ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 نموذج االنحدار الخطي أ االختبار األولي المعدلة لمعالم ومقدرات ثومبسن ذ البسیط عابدةنوره اسام جامعة بغداد،كلیة االدارة واالقتصاد 2011 تموز 13: ، قبل البحث في 2011 ایار 23:استلم البحث في الخالصة ي المقلصه ذف بمقُلتي تعرا یتعلق موضوع هذا البحث بمقدرات ثومبسن مع يدرات االختبار االول المرحلة الواحدة اقترح هذا النوع من المقدرات لتقدیر المعالم . () عن طریق عامل تقلص موزون ةبعض التعدیالت على صیغته العام () 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 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Modified Thompson –Type Testimators for the Parameters o f Simple Linear Regression Model N. U. Abde Economic and Administrative College, Unive rsity of Baghdad Received in:23May 2011, Accepte d in:13July2011 Abstract This p aper is concerned with T hompson-T y p e est imators, which is known as p reliminary test shrinkage est imator with some modification on its form via shrinkage weight factor (). This ty p e of estimators have been considered for estimating the p arameters  of simple linear regression model, when a p rior estimate of the p arameter value () is available, say 0. This 0 has been referred in st atist ical literatures as guess p oint about the parameter . The exp ressions for Bias, M ean Squared Error (M SE) and Relative Efficiency of the p rop osed estimators are obtained. Numerical results are p rovided when the p rop osed 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 p rop osed estimators in the sense of Relative Efficiency and M ean Squared Error. Key Words: Simp le linear regression, least square met hod, Shrinkage estimat or, preliminary t est est imat or, prior estimat or , Bias, Mean Square Error an d Relat ive Efficiency. Introduction Some time we may have a p rior estimate value (p oint guess) of the p arameter to be estimated. If this value is in the vicinity of the true valu e, the shrinkage techniqu e is useful to get an imp roved est imator. T hompson in [14], M ehta and Srinivasan in [8], Singh at el in [12] and others suggested shrunken estimators for different distributions when a prior estimate or guess p oint is available. They showed that these est imators p erform bett er in the term of M ean Square Error when a guess value 0 close to t he true value . Consider the followin g simple lin ear regr ession model: y i =  + (xi – x ) + ei, i = 1,2,…,n (1) where xi is the indep endent variable and y i is the resp onse variable,  =  +  x , x is the mean of xi, ei is the random error which is distribute as normal distribution with zero M ean and Variance  2 and Cov(ei,ej ) = 0 and y i  N[ + (xi – x ), 2 ( i x (x x )1 n SS   )], see [5], [6]. Thompson-T y p e estimator in [14] is considered for estimating the p arameter  ( may refer to  or ) of p revious model when a guess p oint 0 is available about  due the p ast exp erience or similar cases. From the emp irical st udies it has been established that the shrunken est imators p erforms bett er than the usual estimator when our guess p oint be very close to the true value of the p arameter. 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 . ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Thompson suggested shrinking the least square estimator ̂ of  towards the p rior guess p oint 0 and p rop osed the est imator ˆ ˆ ˆ   % , where ˆ represents the exp erimenters belief in the guess p oint 0. He was found the estimator % is more efficient than ̂ if the true value  is close to 0 (H0 accep ted) but may be less efficient otherwise, therefore to resolve the uncertainty that a guess p oint value is app roximately the true value or not, a p reliminary test of significance may be employed. So he take the least square est imator ̂ when  is far a way from 0 (H0 rejected) after he made the p reliminary test. Thus, t he preliminary test shruken estimator has t he following form 0 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ, if R            % (2) where R is the p reliminary test region for accep tance the null hy p othesis 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 st udied a p reliminary test shrunken estimator which is defined in (2) for sp ecial p op ulation by choosing different weight factors ˆ( )  . See for examp le [1], [2], [3], [4], [7], [10], [11] and [13]. The aim of this p aper is to modify the p reliminary test shrunken estimator which is defined in (2) for estimating the p arameters () of the p rop osed simple linear regression model (1). Therefore, the form of the p rop osed p reliminary 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 exp ressions for Bias, M ean Square Error and Relative Efficiency of the estimator PT % above are derived. Numerical results of these exp ressions were made to show t he validity and the usefulness of the p rop osed estimator when it is comp ared 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 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 p rior guess value of , ̂ is a unbiased estimator (L.S.M .) of  and R1 is a p reliminary test region of acceptance of size  for testing the hyp othesis H0: = 0 against the hy p othesis 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, p ut forward 1 ˆ( ) 0   and 2 ˆ( )  = k = e – 10 / n. The exp ressions for Bias and M ean Square Error (M SE) of % are resp ectively 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 01 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 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 prop osed 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 y 1, y 2, …, y n 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 p reliminary test Shrunken est imator: 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 p retest region for accep tance of testing the hy p othesis H00: = 0 vs. t he hyp othesis 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) p ercentile p oint of t he standard normal dist ribution. In the estimator PT% which is defined in (11), we assume that 3 ˆ( )  = 0 and / 2Z 4 ˆ( ) h e       . The exp ressions for Bias and M ean Square Error (M SE) of PT % are resp ectively given as below:- PT 2 PT Bias( , R ) E ( )    % % ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 prop osed estimator PT% relative to est imator ̂ is defined as PT 2 PT 2 ˆMSE( ) R.Eff ( , R ) MSE( , R )       % % …(2 0) Numerical Re sults 1. The comp utation of Relative Efficiency [R.Eff()] and Bias Ratio [B()] were used for the estimator PT % , these comp utations were performed for  = 0.01, 0.05, 0.1, 1 = 0.0(0.1)2 and n = 8, 10, 12, 20. Some of these comp utations are disp lay ed in the att ached table 1 which leads t o the following results. i. The Relative Efficiency [R.Eff()] of PT % are adversely p rop ortional with small v alue 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 reasonab ly small when  = 0 and vice – v ersa otherwise. iv. The Bias R atio [B()] of PT % are increasing function with icr eases va lue of sample size (n). ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 prop osed estimator PT % dominate the usual estimator ̂ with large sample size n. i.e.; n lim  [M SE PT 1( , R )  % – M SE(̂ )]  0. vii. PT % is consistent estimator i.e.; n lim  M SE PT 1( , R )  % = 0. viii. The considered estimator PT % is bett er than the usual estimator and also than the estimator introduced by [1] and [2] in the sense of M ean Squared Error. 2. The comp utation of Relative Efficiency [R.Eff()] and Bias Ratio [B()] of the p rop osed estimator PT% were made on different constants involved in it, some of these comp utations 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 Efficien cy [R.Eff()] of PT% has a maximum value when  very close to 0 (2 = 0) and decr eases with increases value of 2 and h. ii. R.Eff( PT% ) increasing function with small valu e of  [lev el of si gnificance o f accep tance region R2]. iii. The Bias Ratio [B()] of PT% are reasonably small when  close to 0 (2 = 0) and increases ot herwise iv. B( %) are increases when  increases. v. The Effective Interval [The value of 2 which make the R,Eff() of PT% gr eater than 1 ] is [0,1]. vi. The considered est imator PT% is consistent est imator and dominate the usual estimator ̂ . vii. The considered estimator PT% is bett er than the est imator ̂ (least square method) an d some existing estimator e. g. [1] and [2] in terms of h igher Efficiency esp ecially at  ≃ 0. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012 Conclusions From the above discussions it is obvious t hat by using guess p oint value one can imp rove the usual estimator. It can be not ed that if the gu ess p oint 0 is very close to the true value of the parameter  (i.e.; i is app roximate close to one), the p rop osed estimators p erform better than the usual estimator. If one has no confidence in the gu essed value then p rop osed p reliminary test Shrunken estimators can be suggested. We can safely use the p rop osed estimators for small sample size at usual level of sign ifican ce  and moderate valu e of shrunken weight factor ˆ . Re ferences 1. Al-Jubori, A.N. (2000), Preliminary Test Single Stage Shrunken Estimator for the Parameters of Simp le Linear Regression M odel, Ibn Al-Haitham J. for Pure and Ap p lied Sci., 13(3): 65-73. 2. Al-Jubori, A.N., (2002), On Shrunken Estimators for the Parameters of Simp le Linear Regression M odel, Ibn Al-Haitham J. for Pure and Ap p lied Sci.,15(4A):60-67. 3. Al-Jubori, A.N. (2010), Pre-Test Single and Double Stage Shrunken Estimators for the M ean of Normal Dist ribution 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 Dist ribution, 3 rd Conf. of Computer and M athematics College, Al- Qadisy ia Univ., Al-Qadisy ia, Iraq. 5. Al-Kanane, I.H. (1997), Single and Double Stage Shrunken Estimators for the Linear Regression M odels, Ph.D.Thesis, Administ ation and Economic College, Al- M ust ansiriy ah University . 6. Drap er, N.R. and Smith, H. (1981), Ap p lied Regression Analysis, John Wiley and Sons. 7. Kambo, N.S.; Handa, B.R. and Al-Hemyari, Z.A. (1990), On Shrunken Estimator for Exp onential Scale Parameter, Journal of Statist ical Planning and Inference, 24:87-94. 8. M ehta, J.S. and Srinivasan,R. (1971), Estimation of the M ean by Shrinkage to a Point, Jour. Amer. Statist . Assoc., 66:.86-90. 9. Pandey, B.N. (1979), On Shrinkage Estimation of Normal Pop ulation Variance, Communication in Statist ics – T heory and M ethods,.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 Statist ical Research, 40(2):41-54. 11. Saleh, A.K.E., (2006), Theory of Preliminary Test and Stein-Ty p e Estimatoes with Ap p lication, 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, M icroelectron Reliability , .36 (3):435-439. 13. Singh, H.P. and Shukla, S.K., (2000), Esimation in the Two Parameter Weibull Dist tribution with Prior Information, IAPQR Transactions, 25 (2):107-118. 14. Thompson,J.R., (1968), Some Shrinkage Techniques for Estimating the M ean, J. Amer. Statist . Assoc, 63,113-122. ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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) ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or 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 ة مجلة إبن الھیثم للعلوم الصرفة و التطبیقی 2012 السنة 25 المجلد 1 العدد Ibn A l-Haitham Journal f or Pure and Applied Science No. 1 Vol. 25 Year 2012