382 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 Estimate The Mean of Normal Distribution Via Preliminary Test Shrinkage Technique S. T.Abdulrahman Department of Mathematics-Ibn-Al-Haitham College of Education – University of Baghdad. Received in: 18 December 2011 Accepted in: 11January 2012 Abstract This paper is concerned with preliminary test single stage shrinkage estimators for the mean (q) of normal distribution with known variance s2 when a prior estimate (q0) of the actule value (q) is available, using specifying shrinkage weight factor y (◊) as well as pre-test region (R). Expressions for the Bias, Mean Squared Error [MSE(◊)] and Relative Efficiency [R.Eff.(◊)] of proposed estimators are derived. Numerical results and conclusions are drawn about selection different constants including in these expressions. Comparisons between suggested estimators with respect to usual estimators in the sense of Relative Efficiency are given. Furthermore, comparisons with the earlier existing works are drawn to shown the usefulness of the proposed estimators. Key Words: Prior Estimate, Shrinkage Estimator, Shrinkage weight factor, Pre-test Region, Bias Ratio, Mean Squared Error and Relative Efficiency. 1.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 [1], Mehta and Srinivasan in [2], Singh at el in [3] 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 q0 close to the true value q. Assume that x1, x2, …, xn be a random sample of size (n) from a normal population with known variance (s2) and un-known mean (q). In conventional notation, we write x~N(q,s2) …(1) Preliminary test estimator in Thompson [1] is considered for estimating the parameter q in previous model of distribution in (1) when a guess point (prior estimate) q0 is available about q due the past knowledge or similar cases. From the empirical studies it has been established that the shrinkage estimators performs better than the usual estimator when the guess point be very close to the true value of the parameter. Therefore to make sure whether q is closed to q0 or not, we may test H0:q = q0 against H1: q π q0, so we denote by R to the critical region for above test. Thompson suggested shrinking the usual estimator q̂ of q towards the prior guess point q0 and proposed the estimator ˆ ˆ ˆ 0q = y(q)q + (1 - y(q))q% , where ˆ(1 - y(q)) represents the experimenters belief in the guess point q0. He was found the estimator q% is more efficient than q̂ if the true value q is close to q0 (H0 accepted) but may be less efficient otherwise, therefore to resolve the uncertainty that a guess point value is approximately the true value or 383 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 not, a preliminary test of significance may be employed. So he take the usual estimator q̂ when q is far a way from q0 (H0 rejected) after he made the preliminary test. Thus, the preliminary test shrunken estimator has the following form 0 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ, if R Ïy q q + - y q q q ŒÔ q Ì q q œÔÓ =% º(2) where R is the preliminary test region for acceptance the null hypothesis H0 as we mentioned above, q̂ is the usual estimator of q, ˆ( )y q is a shrinkage weight factor such that 0 £ ˆ( )y q £ 1 which may be a function of q̂ or may be a constant (ad hoc basis). Several authors had been studied a preliminary test shrinkage estimator which is defined in (2) for special population by choosing different weight factors ˆ( )y q . See for example [4], [5], [6], [7], [8], [9], [10], [11],[12]and [13]. The aim of this paper is to modify the preliminary test shrunken estimator which is defined in (2) for estimate the parameters (q) of the proposed distribution model (1). Therefore, the form of the proposed modify preliminary test shrinkage estimator is as below:- 1 1 0 PT 2 2 0 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ ˆ ˆ( ) (1 ( )) , if R Ïy q q + - y q q q ŒÔ q Ì y q q + - y q q q œÔÓ =% º(3) where i ˆ( )y q , (i = 1,2) is a shrinkage weight factor such that 0 £ i ˆ( )y q £ 1. The expressions for Bias, Mean Square Error and Relative Efficiency of the estimator PTq% above are derived. Numerical results of these expressions were made to show the validity and the usefulness of the proposed estimator when it compares with the usual and existing estimators. 2. Preliminary Test Single Stage Shrunken Estimator PT1q% In this section, we want to estimate the parameter q using the following preliminary test Shrunken estimator: 1 1 0 2 2 0 PT1 ˆ ˆ ˆ ˆ( ) (1 ( )) , if R ˆ ˆ ˆ ˆ( ) (1 ( )) , if R q Ïy q q + - y q q q ŒÔ Ì y q q + - y q q q œÔÓ =% º(4) where i ˆ( )y q , (i = 1,2) is shrinkage weight factor such that 0 £ i ˆ( )y q £ 1 and q̂ is usual estimator of q as well as R is the pretest region for acceptance of testing the hypothesis H0:q = q0 vs. the hypothesis H1:q π q0 with level of significance a using test statistic 0 0 ˆ ˆ / n T( / ) qq s - q q = . i.e. 2 2 0 / 2 0 / 2R [ Z , Z ]n n s s a a= -q - -q + …(5) where Za/2 is the 100(a/2) percentile point of the standard normal distribution. In the estimator PT1q% which is defined in (4), we assume that 1 ( )y q = 0 and y 2(q) = e – 10 / n. 384 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 The expressions for Bias and Mean Square Error (MSE) of PT1q% are respectively given as below:- 10/ n R R PT1 PT1Bias ( , R) E( ˆ ˆ ˆ ˆ ˆf d e f d q - 0 0 0 qq = ) - q = [-(q - q ) ] (q q) q + [ (q - q ) + (q - q) ] (q q) qÚ Ú % % where R is the complement region of R in real space, q̂ ~ N(m, 2 n s ) and 2 2 2nˆ ˆ ˆf exp[ n( ) / 2 ], , , 0 2 s s s s p 2(q q, ) = - q - q - • < q < • - • < q < • ≥ . The previous expression will result { }10 / n 10 / n 10 / n0 1 1 1 1 1PT1Bias ( , R) [1 e e J (a , b )] e J (a , b )n - - - q s q = -l - + -% …(6) where Jℓ(a1,b1)= 1 2 1 b Z / 2 a 1 Z e dZ, 0,1, 2 2 - = p Ú l l , …(7) and 0 1 / 2 1 / 2 ˆ n ( )n ( ) Z , , a Z , b Za a q - qq - q = l = = -l - = -l + s s …(8) { } 2 PT PT 2 10/ n 2 2 2 10/ n 10/ n 2 2 2 1 1 1 1 1 0 1 1 10/ n 1 1 1 0 1 1 MSE ( , R) E( ) ( e ) (1 ) (2e 1) ( e ) [J (a , b ) 2 J (a , b ) J (a , b )] n 2e [J (a , b ) J (a , b )] - - - - q q = q - q s = + l - l - - + l + l - l + l …(9) 3. Preliminary Test Single Stage Shrunken Estimator PT 2q% In this section, we use the following estimator to estimate the mean q of model (1). 0 0 PT2 ˆ, if ˆ(a / b) (a / b) , if ˆ(a / b) [1 (a / b)] R, ˆ[1 ] R. q Ï qÔ Ì Ô q qÓ q + - q Œ = - q + œ % …(10) i.e.; we put forward 1 ˆ( ) a / by q = and 2 ˆ( ) 1 a / by q = - in equation (3), where a and b are positive real numbers such that a £ b). The expressions for Bias and Mean Squared Error (MSE) of PT 2q are respectively given as follows:- { }1 1 1 0 1 1PT2 , R) ( / n ) (2a / b 1)[J (a , b ) J (a , b )] (a / b)Bias ( q = s - - l + lq% …(11) and, { }2 2 2 22 1 1 0 1 1PT2 , R) ( / n) (2a / b 1)[J (a , b ) J (a , b )] (1 a / b) (a / b)MSE( q = s - - l + - + lq% …(12) There is no doubt to take b=1 and find the value of (a) by minimizing the MSE( 2PT 2 , Rqq% ). 385 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 i.e. 2PT 2 , R) 0 a MSE q = ∂ ∂ (q% . Therefore the value of (a) becomes: 2 2 0 0 R 2 0 0 ˆ ˆ ˆ ˆ ˆMSE( / ) ( )Bias( / ) [( / ) ( ) ]f ( / )d a* ,ˆ ˆMSE( / ) 2( )Bias( / ) ( ) q q - q - q q q - q q + q - q q q q = q q - q - q q q + q - q Ú …(13) by simple calculation, 2 1 1 0 1 1 2 1 J a b J a b a* 1 l l 2- ( , ) + ( , ) = + …(14) 2 2PT 2 2 2 , R) 1 0 a MSE l q = + > ∂ ∂ (q% To be ensure that a Œ [0,1], we take 0 , if a 0 a a , if 0 a 1 1 , if a 1 * * * * Ï £ Ô = < <Ì Ô ≥Ó we denote to the Bias Ratio of PTiq % as PTiB( )q% which is defined as PTi PTi Bias( , R) B( ) / n q q q = s % % for i=1,2 …(15) See [6], [7] and [8]. The Efficiency of the proposed estimator PTi (i 1,2)q = % relative to estimator q̂ is defined as : PTi PTi ˆMSE( ) R.Eff ( , R) , i 1, 2 MSE( , R) q q q q = = q % % …(16) See [1], [6], [7] and [8]. 4. Conclusions and Numerical Results From the expressions of Bias and MSE of PTiq% ,i=1,2, the following could be easily seen :- 1) i. PTiB( , R)q q% is an odd function of l for i=1,2. ii. PTiMSE( , R)q q% is an even function of l for i=1,2 iii. The considered estimator PTiq% is a consistent estimator of q, i.e; n PTi lim MSE( , R) 0 Æ• q q =% for i=1,2. iv. The consider estimator PTiq% dominates ( q̂ ) with large sample size (n) in the term of MSE, i.e.; n PTi ˆlim[MSE( ) MSE )] 0 Æ• q - (q £% , for i=1,2 v. Practically, the consider estimator PTiq% is unbiased when q = q0, i.e.; 0 PTi lim B( R) 0 l Æ q q, =% , for i=1,2 and for each a and l. 386 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 2) The computations of Relative Efficiency [R.Eff( PTiq% )] and Bais Ratio [B(◊)][= PTin ) / ]q sB( % of consider estimator PTiq% (i=1,2) were made on different constants involved in it, some of these computations are given in attached tables (1) and (2) for some samples of these constant e.g. a = 0.02, 0.01,0.05,0.1and l = 0.0(0.1)1,2. The following numerical results from the mentioned tables leads to:- i. Relative Efficiency of PTiq% (i=1,2) is maximum when q ª q0, and decreases with increasing value of l. ii. Relative Efficiency of PTiq% (i=1,2) is maximum when the value of a is small. i.e.; the Relative Efficiency of PTiq% decreases with size a of the pre-test region in neighborhood of q ª q0. iii. The Bias Ratio of PTiq% are reasonably small when q ª q0, for i=1,2. i.e.; The Bias Ratio decreases as l decreases. iv. The Bias Ratio of PTiq% decreases when a increases, for i=1,2. v. The Effective Interval [the value of l that makes R.Eff.(.) greater than one] using proposed estimator PTiq% is [-1,1] for i=1,2. vi. The estimator PT1q% is better than the estimator PT 2q% in the sense of higher Relative efficiency for each a and l. vii. The Relative Efficiency of PT1q% decreases function with increasing of (n) for each a and l. 3) The consider estimator PTiq% (i=1,2) is better than the usual estimator ( q̂ ) and than the existing estimators, for example Thompson (1), Al-Hemyari and Al-Juboori (14) and others in terms of higher Relative Efficiency specially at q ª q0 4) 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 q0 is very close to the true value of the parameter q (i.e.; l is approximate close to zero), the proposed estimators perform better than the usual estimator q̂ . If one has no confidence in the guessed value then proposed preliminary test Shrinkage estimators can be suggested. We can safely use the proposed estimators for small sample size at usual level of significance a and moderate value of shrunken weight factor ˆy(q) . References 1.Thompson,J.R., (1968), "Some Shrinkage Techniques for Estimating the Mean", J. Amer. Statist. Assoc, 63, 113-122. 2.Mehta,J.S. and Srinivasan,R., (1971), "Estimation of the Mean by Shrinkage to a Point", J. Amer. Statist. Assoc., 66, 86-90. 3.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. 4.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. 5.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. 387 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 6.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. 7.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. 8.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. 9.Mehta, J.S. and Srinivasan,R., (1971), Estimation of the Mean by Shrinkage to a Point, Jour. Amer. Statist. Assoc., 66, 86-90. 10.Pandey, B.N., (1979), On Shrinkage Estimation of Normal Population Variance, Communication in Statistics – Theory and Methods,8, 359-365 11.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. 12.Saleh, A.K.E., (2006), Theory of Preliminary Test and Stein-Type Estimators with Application, Wiley and Sons, New York. 13.Singh, H.P. and Shukla, S.K., (2000), Estimation in the Two Parameter Weibull Distribution with Prior Information, IAPQR Transactions, 25 (2), 107-118. 14.Al-Hemyari, Z.A. and AL-Jubori A.N., (1999), "Modifical Single Stage Estimators of the Mean of Normal Population", Al-Fath J. of the College of Education for Puresci and Humanities, 3(5), 45-57. 388 مجلة إبن الهيثم للعلوم الصرفة و التطبيقية 2012 السنة 25 المجلد 2 العدد Ibn Al-Haitham Journal for Pure and Applied S cience No. 2 Vol. 25 Year 2012 طريقة االختبار األولــي المقلصـــة عمالالتوزيع الطبيعي باستتقدير متوسط سها طالب عبد الرحمن جامعة بغداد ¡ابن الهيثم -كلية التربية ¡قسم الرياضيات 2012كانون الثاني 11قبل البحث في 2011كانون االول 18:استلم البحث في الخالصة (q)المرحلة الواحدة لمتوسط التوزيع الطبيعي Çيتعلق موضوع هذا البحث بمقدرات االختبار األولي المقلصة ذ دالة تقلص عمال، باست(q)حول القيمة الحقيقية (q0)معلوماً وعند توافر المعلومات المسبقة s2عندما يكون التباين .(R)مجال االختبار األولي فضال عن (◊)yموزونة ، للمقدرات المقترحة. [(◊)R.Eff]النسبية يةوالكفا [(◊)MSE]اشتقت معادالت التحيز ، متوسط مربعات الخطأ أعطيت بعض االستنتاجات والنتائج العددية الخاصة بالمعادالت أعاله من خالل اختيار بعض القيم للثوابت المتضمنة المقارنات للمقدرات المقترحة مع المقدرات الكالسيكية وبعض البحوث المنجزة حديثاً لبيان فائدة فيها. أجريت بعض ة.يالنسب يةمن حيث الكفا وأفضلية المقدرات المقترحة التقدير األولي،المقدر المقلص،دالة التقلص الموزونة،مجال االختبار األولي، نسبة التحيز ،متوسط الكلمات المفتاحية : النسبية . يةمربعات الخطأ و الكفا 1.Introduction 2. Preliminary Test Single Stage Shrunken Estimator 3. Preliminary Test Single Stage Shrunken Estimator …(16) 4. Conclusions and Numerical Results قسم الرياضيات , كلية التربية- ابن الهيثم, جامعة بغداد يتعلق موضوع هذا البحث بمقدرات الاختبار الأولي المقلصة ذا المرحلة الواحدة لمتوسط التوزيع الطبيعي (() عندما يكون التباين (2 معلوما ً وعند توافر المعلومات المسبقة ((0) حول القيمة الحقيقية (()، باستعمال دالة تقلص موزونة ((() فضلا عن مجال الاختبار الأولي (R). اشتقت معادلات التحيز ، متوسط مربعات الخطأ [MSE(()] والكفاية النسبية [R.Eff(()]، للمقدرات المقترحة. أعطيت بعض الاستنتاجات والنتائج العددية الخاصة بالمعادلات أعلاه من خلال اختيار بعض القيم للثوابت المتضمنة فيها. أجريت بعض المقارنات للمقدرات المق... الكلمات المفتاحية : التقدير الأولي،المقدر المقلص،دالة التقلص الموزونة،مجال الاختبار الأولي، نسبة التحيز ،متوسط مربعات الخطأ و الكفاية النسبية .