 MSE ( , R ) E( ) k (1 ) (2k 1) k [J (a*, b*) 2 J (a *b*) J (a*, b*) ] SS 2k [J (a*,b*) J (a*, b*)] ...(11)                    % % The Efficiency of the proposed estimator PT % relative to ̂ is defined as PT PT 1 ˆMSE( ) R.Eff ( , R) MSE( ,R )       % % …(12) See [3], [4] and [7]. Preliminary Test Single Stage Shrunken Estimator % Let y1, y2, …, yn distribute as normal distribution with mean  and known variance 2, where ˆ y  . In this section, we want to estimate the parameter  using the following preliminary test Shrunken estimator: 3 3 0 2 PT 4 4 0 2 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ ˆ ˆ( ) (1 ( )) , if R                     % (13) where i ˆ( )  , i = 3,4 are shrinkage weight factors such that 0  i ˆ( )   1 and ̂ is an unbiased estimator (L.S.M) of  as well as R2 is the pretest region for acceptance of testing the hypothesis H00: = 0 vs. the hypothesis H11:   0 with level of significance . i.e. 2 2 2 0 /2 0 /2 R [ Z , Z ] n n          , see [5] …(14) where Z/2 is the 100(/2) percentile point of the standard normal distribution. In the estimator PT % which is defined in (11), we assume that 3 ˆ( )  = 0 and / 2Z 4 ˆ( ) h e     . The expressions for Bias and Mean Square Error (MSE) of PT % are respectively given as below:- PT 2 PTBias( ,R ) E( )     % % مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012  2 0 1 1 1 1 1[1 h hJ (a ,b )] hJ (a ,b ) n       …(15) where Jℓ(a1,b1)= 1 2 1 b Z / 2 a 1 Z e dZ, 0,1,2 2     l l …(16) and 0 2 1 2 / 2 1 2 / 2 ˆ n( )n ( ) Z , ,a Z ,b Z                  …(17) we denote to the Bias ratio of PT % as PTB( )% which is defined as PT 2 PT Bias( , R ) B( ) / n      % % …(18) and   2 PT 2 PT 2 2 2 2 2 2 2 2 2 1 1 2 1 1 1 2 0 1 1 2 1 1 1 2 0 1 1 MSE ( ,R ) E( ) h (1 ) (2h 1) h [J (a ,b ) 2 J (a , b ) J (a , b )] n 2h [J (a , b ) J (a ,b )] ...(19)                    % % The Efficiency of the proposed estimator PT % relative to estimator ̂ is defined as PT 2 PT 2 ˆMSE( ) R.Eff ( ,R ) MSE( , R )       % % …(20) Numerical Results 1. The computation of Relative Efficiency [R.Eff()] and Bias Ratio [B()] were used for the estimator PT % , these computations were performed for  = 0.01, 0.05, 0.1, 1 = 0.0(0.1)2 and n = 8, 10, 12, 20. Some of these computations are displayed in the attached table 1 which leads to the following results. i. The Relative Efficiency [R.Eff()] of PT% are adversely proportional with small value of  and those of n and k. ii. R.Eff( PT % ) has a maximum value when  = 0 (1 = 0). iii. The Bias Ratio [B()] of PT% are reasonably small when  = 0 and vice – versa otherwise. iv. The Bias Ratio [B()] of PT% are increasing function with icreases value of sample size (n). مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 v. The Effective Interval [The value of 1 which make the R.Eff() of PT% greater than 1] is [0,1]. vi. The proposed estimator PT% dominate the usual estimator ̂ with large sample size n. i.e.; n lim  [MSE PT 1( , R ) % – MSE(̂ )]  0. vii. PT % is consistent estimator i.e.; n lim  MSE PT 1( , R ) % = 0. viii. The considered estimator PT % is better than the usual estimator and also than the estimator introduced by [1] and [2] in the sense of Mean Squared Error. 2. The computation of Relative Efficiency [R.Eff()] and Bias Ratio [B()] of the proposed estimator PT % were made on different constants involved in it, some of these computations are given in annexed table (2) for samples of these constant e.g.  = 0.01, 0.05, 0.1, n = 8, 10, 12, 20 and 2 = 0.0(0.1)2. The following results from the mentioned table were made i. The Relative Efficiency [R.Eff()] of PT % has a maximum value when  very close to 0 (2 = 0) and decreases with increases value of 2 and h. ii. R.Eff( PT % ) increasing function with small value of  [level of significance of acceptance region R2]. iii. The Bias Ratio [B()] of PT % are reasonably small when  close to 0 (2 = 0) and increases otherwise iv. B( %) are increases when  increases. v. The Effective Interval [The value of 2 which make the R,Eff() of PT % greater than 1] is [0,1]. vi. The considered estimator PT % is consistent estimator and dominate the usual estimator ̂ . vii. The considered estimator PT % is better than the estimator ̂ (least square method) and some existing estimator e.g. [1] and [2] in terms of higher Efficiency especially at  ≃ 0. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Conclusions From the above discussions it is obvious that by using guess point value one can improve the usual estimator. It can be noted that if the guess point 0 is very close to the true value of the parameter  (i.e.; i is approximate close to one), the proposed estimators perform better than the usual estimator. If one has no confidence in the guessed value then proposed preliminary test Shrunken estimators can be suggested. We can safely use the proposed estimators for small sample size at usual level of significance  and moderate value of shrunken weight factor ˆ . References 1. Al-Jubori, A.N. (2000), Preliminary Test Single Stage Shrunken Estimator for the Parameters of Simple Linear Regression Model, Ibn Al-Haitham J. for Pure and Applied Sci., 13(3): 65-73. 2. Al-Jubori, A.N., (2002), On Shrunken Estimators for the Parameters of Simple Linear Regression Model, Ibn Al-Haitham J. for Pure and Applied Sci.,15(4A):60-67. 3. Al-Jubori, A.N. (2010), Pre-Test Single and Double Stage Shrunken Estimators for the Mean of Normal Distribution with Known Variance, Baghdad Journal for Science, 7(4):1432-1441. 4. Al-Jubori, A.N. (2011), On Significance Test Estimator for the Shape Parameter of Generalized Rayleigh Distribution, 3rd Conf. of Computer and Mathematics College, Al- Qadisyia Univ., Al-Qadisyia, Iraq. 5. Al-Kanane, I.H. (1997), Single and Double Stage Shrunken Estimators for the Linear Regression Models, Ph.D.Thesis, Administation and Economic College, Al- Mustansiriyah University. 6. Draper, N.R. and Smith, H. (1981), Applied Regression Analysis, John Wiley and Sons. 7. Kambo, N.S.; Handa, B.R. and Al-Hemyari, Z.A. (1990), On Shrunken Estimator for Exponential Scale Parameter, Journal of Statistical Planning and Inference, 24:87-94. 8. Mehta, J.S. and Srinivasan,R. (1971), Estimation of the Mean by Shrinkage to a Point, Jour. Amer. Statist. Assoc., 66:.86-90. 9. Pandey, B.N. (1979), On Shrinkage Estimation of Normal Population Variance, Communication in Statistics – Theory and Methods,.8:359-365. 10. Prakash, G.; Singh, D.C. and Singh, R.D., (2006), Some Test Estimator for the Scale Parameter of Classical Pareto Distribution, Journals of Statistical Research, 40(2):41-54. 11. Saleh, A.K.E., (2006), Theory of Preliminary Test and Stein-Type Estimatoes with Application, Wiley and Sons, New Yourk. 12. Singh, D.C.; Singh, P. and Singh, P.R. (1996), Shrunken Estimator for the Scale Parameter of Classical Pareto Distribution, Microelectron Reliability , .36 (3):435-439. 13. Singh, H.P. and Shukla, S.K., (2000), Esimation in the Two Parameter Weibull Disttribution with Prior Information, IAPQR Transactions, 25 (2):107-118. 14. Thompson,J.R., (1968), Some Shrinkage Techniques for Estimating the Mean, J. Amer. Statist. Assoc, 63,113-122. مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Table (1) Shown the R.Eff. () and B()of % w.r.t. , n and 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R.Ef f( ) B() 3689.8 (1.7707 e – 15 ) 97.349 (0.099901) 24.831 (0.19979) 11.079 (0.29966) 6.2411 (0.39948) 3.9977 (0.49924) 2.7781 (0.59892) 2.0424 (0.69848) 1.5648 (0.7979) 1.2374 (0.8971) 1.0033 (0.8971) R.Ef f( ) B() 287.39 (2.9328 e – ) 73.968 (0.099156) 22.931 (0.19827) 10.672 (0.29722) 6.1085 (0.39593) 3.9449 (0.49431) 2.7558 (0.59223) 2.0343 (0.68958) 1.5644 (0.786) 1.2417 (0.8819) 1.0108 (0.9766) R.Eff( ) B() 66.572 (6.6945 e – 10 ) 39.599 (0.097423) 17.898 (0.19972) 9.3655 (0.29164) 5.6251 (0.38797) 3.7232 (0.48349) 2.6398 (0.57796) 1.9688 (0.67111) 1.5262 (0.7626) 1.2199 (0.8523) 0.9996 (0.93958) R.Ef f( ) B() 108.11 (5.6271 e – 5) 52.823 (0.096686) 20.867 (0.19326) 10.411 (0.28949) 6.1335 (0.3852) 4.0245 (0.48022) 2.8422 (0.57437) 2.1166 (0.66746) 1.6406 (0.759) 1.3121 (0.8497) 1.0761 (0.9385) R.Ef f( ) B() 30.176 (0.0001948) 23.453 (0.092123) 14.08 (0.18411) 8.4753 (0.27545) 5.4637 (0.36582) 3.7677 (0.4549) 2.7455 (0.54238) 2.0906 (0.62795) 1.649 (0.71131) 1.3387 (0.7922) 1.132 (0.8703) مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012 Table (2) Shown the R.Eff. () and B() of % w.r.t.  and 2 R.Ef f( ) B() 12.33 (0.00011205) 11.021 (0.086432) 8.3709 (0.17248) 5.9936 (0.25754) 4.3075 (0.34113) 3.1811 (0.42277) 2.4262 (0.50201) 1.9085 (0.57841) 1.5433 (0.656) 1.2786 (0.7211) 1.0822 (0.78669) R.Ef f( ) B() 42.278 (0.00020265) 30.698 (0.091496) 16.867 (0.18294) 9.6606 (0.27387) 6.0679 (0.36405) 4.1232 (0.45324) 2.9769 (0.54122) 2.252 (0.6278) 1.7671 (0.7128) 1.4279 (0.79603) 1.1819 (0.87739) R.Ef f( ) B() 15.085 (0.00024397) 13.378 (0.084433) 10 (0.16871 7.061 (0.25219) 5.0266 (0.33449) 3.6898 (0.41527) 2.8035 (0.4942) 2.1997 (0.57098) 1.775 (0.6454) 1.4685 (0.71711) 1.2405 (0.7861) R.Ef f( ) B() 6.9216 (0.0007499) 6.5433 (0.07533) 5.6271 (0.15084) 4.5754 (0.22522) 3.6438 (0.29797) 2.9063 (0.36858) 2.3478 (0.4366) 1.9298 (0.50167) 1.616 (0.5634) 1.3781 (0.62162) 1.1954 (0.67607) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R.Eff( ) B() 2080.9 (4.0707 e – 20 ) 96.418 (0.09936) 24.981 (0.19868) 11.184 (0.29793) 6.3112 (0.39706) 4.048 (0.49604) 2.8167 (0.59482) 2.0736 (0.69335) 1.591 (0.7916) 1.2601 (0.88951) 1.0234 (0.98704) 0.2670 (1.931 R.Eff( ) B() 180.59 (8.8577 e – 19 ) 67.159 (0.096048) 23.31 (0.19197) 11.182 (0.28765) 6.4832 (0.38297) 4.2189 (0.4778) 2.964 (0.57205) 2.1986 (0.6656) 1.6982 (0.75838) 1.3534 (0.8503) 1.1059 (0.9413) 0.3052 (1.798 R.Eff( ) B() 61.117 (1.0505 e – 18 ) 40.007 (0.091493) 19.674 (0.18281) 10.677 (0.27378) 6.5289 (0.36424) 4.3687 (0.45403) 0.1218 (0.54302) 2.3429 (0.63109) 1.8259 (0.71814) 1.4658 (0.8041 1.2053 (0.889) 0.3454 (1.681 مجلة إبن الھیثم للعلوم الصرفة و التطبیقیة 2012 السنة 25 المجلد 1 العدد Ibn Al-Haitham Journal for Pure and Applied Science No. 1 Vol. 25 Year 2012