Microsoft Word - 463-489 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|484    Minimax Shrunken Technique for Estimate Burr X Distribution Shape Parameter Abbas Najim Salman abbasnajim66@yahoo.com Dept. of Mathematics/College of Education for Pure Science (Ibn-Al-Haitham )University of Baghdad Maymona M. Ameen memo_ameen_89@yahoo.com University of Fallujah A. E.Abdul-Nabi ahmedissa432@yahoo.com Dept. of Mathematics/ College of Education for Pure Science/(Ibn-Al- Haitham)/ University of Baghdad Abstract:    The present paper concern with minimax shrinkage estimator technique in order to estimate Burr X distribution shape parameter, when prior information about the real shape obtainable as original estimate while known scale parameter. Derivation for Bias Ratio, Mean squared error and the Relative Efficiency equations. Numerical results and conclusions for the expressions mentioned above were displayed. Comparisons for proposed estimator with most recent works were made. Keywords: Minimax Shrunken Technique, Burr X Distribution, MLE, MSE and Efficiency. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|485    1. Introduction Burr X distribution would be able to use effectively in modeling data and also modeling lifetime data. [2],[4],[8],[9],[10]and [11] . Let X be a R.V. follows Burr X distribution with two parameters  and  . The (C.D.F.) and (P.D.F) of X are represented respectively as below; 𝐹 𝑥; 𝛼, 𝜆 1 𝑒 𝐹𝑜𝑟 𝑥 0, 𝛼 0 , 𝜆 0 … (1) 𝑓 𝑥; 𝛼, 𝜆 2𝛼𝜆 𝑥𝑒 1 𝑒 𝑓𝑜𝑟 𝑥 0, 𝛼 0, 𝜆 0 0 𝑜. 𝑤 … . 2 Wherever,  &  refer respectively to shape and scale parameters. Pretest shrunken estimator of  determination as below: 𝛼  𝛼 𝛼 1  𝛼 𝛼 𝑖𝑓𝛼 ∈ 𝑅  𝛼 𝛼 1  𝛼 𝛼 𝑖𝑓𝛼 ∉ 𝑅 … 3 Anywhere , i iˆ ˆ( ), 0 ( ) 1      , i = 1, 2 refer to weightiness shrunken identifying acceptance of ̂ while (1 – i ()) agreeing confidence of 0, and well-known that i ˆ( )  may be constant or function depend on ̂ (ad hoc basis), and R refer to pretest region for testing Ho:  = 0 opposite H1:   0 via statistics test T ( ̂/0).[1]and [12]. Minimax procedure of estimation is a non-classical approach for estimation presented by Abraham Wald (1945) from the model of game theory. The most significant basics in the minimax approach are the description of the prior distribution and the loss function used. Quadratic loss functions will be considered to find the Minimax estimators of Burr X distribution in present research; [3], [5], [6] and [7]. The goal of this paper, toward suggested shrunken estimator technique α defined by (3) through employment the Minimax estimator (α =α ) instead of the classical estimator (MLE) ̂ to estimate  Burr X distribution with known . Expressions for main statistical indicators of the proposed estimator α were derived and obtained. Discussions and numerical results were achieved and presented in appropriated table. Comparisons were done between the suggested estimators with standard minimax estimator α as well as with some of the last studies. 2. Minimax Estimator of  In this section, we obtain the minimax estimators of the parameter α for the Burr X distribution depends on Lehmann s theorem in (1950). Agreeing with this theorem, consequently obtain the minimax estimator of the parameter 𝛼 as α α n 2 / ∑ ln 1 e of the Burr X distribution in case of quadratic loss function. It is known, the MLE of 𝛼 in Burr X distribution when  equal to one has the following form ; [8]and[9] IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|486    𝛼 ∑ … (4) Reminder, when xi ~ Burr X (,1), then 𝛼 ∑ ln 1 𝑒 follow Gamma population including n shape and 1 scale; G(n,1). [4], [8] and [9]. i.e.; 𝐸 𝛼 𝛼 𝑎𝑛𝑑 𝑣𝑎𝑟 𝛼 𝛼 . By using (4), then Bias α E α α α n 1 𝑉ar α n 1 n . Var α α MSE α Var α Bias α α n 1 3. Minimax Shrinkage Estimator  This section involves the suggestion of special case of pretest shrunken estimator α which is defined in (3) using Minimax estimator 𝛼 for estimate  of Burr X distribution when  = 1, 𝛼 𝛼 , 𝑅 𝛼 , 𝑖𝑓𝛼 ∈ 𝑅 𝑘 𝛼 𝛼 𝛼 , 𝑖𝑓 𝛼 ∉ 𝑅 … 5   When ψ1 𝛼 0 𝑎𝑛𝑑 ψ2 𝛼 𝑘 . ; 𝑐 𝜒 , 100 such that (0 < k < 1) , and R refer to pretest region for testing Ho:  = 0 vs. H1:   0 through level of significance () via statistic test function TT i.e.;R a b …(6) Where a Χ , and b Χ , The derivation of Bias equation for α is 𝐵𝑖𝑎𝑠 𝛼|𝛼; 𝑅 𝐸 𝛼 𝛼 𝛼 𝛼 𝑓 𝛼 𝑑 𝛼 𝑘 𝛼 𝛼 𝛼 𝛼 𝑓 𝛼 𝑑𝛼 Wherever, refers to supplemented region of R , and R IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|487    f α n 2 α αB n 1 .e n 2 α αB Γ n . n 2 α , αB 0 … (7) We conclude, 𝐵𝑖𝑎𝑠 𝛼|𝛼; 𝑅 𝛼 𝑛 2 𝑐 𝜁 𝑛 𝑐 𝜁 𝜁 1 𝑛 2 𝑐 𝜁𝐽 𝑎∗, 𝑏∗ 1 𝑐 𝜁 𝐽 𝑎∗, 𝑏∗   Anywhere, 𝐽 𝑎∗, 𝑏∗ 𝑑𝑦 ; 𝑙 0,1,2, … ∗ ∗ …(8) Also ζ α α , a∗ 2ζ . a , b∗ 2ζ . b and y 𝑛 2 α α … 9 Bias Ratio [B (𝛼 )] well-defined as: B α Bias α |α; R α … 10 Derivation of MSE (𝛼 ) will be as below:- MSE α|α; R E α α 𝛼 𝛼 𝑓 𝛼 𝑑 𝛼 𝑘 𝛼 𝛼 𝛼 𝛼 𝑓 𝛼 𝑑𝛼 𝛼 𝛼 𝑓 𝛼 𝑑 𝛼 𝑛 2 𝛼 𝑐 𝑛 2 𝛼 𝛼𝑐 𝛼 𝛼 𝑓 𝛼 𝑑𝛼 𝑛 2 𝛼 𝑐 𝑛 2 𝛼 𝛼𝑐 𝛼 𝛼 𝑓 𝛼 𝑑𝛼  Some by simplification, the above equation became:  𝑀𝑆𝐸 𝛼|𝛼; 𝑅 𝛼 𝑛 2 𝑐 𝜁 2 𝑛 𝑛 2 𝑐 𝜁 𝑛 𝑛 1 𝑐 𝜁 2 𝑛 2 𝑐 𝜁 𝜁 1 2 𝑛 𝑐 𝜁 𝜁 1 𝜁 1 𝑛 2 𝑐 𝜁 𝐽 𝑎∗, 𝑏∗ 2 𝑛 2 𝑐 𝜁 𝐽 𝑎∗, 𝑏∗ 1 𝑐 𝜁 𝐽 𝑎∗, 𝑏∗ 2 𝑛 2 𝑐 𝜁 𝜁 1 𝐽 𝑎∗, 𝑏∗ 2 1 𝑐 𝜁 𝜁 1 𝐽 𝑎∗, 𝑏∗   …(11) The Relative Efficiency of α to α is : R. Eff α|α; Ri MSE α MSE α|α; R … 12   4. Discussion and Numerical Results The calculations of the equations (10),(11) and (12), were performed for n = 4,6,8,10,12, 𝜁= 0/ = 0.25(0.25)2 and  = 0.01,0.05,0.1, for the estimator α . These calculations are prearranged in attached table and consequence the following: IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|488    i. R.Eff(α ) has opposite relational with small value of  especially when 𝜁 = 1 ii. When 𝜁 =1, the R.Eff (α ) takes height value for each n , , and lessening otherwise (𝜁 1). iii. B(α ) are almost small when 𝜁 = 1 for all n,  , and increases else. iv. Effective interval of α . [Value of 𝜁 that makes R.Eff(α ) larger one] is [0.5, 1.5]. v. R.Eff(α .) is reducing function with n, for each  and 𝜁. vi. The suggested estimator α dominate ̂ especially at 𝜁 = 1. vii.The suggested estimator α dominate the estimators presented by [5],[7] in the sense of minimum mean squared error. Table (1): B(𝛂 ) and R.Eff (𝛂 w.r.t. , n1 and   ζ Δ n R.Eff(- ) B(-) 0.25 0.50 0.75 1 1.25 1.50 1.75 2 0.01 4 R.Eff(- ) B(-) 0.5926898 - 0.7499385 1.3334412 - 0.4999798 5.3336397 - 0.2499928 1.8766999E+6 -2.507439E-5 5.3411594 0.2498139 1.336686 0.4993667 0.594938 0.7485095 0.3352064 0.9971855 6 R.Eff(- ) B(-) 0.3557834 - 0.7497598 0.8002317 - 0.4999276 3.20053 - 0.2499792 8.5776432E+5 -2.524196E-5 3.2066372 0.2497366 0.8031576 0.499006 0.3578644 0.7475579 0.201871 0.9953259 8 R.Eff(- ) B(-) 0.2542997 - 0.7495111 0.5718069 - 0.4998344 2.286422 - 0.2499612 4.9532531E+5 -2.532473E-5 2.2919673 0.2496521 0.5746331 0.4985883 0.2563705 0.7464494 0.1448072 0.9932013 10 R.Eff(- ) B(-) 0.1979233 -0.749256 0.4449856 - 0.4996955 1.778652 - 0.2499383 3.234304E+5 -2.537393E-5 1.7839172 0.2495605 0.4477738 0.4981157 0.2000606 0.7452037 0.1131537 0.9908772 12 R.Eff(- ) B(-) 0.1620404 - 0.7490176 0.3643486 - 0.4995104 1.4555841 - 0.2499104 2.280651E+5 -2.540652E-5 1.4606951 0.2494616 0.3671231 0.4975907 0.1642812 0.7438406 0.0930405 0.9884106 0.05 4 R.Eff(- ) B(-) 0.5927787 - 0.7498823 1.3336565 - 0.4999394 5.3342047 - 0.2499795 5.9151045E+5 -9.939819E-5 5.3539023 0.2495134 1.3401523 0.4987173 0.5965971 0.7474654 0.336158 0.9957716 6 R.Eff(- ) B(-) 0.3558701 - 0.7496684 0.8005953 -0.499814 3.201717 - 0.2499328 2.6800421E+5 -9.980886E-5 3.2167515 0.2493392 0.8060428 0.4981078 0.3592242 0.7461391 0.2026157 0.9934961 8 R.Eff(- ) B(-) 0.2543477 - 0.7494403 0.5722693 - 0.4996323 2.2879665 - 0.2498766 1.5388055E+5 -1.000067E-4 2.3009895 0.2491563 0.5772602 0.4974474 0.2575699 0.7447081 0.1454268 0.991088 10 R.Eff(- ) B(-) 0.1979429 -0.749219 0.4455051 -0.499404 1.78046 -0.249811 1.0007843E+5 -1.001240E-4 1.7923586 0.2489647 0.4502451 0.4967414 0.201143 0.7431971 0.1136772 0.9886024 12 R.Eff(- ) B(-) 0.1620468 - 0.7490028 0.3648894 - 0.4991402 1.4576081 - 0.2497362 7.0362889E+4 -1.002015E-4 1.4687862 0.2487646 0.3694813 0.495995 0.1652655 0.7416269 0.0934839 0.9860785 0.1 4 R.Eff(- ) B(-) 0.592807 - 0.7498643 1.3337917 - 0.4999141 5.334351 -0.249976 3.7785035E+5 -1.770097E-4 5.3636228 0.2492857 1.3423159 0.4983142 0.5974921 0.7469051 0.3366172 0.9950928 6 R.Eff(- ) B(-) 0.3558868 - 0.7496508 0.8008029 - 0.4997492 3.2025271 -0.249901 1.7103776E+5 -1.775499E-4 3.2241357 0.2490514 0.8077279 0.4975869 0.3598975 0.7454414 0.2029403 0.9927035 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1820 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|489      References [1] A.N.Al-Joboori, , 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. 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Assoc,.63,.113-122. 1968 8 R.Eff(- ) B(-) 0.2543533 - 0.7494321 0.5724995 - 0.4995319 2.2890869 - 0.2498152 9.8133771E+4 -1.778086E-4 2.3073582 0.2488095 0.5787131 0.4968218 0.2581224 0.7439125 0.1456745 0.9902491 10 R.Eff(- ) B(-) 0.1979443 - 0.7492164 0.445731 - 0.4992775 1.7817676 - 0.2497191 6.3788371E+4 -1.779616E-4 1.798152 0.2485602 0.4515479 0.4960242 0.2016105 0.7423382 0.1138709 0.987767 12 R.Eff(- ) B(-) 0.1620471 - 0.7490021 0.3650962 - 0.4989989 1.4590432 - 0.2496131 4.4829877E+4 -1.780627E-4 1.474204 0.2483035 0.3706716 0.4951987 0.1656663 0.7407343 0.0936367 0.9852809