Conseguences of soil crude oil pollution on some wood properties of olive trees Mathematics |224 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Comparison of the Suggested loss Function with Generalized Loss Function for One Parameter Inverse Rayleigh Distribution Emad Farhood AL-Shareefi Dept.of Accounting Techniques/ Technical College / Southern Technical University / Dhi_Qar Received in : 13/ April /2017 , Accepted in : 18 /June/ 2017 Abstract The experiences in the life are considered important for many fields, such as industry, medical and others. In literature, researchers are focused on flexible lifetime distribution. In this paper, some Bayesian estimators for the unknown scale parameter of Inverse Rayleigh Distribution have been obtained, of different two loss functions, represented by Suggested and Generalized loss function based on Non-Informative prior using Jeffery's and informative prior represented by Exponential distribution. The performance of estimators is compared empirically with Maximum Likelihood estimator, Using Monte Carlo Simulation depending on the Mean Square Error (MSE). Generally, the preference of Bayesian method of Suggested loss function with Exponential informative prior are the best estimator compared to others. Key words: Inverse Rayleigh Distribution, Bayes estimator, Suggested loss function(SLF), Generalized Loss Function(GLF), Maximum likelihood (MLE), Jeffery prior; Exponential informative prior MSE. Mathematics |225 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Introduction In the term of reliability studies many applications used the Distribution of Inverse Rayleigh. It was introduced in literary (Trayer 1964) of reliability with survival studies, life distribution which characterized via a monotonic failure rate. In 1972 Voda has explained that is lifetimes distribution that related with served types of experimental unite can approximated by the Inverse Rayleigh distribution [1] in this regard let consider to be a randomize sample of independent observation from a one parameter Inverse Rayleigh distribution with probability of density (p.d.f) to scale parameter ( ) as shown in[2]. ( ) (1) The Corresponding Cumulative of distribution Function ( CDF) is: ( ) (2) AL-Shareefi. E. F. (2015). Suggested loss Function in estimating the Scale parameter for Laplace distribution [3] . ( ̂ ) (∑ )( ̂ ) Maximum Likelihood Estimator (MLE). Maximum Likelihood can be obtained for the scale parameter , as following: Suppose to be random sample with density function(1). The likelihood function illustrated by [4]. ( ) ∏ [ ∑ ] (3) R. A. Fisher (1920) proposed The Maximum Likelihood method [5], since then used extensively. This method consider the most popular algorithm to estimate the unknown parameter to specify the probability function ( ) , based on the observation ( ) which were independently sample from the Inverse Rayleigh distribution. In Equation (3). By using the logarithm of the likelihood function and differentiation with respect to ( ), will get: ( ) ∑ ( ) Hence: ̂ ∑ ∑ (4) Some Bayes Estimators 1- Bayes Estimator by using Jeffreys prior Information [6]. We assumed that has non-information prior density, which is defined as: √ ( ) Where ( ) that considered Fisher information which is defined as following: ( ) * ( ) + ( ) √ ( ( ) ) (5) where b is a constant. ( ) ( ) ( ) ( ) Mathematics |226 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 ( ) ( ) Hence, we get: * ( ) + (6) by substitution (6) with (5), will get: ( ) √ The posterior density function is: ( ) ( ) ( ) ∫ ( ) ( ) ( ) ∫ Hence, the posterior density functions of ( ) with Jefferys prior is: ( ) (7) The posterior density function of ( )is recognized as the density of Gamma distribution with parameters n and T i.e: ( ) Hence: ( ) ( ) ∑ 2- Bayes Estimator by using Exponential prior Information [4]. Hence, ( ) reflect the information of prior exponential distribution: ( ) (8) ( ) ( ) ( ) ∫ ( ) ( ) ( ) ∫ ( ) ( ) ( ) ( ) ( ) (9) Notice that: ( ) ∑ 3- Bayesian Estimators under Suggested Loss function A new loss function suggested here which is called Modified Generalized loss function defined as following [4]: ( ̂ ) (∑ )( ̂ ) , C=0, 1, 2, …, n is constant Where ( ̂ ) is modified by Al-Sherefi Then, the Risk function under the Suggested loss function denoted by ( ̂ ) will be: ( ̂ ) [ ( ̂ )] ∫ ( ∑ )( ̂ ) ( ) Mathematics |227 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 ̂ ( ) ̂ ( ) ( ) ̂ ( ) ̂ ( ) ( ) ̂ ( ) ̂ ( ( ) ) ( ( ) ) By taking partial derivative of ( ̂ ) with respect to ̂ and making it equal to zero yields: ( ̂ ) ̂ ̂ ( ) ( ) ̂ ( ) ( ) ̂ ( ) ( ( ) ) ̂ ( ) ̂ ( ) ̂ ( ) ( ) ( ) ( ( ) ) ̂ ( ) ( ) ( ( ) ) ( ) ( ) ( ) (10)  With Jeffrey's prior information According to the posterior density function ( ), we derived ( ) ( ). Since ( ) ( ) ∫ ( ) ( ) ( ) ( ) ∫ ( ) ( ) ( ) Which can be substituted to (10) to obtain Bayes estimator Jeffrey's prior information: 1- When k =1 and c=1 ̂ ( ) ( ) ( ) ( ) (11) 2- When k =2 and c=1 ̂ ( ) ( ) ( ) ( ) ( ) (12) 3- When k =1 and c=2 ̂ ( ) ( ) ( ) (13) 4- When k =2 and c=2 ̂ ( ) ( ) ( ) ( ) ( ) ( ) (14) 5- When k =1 and c=3 ̂ ( ) ( ) ( ) ( ) (15) 6- When k =2 and c=3 ̂ ( ) ( ) ( ) ( ) ( ) (16)  With Exponential prior information According to the posterior density function ( ) , can derived ( ) ( ) and get some estimators for based on Exponential prior as follows: ( ) Mathematics |228 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 ( ) ∫ ( ) ( ) ( ) ( ) ∫ ( ) ( ) ( ) Putting k=1 and c=1 we get: ̂ ( ) ( ) ( ) ( ) (17) Putting k=2 and c=1 we get: ̂ ( ) ( ) ( ) ( ) ( ) ( ) (18) Putting k=1 and c=2 we get: ̂ ( ) ( ) ( ) ( ) (19) Putting k=2 and c=2 we get: ̂ ( ) ( ) ( ) ( ) (20) Putting k=1 and c=3 we get: ̂ ( ) ( ) ( ) ( ) (21) Putting k=2 and c=3 we get: ̂ ( ) ( ) ( ) ( ) ( ) (22) 3.4 Bayesian Estimator under Generalized Loss function. The Generalized loss function [7] can be written as: ( ̂ ) (∑ )( ̂ ) Where aj, j=0, 1,2 , …,k are constant So, Risk function of Generalized loss function, denoted by RGS( ̂ )is: RG( ̂ ) [ ( ̂ )] ∫ ( ̂ ) ( ) ∫( )( ̂ ̂ ) ( ) ̂ ̂ ( ) ( ) ̂ ( ) ̂ ( ) ( ) ̂ ( ) ̂ ( ) ( ) By taking the partial derivative for RGS( ̂ ) with respect to ̂ and make it equal to zero yields: ̂ ( ) ( ) ( ) ( ) ( ) ( ) (23)  With Jeffrey's prior information From the posterior density function ( ) , can show the Bayes estimator of One parameter Inverse Rayleigh distribution under Generalized loss function , ̂ can be ̂ ( ) ( )( ) ( ) ( )( ) ( ) (24) In this paper, the first and second polynomials are used as follows: ̂ ( ) (25) Mathematics |229 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 ̂ ( ) ( )( ) ( ) (26)  With Exponential prior information When consider posterior density function ( ) , the Bayes estimator of One Parameter to Inverse Rayleigh distribution when generalized loss function ̂ can be obtained as follows: ̂ ( ) ( )( ) ( )( ) ( ) ( ) ( ) ( ) In this paper, the first and second polynomials are used as follows: ̂ ( ) ( )( ) ( ) (27) ̂ ( ) ( )( ) ( )( )( ) ( ) ( )( ) (28) Simulation Results In this research Q basic program is used to simulate the results and the tables considered in Monte Carlo simulation study is explained for comparing 17 estimators of the scale parameter with One Parameter Inverse Rayleigh distribution, using Mean Square Error (MSE) of an estimator which is defined as follows: ( ̂) ∑ ( ̂ ) Where, R is the number of replications. we generated R=5000 samples each with size ( n=5, 10, 30, 50, 100) respectively, from the Inverse Rayleigh distribution. With the scale parameter ( ) ( ) and one values of the hyper- parameter of Exponential Prior (b=0.8). Discussion The experimental results of simulation study for estimating the scale parameter of Inverse Rayleigh distribution are summarized and tabulated in (1 and 2) involved both expected values and MSE , s . The result can be summarized as the following important points. 1- In general, Bayesian methods with proposed loss function achieved better performance when compared with generalized one. 2- In term of estimation Bayes method performed good with two different loss functions when consider exponential prior, noted better performance of corresponding estimator when consider Jeffrey's non- informative prior. 3- The values of MSE's for Bayes estimators using Exponential informative prior, are decreasing with using the value of the parameter of Exponential Prior (b=0.8) 4- For all estimates, (MSE's) of the scale parameter is getting high with the increase of the scale in parameter value, with all cases. 5- Tables (1) shows the Bayes estimator performance under suggested loss function with Exponential prior information (ME3) is better estimator, when comparing with other estimator this includes all sample sizes and values of the scale parameter, and followed by the (ME4). 6- In the tables (2), Bayes estimator performance under suggested loss function with Exponential of prior information (ME1) also become best estimator, when comparing with other estimator including all size samples and all values of the scale parameter, and followed by the (ME2). Mathematics |230 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Conclusions The results in table (1) show that, Bayes estimators under suggested loss function (c = 2, k = 1) with Exponential prior (ME3 and ME4) are the best estimators when comparing with other estimators, for all sample sizes. The results in table (2) show that, Bayes estimators under suggested loss function (c = 1, k = 1) with Exponential prior (ME1 and ME2) are the best estimators comparing to other estimators, for all sample sizes. Recommendations 1- Preferably use a suggested loss function with Exponential prior information of Inverse Rayleigh distribution, for all sample size. 2- The researchers in literature of life research field use of Inverse Rayleigh distribution with a Suggested loss function with Exponential prior information (ME3), to get best estimator. Acknowledgment The author would like to thank the Ministry of Higher Education and Scientific Research /Iraq, Southern Technic University\ Technical College in Dhi Qar for Technical Support. References 1. Voda, R. Gh (1972). One Inverse Rayleigh Variable " , Rep. Stat. Res. Vol.19, No.4, PP.15-21. 2. Shawky, A. I. and Badr, M. M. (2012) " Estimation and Prediction from The Inverse Rayleigh Model Based on Lower Record Statistics " life Science Journal, PP.985. 3. AL-Shreefi, E. F. (2015). " Some Bayes Estimator for the Scale Parameter of the Laplace distribution / A comparison study " Thesis submitted to the college of science at AL.Mustansiriya University. 4. Rasheed, H. A. (2015). " A Comparison of the classical Estimators with the Bayes Estimator of one parameter Inverse Rayleigh distribution ". International Journal of Advanced Research, Volume 3, ISSUE, 738-749. 5. Rohatgi, V. K. (1997). " An Introduction to Probability Theory and Mathematical Statistics " . John Wiley and Sons. 6. Chrek, D.J. (1985). " A Comparison of Estimation Techniques for Thither Three Parameter Pareto distribution " , Science in space operation Thesis Faculty of the School of Engineering, Ohio. 7. AL-Shreefi, E. F. and Rasheed, H. A. (2016). " Using a suggested loss function and generalized loss function to estimate the scale parameter of Laplace distribution " conference of Iraqi statistical Association (I.S.A). Mathematics |231 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (1) :Estimated Value and MSE of Different Estimates of when Estimator n Criteria 5 10 30 50 100 MLE EXP 1.003300 1.00212 1.000135 .999999 1.00086 MSE 0.196414 0.99520 0.0332317 0.199787 0.0099792 MJ1 EXP 1.003663 1.002243 1.000171 1.000015 1.00097 MSE 0.1966515 0.99569 0.0323591 0.0199804 0.0099796 MJ2 EXP 1.003804 1.00228 1.000179 1.00001 1.00098 MSE 0.1968111 0.0959146 0.0332375 0.019980 0.0099797 MJ3 EXP 0.8354679 0.9101985 0.9669396 0.97942 0.99007 MSE 0.1629812 0.089998 0.0320929 0.019550 0.0098143 MJ4 EXP 0.836459 0.9112321 0.9680131 0.980509 0.991166 MSE 0.1634007 0.090224 0.0321699 0.019590 0.0098661 MJ5 EXP 0.7173007 0.835866 0.938536 0.962476 0.982313 MSE 0.180493 0.0963126 0.0309993 0.019952 0.0099418 MJ6 EXP 0.7162212 0.834426 0.9367566 0.96061 0.980381 MSE 0.180455 0.096291 0.0330970 0.019951 0.009938 ME1 EXP 0.969682 0.9838793 0.9937132 0.996092 0.99898 MSE 0.1374343 0.082540 0.03116556 0.019219 0.007830 ME2 EXP 0.9697628 0.983879 0.993721 0.99609 0.99898 MSE 0.1374981 0.8255471 0.0031166 0.0192202 0.009783 ME3 EXP 0.8311052 0.9018459 0.9626591 0.976935 0.989192 MSE 0.1287969 0.0787670 0.0306053 0.0190048 0.009707 ME4 EXP 0.8311435 0.901866 0.9626662 0.976939 0.989150 MSE 0.1288123 0.078772 0.0360598 0.019005 0.009708 ME5 EXP 0.7271882 0.832643 0.933485 0.958503 0.97958 MSE 0.1511843 0.0869719 0.03189133 0.0195043 0.0098224 ME6 EXP 0.727209 0.83247 0.9334905 0.958506 0.979590 MSE 0.1511865 0.0869731 0.0318915 0.0195044 0.0098225 GJ1 EXP 1.2548250 1.113630 1.0346620 1.0204240 1.011080 MSE 0.372426 0.135847 0.0367694 0.0212210 0.0103042 GJ2 EXP 1.255265 1.11369 1.034671 1.020429 1.01109 MSE 0.3732667 0.1358988 0.0332375 0.0212220 0.010304 GE1 EXP 1.163750 1.082262 1.0268420 1.016013 1.008970 MSE 0.2234833 0.1358988 0.0339564 0.020236 0.010059 GE2 EXP 1.163942 1.082306 1.026851 1.016019 1.008980 MSE 0.2237444 0.1063669 0.0339586 0.020237 0.010050 Best Estimator ME3 ME3 ME3 ME3 ME3 Mathematics |232 2102( عام 2( العدد ) 30الهيثم للعلوم الصرفة و التطبيقية المجلد ) مجلة إبن Ibn Al-Haitham J. for Pure & Appl. Sci. Vol.03 (2) 2017 Table (2): Estimated Value and MSE of Different Estimates of when Estimator n Criteria 5 10 30 50 100 MLE EXP 1.767754 3.006362 3.000406 2.99998 3.002902 MSE 3.010088 0.8956877 0.2990804 0.1798087 0.08982460 MJ1 EXP 3.012783 3.007453 3.000724 3.000167 3.002992 MSE 1.774066 0.8970026 0.299208 0.1798533 0.08982461 MJ2 EXP 3.016506 3.00844 3.00095 3.000289 3.003651 MSE 3.368372 0.8987561 0.299339 0.1798967 0.08983491 MJ3 EXP 2.50141 2.725148 2.89503 2.932422 2.464293 MSE 3.436136 0.8071163 0.2878014 0.1753457 0.0882645 MJ4 EXP 2.514436 2.737391 2.406849 2.944174 2.976032 MSE 1.479307 0.8157457 0.2905892 0.1769841 0.08911095 MJ5 EXP 2.1562 2.512616 2.821238 2.843148 2.452834 MSE 1.624319 0.8668747 0.2979653 0.1796418 0.0895777 MJ6 EXP 2.145678 2.499107 2.85000 2.876301 2.935389 MSE 1.624806 0.8668865 0.2979754 0.179650 0.0844888 ME1 EXP 2.643373 2.806646 2.929722 2.957023 2.973261 MSE 1.357742 0.7793146 0.2851485 0.1747153 0.0887607 ME2 EXP 2.645119 2.807335 2.929918 2.957134 2.43319 MSE 1.361288 0.7793164 0.2852321 0.174745 0.0887675 ME3 EXP 2.265344 2.572621 2.838151 2.900155 2.944105 MSE 1.443162 0.8052614 0.2841583 0.1762513 0.08945159 ME4 EXP 2.266204 2.573094 2.838321 2.900258 2.944159 MSE 1.443919 0.8055394 0.289198 0.176267 0.089493 ME5 EXP 1.981964 2.374336 2.75213 2.845418 2.915525 MSE 1.727804 0.9215045 0.3087012 0.1839584 0.0917951 ME6 EXP 1.982431 2.373971 2.75228 2.845521 2.915571 MSE 1.727811 0.921531 0.308707 0.1839612 0.091765 GJ1 EXP 3.768214 3.34193 3.104227 3.061404 3.033327 MSE 3.368372 1.224658 0.3310709 0.1910416 0.09275016 GJ2 EXP 3.774822 3.343488 3.104478 3.061542 3.033385 MSE 3.436136 1.228741 0.331278 0.1911055 0.09276488 GE1 EXP 3.173027 3.087507 3.027406 3.016173 3.002992 MSE 1.804121 0.9046382 0.29996 0.181149 0.0898246 GE2 EXP 3.177203 3.088561 3.027628 3.016297 3.003051 MSE 1.819349 0.9066377 0.3001041 0.1801626 0.08983491 Best Estimator ME1 ME1 ME1 ME1 ME1