Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 125 Using Entropy and Linear Exponential Loos Function Estimators the Parameter and Reliability Function of Inverse Rayleigh Distribution Masooma Ali Abod Hazim Mansoor Gorgees Department of Mathematics, College of Education for Pure Science Ibn Al-Haitham , University of Baghdad, Baghdad, Iraq. masoomaaliabod@gmail.com Hazim5656@yahoo.com Abstract This paper is devoted to compare the performance of non-Bayesian estimators represented by the Maximum likelihood estimator of the scale parameter and reliability function of inverse Rayleigh distribution with Bayesian estimators obtained under two types of loss function specifically; the linear, exponential (LINEX) loss function and Entropy loss function, taking into consideration the informative and non-informative priors. The performance of such estimators assessed on the basis of mean square error (MSE) criterion. The Monte Carlo simulation experiments are conducted in order to obtain the required results. Keyword: Inverse Rayleigh distribution, Entropy loss function, LINEX loss function, Prior information. 1. Introduction Inverse Rayleigh distribution (IRD) is one of the comprehensive and relevant lifetime model, and its applications are in reliability and survival data sets .A numerous work has been done in the literature concerning IRD. The distribution was supported by Voda in 1972,who considered its properties and consider MLE estimator for estimate its scale parameter [1]. Next, Gharraph in 1993 developed closed form expressions for the mean, mode, median, harmonic mean and geometric mean of IRD [2]. Furthermore, Soliman, Amin, and Abd-EI Aziz in 2010 estimated the parameter using different traditional and Bayesian estimation's methods [3]. The probability density function of inverse Rayleigh distribution is defined as follows [4]: Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.1.2563 Article history: Received 12,October,2019, Accepted8,January,2020, Published in January 2021 file:///C:/Users/المجلة/Downloads/masoomaaliabod@gmail.com file:///C:/Users/المجلة/Downloads/Hazim5656@yahoo.com 126 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 𝑓(𝑡, 𝜃) = 2𝜃 𝑡3 exp (− 𝜃 𝑡2 ) , 𝑡 > 0, 𝜃 > 0 (1) Where (t) is a random variable that follows IRD and 𝜃 is the scale parameter. The cumulative distribution function is given by [4]: 𝐹(𝑡, 𝜃) = exp (− 𝜃 𝑡2 ) , 𝑡 > 0, 𝜃 > 0 (2) The reliability function of IRD is therefore defined as [4]: 𝑅(𝑡, 𝜃) = 1 − 𝐹(𝑡, 𝜃) = 1 − exp (− 𝜃 𝑡2 ) , 𝑡 > 0, 𝜃 > 0 (3) It is worth mentioning here that the variance and higher order moments not exists in this distribution. In this section, Maximum Likelihood Estimators, Posterior density of the inverse Rayleigh parameter based on (Jeffrey's prior information, exponential prior distribution) and Types of loss functions (Entropy loss function, linear Exponential loss function) will be considered . 2. Maximum Likelihood Estimators Let 𝑡1,𝑡2,………,𝑡𝑛 be random samples drawn from the density given in equation (1), then the likelihood function is defined as 𝐿(𝑡|𝜃) = ∏ 𝑓(𝑛𝑖=1 𝑡𝑖 , 𝜃) = 2 𝑛 𝜃𝑛 ∏ 1 𝑡𝑖 3 𝑛 𝑖=1 𝑒 [−𝜃 ∑ 1 𝑡𝑖, 2] 𝑛 𝑖 (4) Taking the natural logarithm for the likelihood function, we get ln 𝐿 (𝑡|𝜃) = 𝑛𝑙𝑛2 + 𝑛𝑙𝑛𝜃 + ∑ 𝑙𝑛 1 𝑡𝑖 3 𝑛 𝑖=1 − 𝜃 ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 By differentiating the log likelihood function with respect to 𝜃 and then equating the resultant derivative to zero, we get 𝜕𝑙𝑛𝐿(𝑡|𝜃) 𝜕𝜃 = 𝑛 𝜃 − ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 = 0 Hence, the MLE for 𝜃 denoted by 𝜃𝑀𝐿𝐸 is 𝜃𝑀𝐿𝐸 = 𝑛 ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 = 𝑛 𝑇 (5) where 𝑇 = ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 3. Posterior Density of Inverse Rayleigh Parameter Based on Jeffrey's Prior Information Assume that θ has a non-informative prior. Applying Jeffrey's rule [5], we get 𝑔(𝜃) ∝ √𝐼(𝜃) or 𝑔(𝜃) = 𝑐√𝐼(𝜃) 127 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 Where 𝑔(𝜃) represents Jeffrey's prior information, C is the constant of proportionality and 𝐼(𝜃) represents Fisher information, defined as follows: 𝐼(𝜃) = −𝑛𝐸 [ 𝜕2 ln 𝑓(𝑡,𝜃) 𝜕𝜃2 ] (6) Therefore, 𝑔1(𝜃) = 𝑐√−𝑛𝐸 ( 𝜕2 ln 𝑓(𝑡,𝜃) 𝜕𝜃2 ) (7) By taking the logarithm of equation (1), we get 𝑙n 𝑓(𝑡𝑖 , 𝜃) = 𝑙𝑛(2) + 𝑙n(𝜃) + ln ( 1 𝑡3 ) − 𝜃 𝑡2 𝜕 𝑙n 𝑓(𝑡𝑖,𝜃) 𝜕𝜃 = 1 𝜃 − 1 𝑡𝑖 2 Thus, the second derivative is 𝜕2 ln 𝑓(𝑡𝑖 , 𝜃) 𝜕 𝜃2 = − 1 𝜃2 Hence, we get: 𝐸 ( 𝜕2 ln 𝑓 (𝑡𝑖 , 𝜃) 𝜕 𝜃2 ) = − 1 𝜃2 After substitution into (7), we get 𝑔1(𝜃) = 𝑐 𝜃 √𝑛 , 𝜃 > 0 (8) The posterior density function is defined as: ℎ(𝜃|𝑡) = 𝑔(𝜃)𝐿(𝑡|𝜃) ∫ 𝑔(𝜃)𝐿(𝑡|𝜃) ∞ 0 (9) Hence, the posterior density function for θ based on Jeffreys prior will be h1(θ|𝑡) = 𝑐 𝜃 √𝑛 2𝑛𝜃𝑛 ∏ 1 𝑡𝑖 3 𝑛 𝑖=1 exp (−𝜃 ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 ) ∫ 𝑐 𝜃 √𝑛 2𝑛 𝜃𝑛 ∏ 1 𝑡𝑖 3 𝑛 𝑖=1 exp (−𝜃 ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 ) 𝑑𝜃 ∞ 0 h1(θ|𝑡) = 𝜃𝑛−1𝑒 −𝜃𝑇 ∫ 𝜃𝑛−1𝑒 −𝜃𝑇𝑑𝜃 ∞ 0 , 𝑇 = ∑ 1 𝑡2 𝑛 𝑖=1 , 𝜃 > 0 Hence, the posterior density function of 𝜃 with Jeffreys prior can be written as 128 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 = 𝑇𝑛𝜃𝑛−1𝑒 −𝜃𝑇 Γ(n) (10) The posterior density function is recognized as the density of the Gamma distribution, i.e. (θ|t)~𝐺𝑎𝑚𝑚𝑎 (𝑛, 1 𝑇 ), with 𝐸(𝜃) = 𝑛 𝑇 , 𝑉𝑎𝑟(𝜃) = 𝑛 𝑇2 (11) 4. Posterior Density of Inverse Rayleigh Parameter Based on Exponential Prior Distribution Assuming that the inverse Rayleigh parameter 𝜃 follows exponential prior distribution with parameter 𝜆 [6], that is 𝑔2(𝜃) = 𝜆𝑒 −𝜆𝜃 , λ> 0, 𝜃 > 0 (12) Where 𝑔2(𝜃) denotes the exponential prior distribution of the inverse Rayleigh parameter 𝜃. From Bayesian theorem the posterior density function of  denoted by ℎ2(𝜃|𝑡) can be obtained as ℎ2(𝜃|𝑡) = 𝜃𝑛𝑒 −𝜃(𝑇+𝜆) ∫ 𝜃𝑛𝑒 −𝜃(𝑇+𝜆) ∞ 0 𝑑𝜃 (13) where 𝑇 = ∑ 1 𝑡𝑖 2 𝑛 𝑖=1 ℎ2(𝜃|𝑡) = (𝑇+𝜆)𝑛+1𝜃𝑛 𝑒−𝜃(𝑇+𝜆) 𝛤(𝑛+1) 𝜃 > 0 (14) It can easily be noted that θ|x~Gamma (𝑛 + 1 , 1 𝑝 ) where 𝑃 = 𝑇 + 𝜆 with E(𝜃|𝑡) = 𝑛+1 𝑃 , 𝑉𝑎𝑟 (𝜃|𝑡) = 𝑛+1 𝑃2 (15) 5. Types of Loss Functions [7] From the Bayesian viewpoint, the essential step in the estimation and prediction problems was represented by choosing the loss function. In fact, there is no specific analytical procedure to determine the suitable loss function to be employed. In this paper, we consider two types of loss function, the Entropy loss function and Linear Exponential loss function (LINEX), as follows: i) Entropy Squared Loss Function which is defined as below 𝐿(𝜃, 𝜃) = �̂� 𝜃 − 𝑙𝑛 �̂� 𝜃 − 1 (16) ii) Linear Exponential Loss Function (LINEX) which is defined as below 𝐿(𝜃, 𝜃) = 𝑒(�̂�−𝜃) − ( 𝜃 − 𝜃) − 1 (17) 129 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 6. Bayesian Estimation The Bayes estimator of the parameter 𝜃 is the value of 𝜃 that minimize the posterior expectation known as the risk function denoted by 𝑅(𝜃, 𝜃) , that is 𝑅(𝜃, 𝜃) = 𝐸[𝐿(𝜃, 𝜃)] = ∫ 𝐿(𝜃, 𝜃 ∞ 0 )ℎ(𝜃|𝑡)𝑑𝜃 (18) Where ℎ(𝜃|𝑡) is the posterior density of 𝜃|𝑡 7. Bayes Estimator of Parameter 𝜽 and Reliability Function of IRD under Entropy Loss Function [8, 9]. If entropy loss function is chosen, then according to equation (18), we have 𝑅(𝜃, 𝜃) = ∫ [ �̂� 𝜃 − 𝑙𝑛 �̂� 𝜃 − 1 ] ℎ(𝜃|𝑡)𝑑𝜃 ∞ 0 By differentiating R(θ̂, θ) with respect to 𝜃 and setting the resultant, derivative equal to zero, then solving for 𝜃 , we get 𝜃𝐸𝑛 = 1 ∫ 1 𝜃 ℎ(𝜃|𝑡)𝑑𝜃 ∞ 0 (19) On the basis of non-informative prior and according to equation (10), the Bayes estimator of inverse Rayleigh parameter 𝜃 denoted as 𝜃𝐸𝑛(𝐽) is given by 𝜃𝐸𝑛(𝐽) = 𝑛−1 𝑇 (20) If the inverse Rayleigh parameter follows the exponential prior distribution, then by equation (14) we conclude that 𝜃𝐸𝑛(𝐸) = 𝑛 𝑝 , where p=T+λ (21) The estimator of the reliability function based on Jeffrey's prior can be approximated as �̂�(𝑡)𝐸𝑛(𝐽) ≅ 1 − 𝑒 − �̂�𝐸𝑛(𝐽) 𝑡2 �̂�(𝑡)𝐸𝑛(𝐽) ≅ 1 − 𝑒 − 𝑛−1 𝑇𝑡2 (22) The estimator of the reliability function based on exponential prior can be approximated as �̂�(𝑡)𝐸𝑛(𝐸) ≅ 1 − 𝑒 − �̂�𝐸𝑛 𝑡2 �̂�(𝑡)𝐸𝑛(𝐸) ≅ 1 − 𝑒 − 𝑛 𝑝𝑡2 (23) 130 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 8. Bayes estimator of the parameter 𝜽 and reliability function under LINEX By substituting from 𝐿(𝜃, 𝜃) given in equation (17) into equation (18), we get 𝑅(𝜃, 𝜃) = ∫ [𝑒(�̂�−𝜃) − (𝜃 − 𝜃) − 1]ℎ(𝜃|𝑡)𝑑𝜃 ∞ 0 By simplification, we get 𝑒�̂� ∫ 𝑒−𝜃 ∞ 0 ℎ(𝜃|𝑡)𝑑𝜃 = 1 By differentiating 𝑅(𝜃, 𝜃) with respect to 𝜃 then equating the resultant derivative to zero and solving for 𝜃,̂ we get the Bayes estimator of 𝜃 under linear Exponential loss function denoted by 𝜃𝐿 as follows 𝜃𝐿𝐸 = −𝑙𝑛 ∫ 𝑒 −𝜃∞ 0 ℎ(𝜃|𝑡)𝑑𝜃 on the basis of non-informative prior, the Bayes estimator of the inverse Rayleigh parameter 𝜃 denoted as 𝜃𝐿𝐸(𝐽) is given by 𝜃𝐿𝐸(𝐽) = − ln ( 𝑇 1+𝑇 ) 𝑛 (24) If the inverse Rayleigh parameter follows the exponential prior distribution 𝜃𝐿𝐸(𝐸) = − ln ( 𝑃 1+𝑃 ) 𝑛+1 , (25) the estimator of the reliability function based on Jeffrey's prior can be approximated as �̂�(𝑡)𝐿𝐸(𝐽) ≅ 1 − 𝑒 − �̂�𝐿𝐸(𝐽) 𝑡2 Which implies that �̂�(𝑡)𝐿𝐸(𝐽) ≅ 1 − 𝑒 − 𝑙𝑛( 𝑇 1+𝑇 ) 𝑛 𝑡2 �̂�(𝑡)𝐿𝐸(𝐽) ≅ 1 − ( 𝑇 1+𝑇 ) 𝑛 𝑡2 (26) The estimator of the reliability function based on exponential prior can be approximated as �̂�(𝑡)𝐿𝐸(𝐸) ≅ 1 − 𝑒 − �̂�𝐿𝐸(𝐸) 𝑡2 Which implies that �̂�(𝑡)𝐿𝐸(𝐸) ≅ 1 − 𝑒 − 𝑙𝑛( 𝑃 1+𝑃 ) 𝑛+1 𝑡2 �̂�(𝑡)𝐿𝐸(𝐸) ≅ 1 − ( 𝑃 1+𝑃 ) 𝑛+1 𝑡2 (27) 131 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 10. Simulation Study In our simulation study, number of repetitions L=2000 sample of size n=10,50,100 and 200 are generated in order to represent, small, moderate, large and very large sample sizes from inverse Rayleigh distribution with two values of the scale parameter (𝛳 = 0.5 , 𝛳 = 1.5), the scale parameter λ of exponential prior was chosen to be (λ=0.5, λ=1) and mean square error (MSE) is employed to compare the performance of different methods for estimation of the scale parameter and reliability function of IRD where 𝑀𝑆𝐸(𝜃) = 1 𝐿 ∑ (𝜃𝑖 − 𝜃) 2𝐿 𝑖=1 (28) 𝑀𝑆𝐸[�̂�(𝑡)] = 1 𝐿 ∑ [�̂�𝑖 (𝑡) − 𝑅(𝑡)] 2𝐿 𝑖=1 (29) The results are presented in the following Tables (1-8). Table 1. MSE for parameter 𝜃 by using Jeffrey's prior information at 𝜃 = 0.5 n Estimator 10 50 100 200 MLE 0.0043 0.0001082 0.000026176 0.000006513 Ent 0.0033 0.0001019 0.000025396 0.000006412 Lin 0.0250 0.0050 0.0025 0.0013 Best Ent Ent Ent Ent Table 2. MSE values of the Reliability function estimators by using Jeffrey's prior information at 𝜃 = 0.5 n Estimator 10 50 100 200 MLE 0.000321 0.0000101 0.0000024924 0.0000006246 Ent 0.000269 0.0000097 0.0000024460 0.0000006185 Lin 0.000283 0.0000098 0.0000024635 0.0000006209 Best Ent Ent Ent Ent 132 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 Table 3. MSE for parameter 𝜃 by using exponential prior information at 𝜃 = 0.5 n Estimator 10 50 100 200 Ent 𝜆 = 0.5 0.2810 0.0472 0.0231 0.0114 𝜆 = 1 0.2968 0.0479 0.0232 0.0114 Lin 𝜆 = 0.5 0.0250 0.0050 0.0025 0.0013 𝜆 = 1 0.0250 0.0050 0.0025 0.0013 Best Lin Lin Lin Lin Table 4. MSE values of the Reliability function estimators by using Exponential prior information at 𝜃 = 0.5 n Estimator 10 50 100 200 Ent 𝜆 = 0.5 0.0090 0.0018 0.00087303 0.00043568 𝜆 = 1 0.0094 0.0018 0.00087669 0.00043659 Lin 𝜆 = 0.5 0.000317 0.0000101 0.0000025078 0.0000006267 𝜆 = 1 0.000278 0.0000098 0.0000024731 0.0000006223 Best Lin Lin Lin Lin Table 5. MSE for parameter 𝜃 by using Jeffrey's prior information at 𝜃 = 1.5 n Estimator 10 50 100 200 MLE 0.0391 0.00097346 0.00023557 0.00005862 Ent 0.0293 0.00091708 0.00022856 0.000057715 Lin 0.2250 0.0450 0.0225 0.0113 Best Ent Ent Ent Ent 133 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 Table 6. MSE values of the Reliability function estimators by using Jeffrey's prior information at 𝜃 = 1.5 n Estimator 10 50 100 200 MLE 0.000691 0.000024581 0.0000061457 0.0000015451 Ent 0.000647 0.000024264 0.0000061043 0.0000015393 Lin 0.000551 0.000023534 0.0000060124 0.0000015278 Best Lin Lin Lin Lin Table 7. MSE for parameter 𝜃 by using exponential prior information at 𝜃 = 1.5 n Estimator 10 50 100 200 Ent 𝜆 = 0.5 0.0660 0.0137 0.0069 0.0035 𝜆 = 1 0.0584 0.0134 0.0068 0.0034 Lin 𝜆 = 0.5 0.2250 0.0450 0.0225 0.0113 𝜆 = 1 0.2250 0.0450 0.0225 0.0113 Best Ent Ent Ent Ent Table 8. MSE values of the Reliability function estimators by using exponential prior information at 𝜃 = 1.5 n Estimator 10 50 100 200 Ent 𝜆 = 0.5 0.00310 0.00061919 0.00030974 0.00015491 𝜆 = 1 0.00260 0.00060029 0.00030498 0.0000015371 Lin 𝜆 = 0.5 0.000474 0.000022883 0.0000059292 0.000001517 𝜆 = 1 0.000423 0.00002227 0.0000058461 0.000001506 Best Lin Lin Lin Lin 11. Simulation Results and Conclusions From our simulation study, the following conclusions are pointed out : 1. When 𝜃 = 0.5, the Bayes estimators of the scale parameter and reliability function under Entropy loss function with Jeffrey's prior is the best for all cases as shown in 134 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (1) 2021 tables (1 ̶ 4). While the estimators under Linear Exponential loss function (LINEX) are the best when the prior information is exponential. 2. 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