369 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 On Double Stage Shrinkage-Bayesian Estimator for the Scale Parameter of Exponential Distribution A. N. Salman, M. D. Salman Department of Mathematics-Ibn-Al-Haitham College of Education - University of Baghdad Received in : 19 October 2011 Accepted in : 7 December 2011 Abstract This paper is concerned with Double Stage Shrinkage Bayesian (DSSB) Estimator for lowering the mean squared error of classical estimator q̂ for the scale parameter (q) of an exponential distribution in a region (R) around available prior knowledge (q0) about the actual value (q) as initial estimate as well as to reduce the cost of experimentations. In situation where the experimentations are time consuming or very costly, a Double Stage procedure can be used to reduce the expected sample size needed to obtain the estimator. This estimator is shown to have smaller mean squared error for certain choice of the shrinkage weight factor y(◊) and for acceptance region R. Expression for Bias, Mean Square Error (MSE), Expected sample size [E(n/q,R)], Expected sample size proportion [E(n/q,R)/n], probability for avoiding the second sample 1ˆ[p( R)]q Œ and percentage of overall sample saved 2 1 n ˆ[ p[ R) 100] n q Œ * for the proposed estimator are derived. Numerical results and conclusions are established when the consider estimator (DSSB) are testimator of level of significance a. Comparisons with the classical estimator as well as with some existing studies were made to show the usefulness of the proposed estimator. Key Words: Exponential distribution, Maximum likelihood estimator, Bayesian estimator, Double stage shrinkage estimator, Mean square error, Relative Efficiency. Introduction 1.1 The Model: Exponential distribution is one of the most useful and widely exploited model, Epstein [1] remarks that the exponential distribution plays as important a role in life experiments as the part played by the normal distribution in agricultural experiments. It is applied in a very wide variety of statistical procedures. Among the most prominent applications are those in the field of life testing and reliability theory. The scale parameter (q) is known as mean life time. The maximum likelihood estimator (MLE; q̂ ) is the sample mean which is the minimum variance unbiased estimator. The one parameter exponential distribution has the following probability density function (p.d.f.) exp( t) , t 0, 0 f (t; ) 0 , o.w. b -b ≥ b >Ï b = Ì Ó …(1) Also, the above p.d.f can be written as below:- 370 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 1 t exp( ) , t 0, 0 f (t; ) 0 , o.w. -Ï ≥ q >Ô q = q qÌ ÔÓ …(2) where q is the average or the mean life or mean time to failure (MTTF) and it also acts as scale parameter, while b = 1/q is called the hazard rate or the mean arrival rate (MAR), see [1]. Furthermore, the Reliability function R(t) is defined as: R(t) = exp(- t/q), t > 0, q >0. Note that the maximum likelihood estimator q̂ of the scale parameter q of the mentioned distribution is n i i 1 t t n =  = . 1.2 Bayesian Estimator [2] Consider the one parameter exponential distribution 1 t exp( ) , for t 0, 0 f (t; ) 0 , o.w. -Ï ≥ q >Ô q = q qÌ ÔÓ Based on the rule proposed by Jeffery, one can get the prior distribution of q [g(q)] as below, g(q) µ ( )I q , where I(q) is fisher information such that 2 2 ln f (t, ) n ( ) n E 2 Ê ˆ∂ q I q = - =Á ˜∂q qË ¯ , see [2] \ g(q) µ n ng( ) kfi q = q q L(t1, t2,…,tn) = n i 1= ’ f (ti Ôq) = n i i 1 n t 1 exp = Ê ˆ Á ˜ Á ˜- q qÁ ˜ Á ˜ Ë ¯  The joint probability density function H(t1, t2,…,tn,q) is given by H(t1, t2,…,tn,q) = n i 1= ’ f (ti,q) g(q) = L(t1, t2,…,tnÔq) g(q) n i i 1 n t 1 k n exp = Ê ˆ Á ˜ Á ˜= - q q qÁ ˜ Á ˜ Ë ¯  371 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 n i i 1 n 1 t k n exp = + Ê ˆ Á ˜ Á ˜= - q qÁ ˜ Á ˜ Ë ¯  …(3) the marginal probability density function of (t1, t2,…,tn) is given by p(t1, t2,…,tn) = H q Ú (t1, t2,…,tn,q) dq n i i 1 n 1 0 t k n exp d • = + Ê ˆ Á ˜ Á ˜= - q q qÁ ˜ Á ˜ Ë ¯ Â Ú ( ) nn i i 1 k n (n 1)! = - = Ê ˆ Á ˜ Ë ¯ Ât And the condition probability density function of q given the data (t1, t2,…,tn) is given by ’(qÔt1, t2,…,tn) 1 2 n 1 2 n H(t , t ,..., t , ) p(t , t ,..., t ) q = n ni n i 1 i i 1 n 1 t exp t (n 1)! = = + È ˘ Í ˙ È ˘ Í ˙- Í ˙q Î ˚Í ˙ Í ˙Î ˚= q -   using squared error loss function 2ˆ ˆL( , ) c( )q q = q - q We can give Risk function, such that 1 2 n 0 ˆ ˆR( , ) E[L( , )] ˆL( , ) ( t , t ,..., t )d • q q = q q = q q ’ q qÚ 372 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 n ni n i 1 i i 1 2 n 1 0 nn n i i i 12 n i 1 0 t exp t ˆc( ) d (n 1)! t t ˆ ˆc 2c exp d ( ) (n 1)! = = • + • = - = Ê ˆ Á ˜Ê ˆ Á ˜- Á ˜qÁ ˜Ë ¯ Á ˜ Ë ¯= q - q q q - Ê ˆ Ê ˆ Á ˜ Á ˜Ë ¯ Á ˜= q - q q - q + f q - qÁ ˜ Á ˜ Ë ¯ Â Â Ú Â Â Ú nn n i i i 1 n i 1 0 t tˆR( , ) ˆ2c 2c exp d zeroˆ (n 1)! • = - = Ê ˆ Ê ˆ Á ˜ Á ˜∂ q q Ë ¯ Á ˜= q - q - q + - q∂q Á ˜ Á ˜ Ë ¯ Â Â Ú Let ˆR( , ) 0ˆ ∂ q q = ∂q , then nn n i i i 1 n i 1 0 n nn n n i i i i 1 i 1 i 1 2 0 n i i 1 n 2 0 t t ˆ exp d (n 1)! t t t exp( y) dy (n 1)! y y t y exp( y)dy (n 1)! q • = - = B - • = = = • = - Ê ˆ È ˘ Á ˜ Í ˙Ë ¯ Í ˙q = q - q - Í ˙ Í ˙Î ˚ Ê ˆ È ˘ Ê ˆ Á ˜ Á ˜Í ˙Ë ¯ Á ˜Í ˙= - - Á ˜Í ˙ Á ˜Í ˙Î ˚ Ë ¯ Ê ˆ Á ˜ Ë ¯= - - Â Â Ú Â Â Â Ú Â Ú n i i 1 t (n 2)! ˆ (n 1)! = B Ê ˆ -Á ˜ Ë ¯q = -  \ n i i 1 t ˆ n 1 = Bq = -  (Bayes estimator) …(5) where nˆE( ) n 1B q = q - Biased ˆ ˆ( ) E ( ) n 1B B q q = q - q = - …(6) 373 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 MSE 2 2 2 (n 1)ˆ ˆ( ) E ( ) (n 1)B B + q = q - q = q - …(7) 1.3 Double Stage Shrinkage Estimator [3], [4], [5], [6] A Double Stage Shrinkage Estimator is defined as follows Let x1i; i = 1, 2, …, n1 be a random sample of n1 from exponential distribution and 1q̂ be a classical estimator (MLE) of q based on n1 observation. Construct a preliminary test region (R) in the parameter space based on prior estimate q0 and an appropriate criterion. If 1q̂ Œ R shrink 1â towards a0 by shrinkage weight factor 1ˆ )y(q ; 0 £ 1ˆ )y(q £ 1 and use the shrinkage estimator 1ˆ )y(q 1q̂ + (1 – 1ˆ )y(q )q0, to estimate q, see [5]. If 1q̂ œ R, obtain x2i; i = 1, 2, …, n2, an additional sample of size n2 and use a pooled estimator pâ of a based on combined sample of size n = n1 + n2, i.e.; 1 1 2 2 p ˆ ˆn nˆ n q + q q = . Thus, the Double Stage Shrinkage Estimator (DSSE) will be 1 1 1 0 1 p 1 ˆ ˆ ˆ ˆ( ) (1 ( )) ; if R ˆ ˆ; if R Ïy q q + - y q q q ŒÔ q = Ì q q œÔÓ % …(8) To motivation of this study was provided by the work of [3], [4], [5], [6], [7], [8] and [9] and others. The aim of this paper is to employ Bayesian estimator which is defined in (5) in double stage shrinkage estimator (DSSE) which is defined in (8) to estimate the scale parameter (q) of exponential distribution. The expressions for Bias, Mean Square Error [MSE], Relative Efficiency [R.Eff(◊)], Expected sample size, Expected sample size proportion, probability for avoiding the second sample and percentage of overall sample saved are derived and obtained for the proposed estimator. Numerical results and conclusions due mentioned expressions including some constants are performed and displayed in annexed tables. Comparisons between the proposed estimator with the classical estimator ˆ(q) and with some of the last studies are demonstrated. 2. Double Stage Shrinkage-Bayesian Estimator This section is concerned with pooling approach between shrinkage estimation that uses a prior information about unknown parameter as initial value and Bayesian estimation that uses a prior information about unknown parameter as a prior distribution for the scale parameter (q) of exponential distribution using different shrinkage weight factors as well as pretest region R when a prior information about (q) is available as initial value (q0). 374 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 A proposed Double Stage Shrinkage-Bayesian Estimator (DSSBE) has the following form 1 1B 1 0 1 SB 1 1 2 2 p 1 ˆ ˆ ˆ ˆ( ) (1 ( )) ; if R ˆ ˆn nˆ ˆ; if R n Ïy q q + - y q q q Œ Ô q = Ì q + q q = q œÔ Ó % …(9) where 1Bq̂ represent to Bayes estimator for q on n1 observation, R is suitable region (say pretest region) and 1ˆ )y(q ; 0 £ 1ˆ )y(q £ 1 is shrinkage weight factor which may be a function of 1q̂ or constant, see [3], [4] and [9]. 2.1 DSSBE( SBq% ) Using Constant Shrinkage Weight Factor Using the form (9), the proposed DSSBE SBq% has the following forms: SB ˆ ˆ, if R ˆ ˆp , if R 1B 1 1 Ïq q ŒÔ q = Ì q q œÔÓ % …(10) i.e. 1 ˆ( 01y q ) = (constant). where R is pretest region of acceptance of size a for testing the hypothesis H0: q = q0 Vs. the hypothesis HA : q π q0 using test statistic 1 ˆ2nˆ ) 11 0 q T(q q = q In that, 1 1 2 20 0 1 / 2,2 n / 2,2 n 1 1 R X , X 2n 2n-a a È ˘q q = Í ˙ Î ˚ …(11) Assume that, R=[a,b], a < b. i.e. 1 20 1 / 2,2n 1 X 2n a -a q = and 1 20 / 2,2n 1 X 2n b a q = …(12) where 1 1 2 2 1 / 2,2n / 2,2nX and X-a a are respectively lower and upper 100(a/2) percentile point of Chi-square distribution with degree of freedom (2n1). The expression for Bias is given below SB SB p R R Bias( , R) E( ˆ ˆ ˆ ˆ ˆ ˆ( )f ( ; )d ( )f ( ; )d1B 0 1 1 1 1 q 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 and ˆf ( ; )1q q is a p.d.f. of ˆ1q which has the following form:- 1 1 n -1 1 n 1 1 ˆ ˆ[ exp[ n ˆ, for 0ˆ( ; ) n θ/n 0 , o.w f 1 1 1 1 Ï q ] - q /q] < q < •Ô q q = G( )( )Ì Ô Ó we conclude: 375 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 [ ]1SB 1 0 1 0 1 n 1 Bias( , R) J ( *, *) J ( *, *) J ( *, *) J ( *, *) n 1 1 u a b a b a b a b Ï ¸ q q = q - - -Ì ˝ - +Ó ˛ % …(13) where, l = q0 / q, y = n1 ˆ 1q q, u = n2/n1, n = n1 + n2, n 1 y 1 1 11J ( *, *) y y e dy n (n ) b* a* a b - -= G Ú l l l …(14) and 1 1 2 2 1 / 2,2 n / 2,2 nX , Xa* b*-a a= l = l …(15) …(16) The Bias ratio B(◊) of SBq% is defined as below SB SB Bias( , R) B( , R) q q q q = q % % See [6] and [9]. The expression of mean square error [MSE] of SBq% is as follows ….17 . [ ] SB SB 2 1 1 2 1 0 1 1 2 2 2 2 1 0 1 1 2 0 1 MSE( , R) E( n n J ( *, *) 2 J ( *, *) J ( *, *) n 1 n 1 1 1 u 1 1 J ( *, *) 2J ( *, *) J ( *, *) 1 u n 1 u n u 1 u u 1 J ( *, *) 1 u n u a b a b a b a b a b a b a b 2 2 q q = q - q) Ï Ê ˆÔ = q - + +Ì Á ˜- -Ë ¯Ô Ó È ˘Ê ˆ Ê ˆ Ê ˆ+ - - + -Í ˙Á ˜ Á ˜ Á ˜+ + +Ë ¯ Ë ¯ Ë ¯Î ˚ ¸ ÔÊ ˆ ˝Á ˜+Ë ¯ Ô ˛ % % The Relative Efficiency of estimator SBq% with respect to classical estimator ( ˆ1q ) is defined as below SB SB ˆMSE( R.Eff ( , R) S , R)[E(n , R) / n] q) q q = M E(q q q % % …(18) where E(nÔa,R) is the Expected sample size, which is defined as: 0 u E(n , R) n 1 J ( *, *) 1 u a bÈ ˘q = -Í ˙+Î ˚ . See for example [3], [10], [11], [12] and [13]. As well as, the Expected sample size proportion E(nÔa,R)/n equal to 0 u 1 J ( *, *) 1 u a b- + …(19) 376 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 See [6] and [9]. Also, we have to define the percentage of the overall sample saved (P.O.S.S) of SBq% as: 2 0 n P.O.S.S. J ( *, *) 100 n a b= * …(20) See [6] and [8]. And, finally, 1ˆp R)( q Œ represent the probability of avoiding the second sample. 3. C on clusions and Numerical Results The computations of Relative Efficiency [R.Eff(◊)] and Bias Ratio [B(◊)], Expected sample size [E(nÔq,R)], Expected sample size proportion [E(nÔq,R)/n], Percentage of the overall sample saved (P.O.S.S.) and probability of a voiding the second sample 1 ˆp( R)q Œ were used for the estimator SBq% . These computations (using Mat.LAB programs) were performed for n1 = 4, 6, 8, 10, 12, 16, u = (n2/n1) = 0.5, 1, 2, 3, 9, 12, l = (q0/q) = 0.25(0.25)2, a = 0.01,0.05,0.1. Some of these computations are given in tables (1)-(6). The observation mentioned in the tables leads to the following results: i.The Relative Efficiency [R.Eff(◊)] of SBq% are adversely proportional with small value of a especially when l = 1, i.e. a = 0.01 yield highest efficiency. ii.The Relative Efficiency [R.Eff(◊)] of SBq% has maximum value when q=q0(l=1), for each n1 and a, and decreasing otherwise (lπ1). This feature showed the important usefulness of prior knowledge which gave higher effects of proposed estimator as well as the important role of shrinkage technique and its philosophy. iii.Bias ratio [B(◊)] of SBq% are reasonably small when q=q0 for each n1, a, and increases otherwise. This property showed that the proposed estimator SBq% is very closely to unbiasedness property especially when q=q0. iv.The Effective interval of SBq% [the value of l which makes R.Eff.(◊) of SBq% greater than one] is approximately [0.75,1.25]. v.Bias ratio [B(◊)] of SBq% are reasonably small with small value of u. vi.R.Eff( SBq% ) is decreasing function with increasing of the first sample size n1, for each a and l. vii.The Expected value of sample size of SBq% is close to n1, especially when 0.5 £ l < 1 and start faraway otherwise. viii.Percentage of the overall sample saved 2 0 n J ( *, *) 100 n a bÈ ˘*Í ˙ Î ˚ is increasing value with increasing value of u(u = n2/n1) and decreasing value with increasing value of l ≥ 0.5. ix.R.Eff( SBq% ) is an increasing function with respect to u. This property showed the effective of proposed estimator using small n1 relative to n2 (or large n2) which gave higher efficiency and reduce the observation cost. x. The considered estimator SBq% is better that the classical estimator especially when qªq0, this will gave the effective of SBq% relative to â and also gave an important weight of prior knowledge, and the augmentation of efficiency may reach to tens times. 377 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 xi. The considered estimator SBq% is more efficient than the estimators introduced by [6] and [9] in the sense of higher efficiency. References 1. Epstein, B. and Sobcl, M., (1984), Some Theorems to Life Testing from an Exponential Distribution, Annals of Mathematical Statistics, 25. 2. Al-Kutobi, H.S., (2005), On Comparison Estimation Procedures for Parameter and Survival Function Exponential Distribution Using Simulation, Ph.D. Thesis, Baghdad University, College of Education (Ibn-Al-Haitham), Baghdad, Iraq. 3. Waikar,V.B., Schuurmann,F.J. and Raghunathan,T.E., (1984), On a Two-Stage Shrinkage Testimator of the Mean of a Normal Distribution, Commum. Statist-Theor. Meth., A13(15), 1901-1913. 4. Al-Joboori, 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),.1432-1442. 5. Thompson, J.R., (1968), Some Shrinkage Techniques for Estimating the Mean, J. Amer. Statist. Assoc, 63:113-122. 6. Al-Joboori, A.N., Khalaf, B.A. and Hamza, S., (2010), Estimate the Scale Parameter of Exponential Distribution Via Modified Two Stage Shrinkage Technique, Journal College of Education, No.(6) 62-75. 7. Katti,S.K., (1962), Use of Some a Prior Knowledge in the Estimation of Means from Double Samples, Biometrics, 18: 139-147. 8. Arnold, J.C. and Al-Bayyati, H.A., (1970), On Double Stage Estimation of the Mean Using Prior Knowledge, The Biometric Society, 26, No.4:782-800. 9. Handa, B.R., Kambo, N.S. and Al-Hemyari, Z.A., (1988), On Double Stage Shrunken Estimator of the Mean of Exponential Distribution, IAPQR, Trans. 13 No(1)19-33 10. Table (1): Showed Bias Ratio [B(◊)] and Relative Efficiency [R.Eff.(◊)] of SBq% w.r.t. u, n1 and l when a =0.01 u n 1 l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 4 R.Eff.(◊) B(◊) 0.3037 – 0.4257 0.7751 – 0.4877 2.0388 – 0.2489 3.8362 – 0.0078 1.8481 0.2163 0.7603 0.4145 0.3935 0.5809 0.2417 0.7124 6 R.Eff.(◊) B(◊) 0.2487 – 0.3848 0.5995 – 0.4906 1.9277 – 0.2488 5.4715 – 0.0142 1.6515 0.1830 0.5951 0.3256 0.3188 0.4093 0.2177 0.4416 8 R.Eff.(◊) B(◊) 0.2110 – 0.3351 0.4712 – 0.4918 1.6296 – 0.2477 5.6753 – 0.0193 1.3781 0.1493 0.5233 0.2398 0.3254 0.2643 0.2702 0.2500 3 4 R.Eff.(◊) B(◊) 0.2927 – 0.5147 0.8492 – 0.4915 2.4385 – 0.2473 5.8772 – 0.0046 2.0368 0.2205 0.6782 0.4186 0.3121 0.5830 0.1761 0.7102 6 R.Eff.(◊) B(◊) 0.2012 – 0.4599 0.6156 – 0.4928 2.0286 – 0.2458 7.6321 – 0.0076 1.4868 0.1915 0.4362 0.3319 0.2089 0.4081 0.1332 0.4285 8 R.Eff.(◊) 0.1543 0.4743 1.6192 7.4731 1.0937 0.3436 0.1978 0.1610 378 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 Table (2): Showed Expected Sample Size [E] and Expected Sample Size Proportion [Ep] of SBq% w.r.t. u and l when n1 = 4, a =0.01 u l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 EP E 0.6017 4.0400 0.5050 0.0400 0.5098 4.0783 0.5238 4.1906 0.5451 4.3612 0.5729 4.5835 0.6057 4.8458 0.6417 5.1338 3 EP E 0.4025 6.4406 0.2575 4.1200 0.2647 4.2348 0.2857 4.5719 0.3177 5.0835 0.3594 5.7506 0.4086 6.5374 0.462 7.4013 9 EP E 0.2830 11.3218 0.1090 4.3600 0.1176 4.7043 0.1429 5.7156 0.1813 7.2506 0.2313 9.2518 0.2903 11.6122 0.3551 14.2038 12 EP E 0.264 13.7624 0.0862 4.4800 0.0950 4.9391 0.1209 6.2875 0.1603 8.3341 0.2116 11.0024 0.2721 14.1496 0.3386 17.601 Table (3): Showed Expected Sample Size [E] and Expected Sample Size Proportion [Ep] of SBq% w.r.t. u and l when n1 = 6, a=0.01 u l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 B(◊) – 0.4023 – 0.4934 – 0.2436 – 0.0101 0.1591 0.2417 0.2516 0.2203 9 4 R.Eff.(◊) B(◊) 0.1863 – 0.5647 0.8696 – 0.4936 2.7523 – 0.2460 10.5122 – 0.0019 1.8503 0.2245 0.4648 0.4234 0.1855 0.5877 0.0953 0.7136 6 R.Eff.(◊) B(◊) 0.1116 – 0.5030 0.6019 – 0.4941 1.9471 – 0.2438 11.9403 – 0.0031 1.0260 0.1979 0.2380 0.3377 0.1023 0.4103 0.0614 0.4244 8 R.Eff.(◊) B(◊) 0.0790 – 0.4414 0.4568 – 0.4944 1.4155 – 0.2410 10.4676 – 0.0041 0.6523 0.1660 0.1695 0.2446 0.0911 0.2462 0.0726 0.2053 12 4 R.Eff.(◊) B(◊) 0.1555 – 0.5723 0.8581 – 0.4939 2.7448 – 0.2458 12.2188 – 0.0015 1.6977 0.2252 0.3980 0.4243 0.1538 0.5885 0.0774 0.7142 6 R.Eff.(◊) B(◊) 0.0907 – 0.5096 0.5891 – 0.4943 1.8572 – 0.2435 13.2757 – 0.0024 0.8771 0.1989 0.1935 0.3387 0.0814 0.4108 0.0484 0.4239 8 R.Eff.(◊) B(◊) 0.0633 – 0.4474 0.4460 – 0.4945 1.3121 – 0.2406 11.2646 – 0.0032 0.5400 0.1671 0.1352 0.2451 0.0718 0.2454 0.0570 0.2030 379 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 1 EP E 0.6457 7.7484 0.5050 6.0600 0.5151 6.1818 0.5458 6.5492 0.5954 7.1444 0.6580 7.8965 0.7251 8.7016 0.7889 9.4664 3 EP E 0.4685 11.2451 0.2575 6.1800 0.2727 6.5453 0.3187 7.6477 0.3930 9.4331 0.4871 11.6896 0.5877 14.1047 0.6833 16.3991 9 EP E 0.3623 21.7354 0.1090 6.5400 0.1273 7.6358 0.1824 10.9432 0.2717 16.2992 0.3845 23.0687 0.5052 30.3141 0.6200 37.1973 12 EP E 0.3459 26.9806 0.0862 6.7200 0.1049 8.1810 0.1614 12.5909 0.2530 19.7323 0.3687 28.7583 0.4925 38.4189 0.6102 47.5964 Table (4): Showed Expected Sample Size [E] and Expected Sample Size Proportion [Ep] of SBq% w.r.t. u and l when n1 = 8, a =0.01 u l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 EP E 0.6885 11.0165 0.5050 8.0800 0.5215 8.3445 0.5743 9.1891 0.6584 10.5337 0.7532 12.0509 0.8380 13.4077 0.9022 14.4345 3 EP E 0.5328 17.0495 0.2575 8.2400 0.2823 9.0336 0.3615 11.5674 0.4875 15.6010 0.6298 20.1526 0.7570 24.2232 0.8532 27.3035 9 EP E 0.4394 35.1483 0.1090 8.7200 0.1388 11.1008 0.2338 18.7021 0.3850 30.8029 0.5557 44.4577 0.7084 56.6697 0.8239 65.9105 12 EP E 0.4250 44.1980 0.0862 8.9600 0.1167 12.1344 0.2141 22.2694 0.3693 38.4039 0.5443 56.6102 0.7009 72.8930 0.8194 85.2141 Table (5): Showed the Percentage of overall Sample Saved (P.O.S.S.) of SBq% w.r.t. u, n1 and l when a =0.01 u n 1 l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 4 P OSS 39.8308 49.5000 49.0218 47.6172 45.4853 42.7058 39.4275 35.8280 6 35.4302 49.5000 48.4854 45.4230 40.4637 34.1956 27.4869 21.1136 8 31.1469 49.5000 47.8467 42.5680 34.1646 24.6822 16.7843 9.7843 12 23.3572 49.5001 46.2829 35.4833 21.0689 9.8173 3.7270 1.1963 3 4 P OSS 59.7462 74.2500 73.5327 71.4259 68.2279 64.0587 59.1412 53.7420 6 53.1453 74.2500 72.7281 68.1344 60.6956 51.2934 41.2304 31.6704 8 46.7203 74.2500 71.7700 63.8520 51.2469 37.0233 24.3024 14.6765 380 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 Table (6): Showed the Probability of a Voiding Second Sample [Av] w.r.t. u, n1 and l when a =0.01 12 35.0358 74.2501 69.4244 53.2249 31.6033 14.7260 5.5905 1.7944 9 4 P OSS 71.6954 89.1000 88.2392 85.7110 81.8735 76.8704 70.9694 64.4904 6 63.7743 89.1000 87.2737 81.7613 72.8347 61.5521 49.4764 38.0045 8 56.0643 89.1000 86.1240 76.6224 61.4963 44.4279 29.1628 17.6118 12 42.0430 89.1001 83.3093 63.8699 37.9240 17.6712 6.7086 2.1533 12 4 P OSS 73.5338 91.3846 90.5017 87.9087 83.9728 78.8414 72.7892 66.1440 6 65.4095 91.3846 89.5115 83.8578 74.7022 63.1304 50.7451 38.9790 8 57.5019 91.3846 88.3323 78.5871 63.0731 45.5671 29.9106 18.0634 12 43.1210 91.3848 85.4454 65.5076 38.8964 18.1243 6.8807 2.2085 u n1 l 0.25 0.5 0.75 1 1.25 1.5 1.75 2 1 4 AV 0.7966 0.9900 0.9804 0.9523 0.9097 0.8541 0.7885 0.7166 6 0.7086 0.9900 0.9697 0.9085 0.8093 0.6839 0.5497 0.4223 8 0.6229 0.9900 0.9569 0.8514 0.6833 0.4936 0.3240 0.1957 12 0.4671 0.9900 0.9257 0.7097 0.4214 0.1963 0.0745 0.0239 3 4 AV 0.766 0.9900 0.9804 0.9523 0.9097 0.8541 0.7885 0.7166 6 0.7086 0.9900 0.9697 0.9085 0.8093 0.6839 0.5497 0.4223 8 0.6229 0.9900 0.9569 0.8514 0.6833 0.4936 0.3240 0.1957 12 0.4671 0.9900 0.9257 0.7097 0.4214 0.1963 0.0745 0.0239 9 4 AV 0.7966 0.9900 0.9804 0.9523 0.9097 0.8541 0.7885 0.7166 6 0.7086 0.9900 0.9697 0.9085 0.8093 0.6839 0.5497 0.4223 8 0.6229 0.9900 0.9569 0.8514 0.6833 0.4936 0.3240 0.1957 12 0.4671 0.9900 0.9257 0.7097 0.4214 0.1963 0.0745 0.0239 12 4 AV 0.7966 0.9900 0.9804 0.9523 0.9097 0.8541 0.7885 0.7166 6 0.7086 0.9900 0.9697 0.9085 0.8093 0.6839 0.5497 0.4223 8 0.6229 0.9900 0.9569 0.8514 0.6833 0.4936 0.3240 0.1957 12 0.4671 0.9900 0.9257 0.7097 0.4214 0.1963 0.0745 0.0239 381 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 البيزي –حول مقدر التقلص ذو المرحلتين لمعلمة القياس للتوزيع االسي عباس نجم سلمان ، منى داود سلمان جامعة بغداد -كلية التربية ابن الهيثم -قسم الرياضيات 2011األول كانون 7قبل البحث في : 2011 تشرين األول 19استلم البحث في : الخالصة (MSE)لتقليــل متوســط مربعــات الخطــأ (DSSBE)البيــزي ذي المــرحلتين –يتعلــق موضــوع البحــث بمقــدر الــتقلص المتــوافرة حـــول q0حــول المعلومــات المســبقة (R)للتوزيــع االســـي عنــد المنطقــة qلمعلمــة القيــاس q̂لمقــدر االمكــان االعظــم بشكل تقدير ابتدائي فضالً عن تقليل كلفة المعينة والتجارب. (q)المعلمة الحقيقية ة عنـــدما يكـــون اســــتهالك الوقـــت او كلفـــة المعاينــــة او التجـــارب عاليـــاً جــــداً فـــان طريقـــة الــــتقلص ذا المـــرحلتين تكـــون مناســــب للحصــول علـــى مقـــدر يقلــل حجـــم العينـــة المتوقــع ومـــن ثـــم التقليــل مـــن هـــذه الكلــف. ومـــن خـــواص هــذا المقـــدر ايضـــاً انـــه ذو بشكل مناسب. Rومنطقة قبول (◊)yصغير السيما عند اختيار عامل تقلص موزون (MSE)متوسط مربعات خطأ ، وحجـــــــم العينـــــــة المتوقـــــــع [(◊)R.Eff]، والكفايـــــــة النســـــــبية (MSE)اشــــــتقت معـــــــادالت التحيـــــــز، ومتوســـــــط مربعـــــــات الخطـــــــأ [E(n/q,R)] وحجـم العينــة المتوقــع النســبي ،[E(n,q,R)/n] 1، واحتماليــة تجنـب العينــة الثانيــةˆp( R)q Œ ونســبة االدخــار ، 2الكلي المئوية للعينة 1 n ˆ[ p[ R) 100] n q Œ .(DSSBE)للمقدر المقترح * المقتـــرح عنــدما يكــون المقـــدر المقتــرح هــو مقـــدر االختبــار األولـــي (DSSBE)أعطيــت النتــائج العدديـــة واالســتنتاجات للمقــدر .aلمستوى معنوية اجريت المقارنات مع المقدر الكالسيكي وبعض المقدرات المقترحة في الدراسات االخيرة لبيان فائدة المقدر المقترح. التوزيــــع االســـي، مقـــدر االمكــــان األعظـــم، المقــــدر البيـــزي، مقـــدر الــــتقلص ذو المـــرحلتين، متوســــط الكلمـــات المفتاحيــــة: مربعات الخطأ، الكفاية النسبية. 1.1 The Model: This section is concerned with pooling approach between shrinkage estimation that uses a prior information about unknown parameter as initial value and Bayesian estimation that uses a prior information about unknown parameter as a prior distri... A proposed Double Stage Shrinkage-Bayesian Estimator (DSSBE) has the following form where represent to Bayes estimator for ( on n1 observation, R is suitable region (say pretest region) and ; 0 ( ( 1 is shrinkage weight factor which may be a function of or constant, see [3], [4] and [9]. 2.1 DSSBE() Using Constant Shrinkage Weight Factor قسم الرياضيات - كلية التربية ابن الهيثم - جامعة بغداد استلم البحث في : 19 تشرين الأول 2011 قبل البحث في : 7 كانون الأول 2011 يتعلق موضوع البحث بمقدر التقلص – البيزي ذي المرحلتين (DSSBE) لتقليل متوسط مربعات الخطأ (MSE) لمقدر الامكان الاعظم لمعلمة القياس ( للتوزيع الاسي عند المنطقة (R) حول المعلومات المسبقة (0 المتوافرة حول المعلمة الحقيقية (() بشكل تقدير ابتدائي فضل... عندما يكون استهلاك الوقت او كلفة المعاينة او التجارب عاليا ً جدا ً فان طريقة التقلص ذا المرحلتين تكون مناسبة للحصول على مقدر يقلل حجم العينة المتوقع ومن ثم التقليل من هذه الكلف. ومن خواص هذا المقدر ايضا ً انه ذو متوسط مربعات خطأ (MSE) صغير لاسيما عند ... اشتقت معادلات التحيز، ومتوسط مربعات الخطأ (MSE)، والكفاية النسبية [R.Eff(()]، وحجم العينة المتوقع [E(n/(,R)]، وحجم العينة المتوقع النسبي [E(n,(,R)/n]، واحتمالية تجنب العينة الثانية ، ونسبة الادخار الكلي المئوية للعينة للمقدر المقترح (DSSBE). أعطيت النتائج العددية والاستنتاجات للمقدر (DSSBE) المقترح عندما يكون المقدر المقترح هو مقدر الاختبار الأولي لمستوى معنوية (. اجريت المقارنات مع المقدر الكلاسيكي وبعض المقدرات المقترحة في الدراسات الاخيرة لبيان فائدة المقدر المقترح.