Bayes Estimators for the Reliability Function of Pareto Type I ...

Al- Mustansiriyah J. Sci.

Vol. 25, No 4, 2014

Bayes Estimators for the Reliability Function of Pareto Type I Distribution Under Squared-Log Error Loss Function Huda, A. Rasheed and Najam A. Aleawy Al-Gazi Department of mathematics, Collage of Science, Al-Mustansiriya University Received 6/4/2014 – Accepted 29/9/2014

‫الخالصة‬ ‫أوجدنا في هذا البحث مقدرات بيز لدالة المعولية لتوزيع باريتو من النوع األول تحت دالة الخسارة اللوغاريتمية‬ ‫ وبغية الحصول على افضل فهم لتحليلنا البيزي فقد افترضنا حالة عدم وجود معلومات مسبقة عن‬.‫التربيعية‬ ‫ تمت مقارنة‬.‫معلمة الشكل باستخدام دالة جيفري للمعلومات كذلك وجود معلومات مسبقة متمثلة بالتوزيع اآلسي‬ ‫مقدر بيز لدالة المعولية لتوزيع باريتو من النوع األول تحت دالة الخسارة اللوغاريتمية التربيعية مع بعض‬ ‫ المقدر المنتظم غير المتحيز ذو اقل تباين وأقل متوسط مربعات‬،‫المقدرات التقليدية مثل مقدر االمكان األعظم‬ ‫كارلو للمحاكاة فقد تمت مقارنة أداء هذه المقدرات باالعتماد على متوسط‬-‫ استناداً الى دراسة مونت‬.‫الخطأ‬ . (IMSE’s) ‫مربعات الخطأ التكاملي‬

ABSTRACT In this paper we obtained Basyian estimators of the reliability function of the Pareto type I distribution under Squared-Log error loss function. In order to get better understanding of our Bayesian analysis we consider non-informative prior for the shape parameter Using Jeffery prior Information as well as informative prior density represented by Exponential distribution. The Bayes estimator of the reliability function of the Pareto type I distribution under Squared-Log error loss function is compared with Some classical estimators such as, the Maximum Likelihood Estimator (MLE), the Uniformly Minimum Variance Unbiased Estimator (UMVUE), and the Minimum Mean Squared Error (MinMSE) estimator according to MonteCarlo simulation study. The performance of these estimators is compared depending on the Integrated mean squared errors (IMSE’s). Key words: Pareto distribution, MLE, Bayes estimator, Squared-Log error loss function, Jeffery prior and Exponential prior.

INTRODUCTION The Pareto distribution is named after the economist Vilfredo Pareto (1848-1923), this distribution is first used as a model for distributing incomes of model for city population within a given area, failure model in reliability theory [2], and a queuing model in operation research [6]. A random variable X, is said to follow the two parameters of Pareto distribution if its pdf is given by: 𝜃𝛼 𝜃 𝑓(𝑥; 𝛼, 𝜃) = 𝜃+1 ; x ≥ α ,α > 0 ,𝜃 > 0 (1) 𝑥 Where 𝛼 and 𝜃 are the scale and shape parameters respectively. The corresponding cumulative distribution function (CDF) is given by α θ 𝐹(𝑥; 𝛼, 𝜃) = 1 − ( ) , 𝑥 ≥ 𝛼; 𝛼, θ > 0 (2) x So, the reliability function is: α θ R(t) = ( ) (3) t SOME CLASSICAL ESTIMATION

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Bayes Estimators for the Reliability Function of Pareto Type I Distribution Under Squared-Log Error Loss Function Huda and Najam

In this section, we obtain some classical estimators of the shape parameter for the Pareto distribution represented by Maximum likelihood estimator, Uniformly Minimum Variance Unbiased Estimator and Minimum mean square error estimator. Given x1,x2,…, xn a random sample of size n from Pareto distribution, we consider estimation using method of Maximum likelihood as follows: 𝑛

𝐿(𝑥1 , … , 𝑥 𝑛 |𝜃) = ∏ 𝑓(𝑥𝑛 ; 𝜃) 𝑖=1 𝑛 𝑛𝜃 −(𝜃+1) ∑𝑛 𝑖=1 𝑙𝑛 𝑥𝑖

𝐿(𝑥1 , … , 𝑥 𝑛 |𝜃) = 𝜃 𝛼 𝑒 The Log- likelihood function is given by

𝑛

𝑙𝑛 𝐿(𝑥1 , … , 𝑥𝑛 |𝜃) = 𝑛 𝑙𝑛 𝜃 + 𝑛𝜃𝑙𝑛𝛼 − (𝜃 + 1) ∑ 𝑙𝑛 𝑥𝑖 𝑖=1

Differentiating the log likelihood with respect to 𝜃 : n

𝜕[𝑙𝑛 𝐿(𝑥1 , … , 𝑥𝑛 |𝜃)] 𝑛 = + nlnα − ∑ 𝑙𝑛 xi 𝜕𝜃 θ 𝑖=1

Setting this expression to zero and solving the equation yields the Maximum likelihood estimator of 𝜃: 𝑛 𝜃̂ = n ∑𝑖=1 𝑙𝑛 xi − 𝑛𝑙𝑛𝛼 n n (4) θ̂ML = , 𝑤ℎ𝑒𝑟𝑒 𝑇 = ∑ 𝑙𝑛 xi T − nlnα 𝑖=1

Using the invariance property, the MLE for R(t) denoted by 𝑅̂𝑀𝐿 (𝑡) may be obtained by replacing θ by its MLE θ̂ML in (3) [7] 𝛼 𝜃̂𝑀𝐿 (5) 𝑅̂𝑀𝐿 (𝑡) = ( ) 𝑡 Here, we obtain the Uniformly Minimum Variance Unbiased Estimator (UMVUE) of 𝜃. Since the family of density (1) belongs to the Exponential family, therefore, statistic T is a complete sufficient statistic for 𝜃. We have: 𝑡 𝑡 ln ( 𝑖 ) ~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙(𝜃) ⇒ T = ∑𝑛𝑖=1 𝑙𝑛 ( 𝑖) ~ Gamma (𝑛, 𝜃), with the 𝛼 𝛼 density: 𝜃 𝑛 𝑛−1 −𝜃𝑡 𝑔(𝑡) = 𝑡 𝑒 , 𝑡 ≥ 0 ,𝜃 > 0 ᴦ(𝑛) Thus, 1 𝐸[θ̂ML ] = 𝑛𝐸 ( ) 𝑇 114

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∞

1 𝜃 𝑛 𝑡 𝑛−2 𝑒 −𝜃𝑡 𝐸( ) = ∫ 𝑑𝑡 𝑇 ᴦ(𝑛) 0

∞

𝜃 𝜃 𝑛−1 𝑡 𝑛−2 𝑒 −𝜃𝑡 = ∫ 𝑑𝑡 𝑛−1 ᴦ(𝑛 − 1) 0

1 𝜃 𝐸( ) = (6) 𝑇 𝑛−1 𝑛𝜃 𝐸[θ̂ML ] = (7) 𝑛−1 Recall that, T is a complete sufficient statistics for 𝜃 . Thus, 𝜃̂𝑈𝑀𝑉𝑈𝐸 is a Uniformly minimum variance unbiased estimator (UMVUE) of 𝜃.[19] 𝑛−1 𝜃̂𝑈𝑀𝑉𝑈𝐸 = (8) 𝑇 We can find the Minimum Mean Squared Error (MinMSE) estimator in 𝑐 the class of estimators of the form . Therefore 𝑐

𝑇 1

2

𝑐

1

𝑀𝑆𝐸𝜃 ( ) = 𝐸 [( − 𝜃) ] = 𝑐 2 𝐸 ( 2) − 2𝑐𝜃𝐸 ( ) + 𝜃 2 𝑇 𝑇 𝑇 𝑇 Taking the derivative with respect to c on the both sides leads to 𝜕 𝑐 1 1 MSE𝜃 ( ) = 2𝑐𝐸 ( 2 ) − 2𝜃𝐸 ( ) 𝜕𝑐 𝑇 𝑇 𝑇 𝜕 𝑐 Let MSE𝜃 ( ) = 0 , so 𝜕𝑐 𝑇 1 𝜃2 𝜃𝐸 ( ) 𝑇 = 𝑛−1 𝑐= =𝑛−2 1 𝜃2 𝐸 ( 2) 𝑇 (𝑛 − 1)(𝑛 − 2) Therefore, we get c = n-2 and hence 𝑛−2 𝜃̂𝑀𝑖𝑛𝑀𝑆𝐸 = (9) 𝑇 Is the Minimum mean square error estimator for 𝜃. 2 𝑛 𝑛 𝑛 2 ̂ 𝑀𝑆𝐸(𝜃𝑀𝐿𝐸 ) = 𝐸 [( − 𝜃) ] = 𝑣𝑎𝑟 ( ) + [𝐸 ( ) − 𝜃] (10) 𝑇 𝑇 𝑇 𝑛 1 𝑣𝑎𝑟 ( ) = 𝑛2 𝑣𝑎𝑟 ( ) 𝑇 𝑇 2 1 1 1 2 𝑣𝑎𝑟 ( ) = 𝐸 [( ) ] − [𝐸 ( )] (11) 𝑇 𝑇 𝑇 ∞

1 2 𝜃2 𝜃 𝑛−2 𝑡 𝑛−3 𝑒 −𝜃𝑡 𝐸 [( ) ] = ∫ 𝑑𝑡 (𝑛 − 1)(𝑛 − 2) 𝑇 ᴦ(𝑛 − 2) 2

=

0

𝜃 (𝑛 − 1)(𝑛 − 2)

(12)

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Bayes Estimators for the Reliability Function of Pareto Type I Distribution Under Squared-Log Error Loss Function Huda and Najam

Substituting (6) and (12) into (11) 1 𝜃2 𝜃 2 𝑣𝑎𝑟 ( ) = −[ ] (𝑛 − 1)(𝑛 − 2) 𝑇 𝑛−1 1 𝜃2 𝑣𝑎𝑟 ( ) = (𝑛 − 1)2 (𝑛 − 2) 𝑇 𝑛

𝑛2 𝜃 2

(13)

Hence, 𝑣𝑎𝑟 ( ) = (𝑛−1)2(𝑛−2) 𝑇 2 𝑛 𝑛 2 𝑛 [𝐸 ( ) − 𝜃] = [𝐸 ( )] + 2𝜃𝐸 ( ) + 𝜃 2 𝑇 𝑇 𝑇 𝑛𝜃 2 𝑛𝜃 =[ ] − 2𝜃 [ ] + 𝜃2 𝑛−1 𝑛−1 𝑛2 𝜃 2 − 2𝑛𝜃 2 (𝑛 − 1) + 𝜃 2 (𝑛 − 1)2 = (𝑛 − 1)2 2 𝑛 𝜃2 [𝐸 ( ) − 𝜃] = (14) (𝑛 − 1)2 𝑇 Substituting (13) and (14) into (10) 𝑛2 𝜃 2 𝜃2 𝜃 2 (𝑛 + 2)(𝑛 − 1) ̂ 𝑀𝑆𝐸(𝜃𝑀𝐿𝐸 ) = + = (𝑛 − 1)2 (𝑛 − 2) (𝑛 − 1)2 (𝑛 − 1)2 (𝑛 − 2) 𝜃 2 (𝑛 + 2) ̂ 𝑀𝑆𝐸(𝜃𝑀𝐿𝐸 ) = (15) (𝑛 − 1)(𝑛 − 2) 2 𝑛−1 ̂ 𝑀𝑆𝐸(𝜃𝑈𝑀𝑉𝑈𝐸 ) = 𝐸 [( − 𝜃) ] 𝑇 2 𝑛−1 𝑛−1 𝑀𝑆𝐸(𝜃̂𝑈𝑀𝑉𝑈𝐸 ) = 𝑣𝑎𝑟 ( (16) ) + [𝐸 ( ) − 𝜃] 𝑇 𝑇 𝑛−1 1 𝑣𝑎𝑟 ( (17) ) = (𝑛 − 1)2 𝑣𝑎𝑟 ( ) 𝑇 𝑇 Substituting (13) into (17), we get 𝑛−1 (𝑛 − 1)2 𝜃 2 𝜃2 𝑣𝑎𝑟 ( = (18) )= (𝑛 − 1)2 (𝑛 − 2) (𝑛 − 2) 𝑇 We have, 𝑛−1 𝐸( (19) )=𝜃 𝑇 Substituting (18) and (19) into (16) 𝜃2 ̂ 𝑀𝑆𝐸(𝜃𝑈𝑀𝑉𝑈𝐸 ) = (20) (𝑛 − 2) 2 𝑛−2 𝑀𝑆𝐸(𝜃̂𝑀𝑖𝑛𝑀𝑆𝐸 ) = 𝐸 [( − 𝜃) ] 𝑇 2 𝑛−2 𝑛−2 ̂ 𝑀𝑆𝐸(𝜃𝑀𝑖𝑛𝑀𝑆𝐸 ) = 𝑣𝑎𝑟 ( (21) ) + [𝐸 ( ) − 𝜃] 𝑇 𝑇

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𝑛−2 1 (22) ) = (𝑛 − 2)2 𝑣𝑎𝑟 ( ) 𝑇 𝑇 Substituting (13) into (22) 𝑛−2 (𝑛 − 2)𝜃 2 𝑣𝑎𝑟 ( (23) )= (𝑛 − 1)2 𝑇 (𝑛 − 2)𝜃 𝑛−2 𝐸( (24) )= 𝑇 (𝑛 − 1) Substituting (23) and (24) into (21) (𝑛 − 2)𝜃 2 𝜃2 𝜃2 ̂ 𝑀𝑆𝐸(𝜃𝑀𝑖𝑛𝑀𝑆𝐸 ) = + = (25) (𝑛 − 1)2 (𝑛 − 1)2 (𝑛 − 1) From (15), (20) and (25), we find that: 𝑀𝑆𝐸(𝜃̂𝑀𝑖𝑛𝑀𝑆𝐸 ) ≤ 𝑀𝑆𝐸(𝜃̂𝑈𝑀𝑉𝑈𝐸 ) ≤ 𝑀𝑆𝐸(𝜃̂𝑀𝐿 ) New, the MinMSE estimator of the Reliability function, denoted by 𝑅̂(𝑡)MinMSE approximately, will be: 𝛼 𝜃̂𝑀𝑖𝑛𝑀𝑆𝐸 ̂ 𝑅(𝑡)MinMSE = ( ) (26) 𝑡 BAYES ESTIMATOR UNDR SQUARED-LOG ERROR LOSS FUNCTION Bayes estimators for the shape parameter 𝜃 and Reliability function were considered under squared-log error loss function with non-Informative prior which is represented by Jeffrey prior and informative loss function is represented by Exponential prior where the squared-log error loss function is of the form: [4] 𝑣𝑎𝑟 (

2    L2  , 1    ln 1  ln     ln 1    

2

Which is balanced with lim L2 ( , 1 )   as 1  0 or . A balanced loss function takes both error of estimation and goodness of fit into account but the unbalanced loss function considers only error of estimation. This loss function is convex for

1  e and concave otherwise, but its risk function 

has a unique minimum with respect to 1 .[4] According to the above mentioned loss functions, we drive the corresponding Bayes estimators for θ using Risk function 𝑅(𝜃̂ − 𝜃), which minimizes the posterior risk ∞

2 𝑅(𝜃̂ − 𝜃) = 𝐸 [𝐿(𝜃̂, 𝜃)] = ∫ (ln θ̂ − ln θ) h(θ|x1 … … … . . xn )dθ 0

𝑅(𝜃̂ − 𝜃) = (ln 𝜃̂)2 − 2(𝑙𝑛𝜃̂) 𝐸(ln 𝜃|𝑥) + 𝐸((ln 𝜃)2 |𝑥) 𝜕Rsik 1 2 = 2(ln θ̂) − E((ln θ)|x) ∂θ̂ θ̂ 𝜃̂

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Bayes Estimators for the Reliability Function of Pareto Type I Distribution Under Squared-Log Error Loss Function Huda and Najam

𝜕𝑅2 (𝜃̂ − 𝜃) By letting =0 𝜕𝜃̂ The Bayes estimator for the parameter θ of Pareto distribution under the squared-log error loss function is: 𝜃̂ = Exp[E(lnθ|x)] (27) According to the Squared-Log error loss function, the corresponding Bayes estimator for the reliability function will be: 𝑅̂(𝑡) = 𝐸𝑥𝑝[𝐸(𝑙𝑛𝑅(𝑡)|𝑡)] (28) ∞

𝐸 (𝑙𝑛𝑅(𝑡)|𝑡) = ∫ 𝑙𝑛𝑅(𝑡) ℎ(𝜃|𝑥1 , … , 𝑥 𝑛 )𝑑𝜃 0

𝛼 𝜃

We have 𝑅(𝑡) = ( ) 𝑡 Hence, 𝛼 𝐸[𝑙𝑛𝑅(𝑡)] = ln ( ) 𝐸[𝜃] 𝑡 Substituting (29) into (28), we get: 𝛼 𝑅̂(𝑡) = 𝐸𝑥𝑝 [ln ( ) 𝐸[𝜃]] 𝑡

(29) (30)

PRIOR AND POSTERIOR DISTRIBUTIONS In this paper we consider informative as well as non-informative prior density for 𝜃in order to get better understanding of our Bayesian analysis as follows: (i)Bayes Estimator Using Jeffery Prior Information: Let us assume that θ has non-informative prior density defined by using Jeffrey prior information 𝑔(𝜃) which given by: 𝑔1 (𝜃) ∝ √𝐼(𝜃) Where 𝐼(𝜃) represents Fisher information which defined as follows [1]: 𝜕 2 𝑙𝑛𝑓 𝐼(𝜃) = −𝑛𝐸( ) 𝜕𝜃 2 Hence, 𝑔1 (𝜃) = 𝑐√−𝑛𝐸(

𝜕2 𝑙𝑛𝑓 𝜕𝜃2

) , where c is a constant

𝑙𝑛𝑓(𝑥; 𝜃) = 𝑙𝑛𝜃 + 𝑛𝜃𝑙𝑛𝛼 − (𝜃 + 1)𝑙𝑛𝑥 𝜕𝑙𝑛𝑓 1 = + 𝑛𝑙𝑛𝛼 − 𝑙𝑛𝑥 𝜕𝜃 𝜃 2 𝜕 𝑙𝑛𝑓 1 =− 2 2 𝜕𝜃 𝜃 𝜕 2 𝑙𝑛𝑓 1 𝐸( = − ) 𝜕𝜃 2 𝜃2 After substitution into (10), we find that:

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𝑔1 (𝜃) =

𝑐 √𝑛 𝜃

ℎ1 (𝜃|𝑥1 , … , 𝑥𝑛 ) =

Vol. 25, No 4, 2014

𝐿(𝑥1 , … , 𝑥𝑛 |𝜃)𝑔1 (𝜃) ∞ ∫0 𝐿(𝑥1 , … , 𝑥𝑛 |𝜃)𝑔1 (𝜃)𝑑𝜃 𝑛 𝑛𝜃 −(𝜃+1) ∑ 𝑙𝑛𝑥 𝑛 𝑛𝜃𝑙𝑛𝛼 −(𝜃+1) ∑ 𝑙𝑛𝑥

Where 𝐿(𝑥; 𝛼, 𝜃) = 𝜃 𝛼 𝑒 =𝜃 𝑒 𝑒 ℎ1 (𝜃|𝑥1 , … , 𝑥𝑛 ) 𝑐 𝜃 𝑛 𝑒 𝑛𝜃𝑙𝑛𝛼 𝑒 −(𝜃+1) ∑ 𝑙𝑛𝑥 √𝑛 𝜃 = ∞ 𝑤ℎ𝑒𝑟𝑒, 𝛼 𝑖𝑠 𝑎 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝑐 ∫0 𝜃 𝑛 𝑒 𝑛𝜃𝑙𝑛𝛼 𝑒 −(𝜃+1) ∑ 𝑙𝑛𝑥 𝜃 √𝑛 𝑑𝜃 𝑐 √𝑛𝑒 ∑ 𝑙𝑛𝑥 𝜃 𝑛−1 𝑒 −𝜃(∑ 𝑙𝑛𝑥−𝑛𝑙𝑛𝛼) = ∞ 𝑐√𝑛𝑒 ∑ 𝑙𝑛𝑥 ∫0 𝜃 𝑛−1 𝑒 −𝜃(∑ 𝑙𝑛𝑥−𝑛𝑙𝑛𝛼) 𝑑𝜃 𝜃 𝑛−1 𝑒 −𝜃(∑ 𝑙𝑛𝑥−𝑛𝑙𝑛𝛼) = ∞ ∫0 𝜃 𝑛−1 𝑒 −𝜃(∑ 𝑙𝑛𝑥−𝑛𝑙𝑛𝛼) 𝑑𝜃 𝜃 𝑛−1 𝑒 −𝜃(𝑇−𝑛𝑙𝑛𝛼) = ∞ [𝑇 − 𝑛𝑙𝑛𝛼]𝑛 𝑛−1 −𝜃(𝑇−𝑛𝑙𝑛𝛼) Г(𝑛) 𝜃 𝑒 𝑑𝜃 ∫ [𝑇 − 𝑛𝑙𝑛𝛼]𝑛 0 Г(𝑛) [𝑇 − 𝑛𝑙𝑛𝛼]𝑛 𝜃 𝑛−1 𝑒 −𝜃(𝑇−𝑛𝑙𝑛𝛼) = Г(𝑛) This posterior density is recognized as the density of the Gamma distribution: 𝜃~𝐺𝑎𝑚𝑚𝑎(𝑛, (𝑇 − 𝑛𝑙𝑛𝛼)) , with: 𝑛 𝑛 𝐸(𝜃) = , 𝑣𝑒𝑟(𝜃) = 𝑇 − 𝑛𝑙𝑛𝛼 (𝑇 − 𝑛𝑙𝑛𝛼)2 (𝑇 − 𝑛𝑙𝑛𝛼)n ∞ E(lnθ|x) = ∫ ln θ θn−1 e−θ[𝑇−𝑛𝑙𝑛𝛼] dθ Г(n) 0 Let 𝑦 = 𝜃(𝑇 − 𝑛𝑙𝑛𝛼) 𝑦 𝑑𝑦 ⇒ 𝜃= , 𝑑𝜃 = 𝑇 − 𝑛𝑙𝑛𝛼 𝑇 − 𝑛𝑙𝑛𝛼 Hence, n−1 (e−y )n ∞ y y e−y E(lnθ|x) = ∫ ln ( 𝑑𝑦 )( ) Г(n) 0 𝑇 − 𝑛𝑙𝑛𝛼 𝑇 − 𝑛𝑙𝑛𝛼 𝑇 − 𝑛𝑙𝑛𝛼 ∞ (𝑇 − 𝑛𝑙𝑛𝛼)𝑛 = ∫ [ln 𝑦 − ln(𝑇 − 𝑛𝑙𝑛𝛼)]𝑦 𝑛−1 𝑒 −𝑦 𝑑𝑦 Г(𝑛)(𝑇 − 𝑛𝑙𝑛𝛼)𝑛 0 ∞ (𝑙𝑛𝑦) y n−1 e−y 𝑙𝑛(𝑇 − 𝑛𝑙𝑛𝛼) ∞ 𝑛−1 −𝑦 =∫ 𝑑𝑦 − ∫ 𝑦 𝑒 𝑑𝑦 Г(𝑛) Г(𝑛) 0 0 E(lnθ|x) = 𝜑(𝑛) − ln[T − nlnα] (32) Where, 𝜑(𝑛) =

́ Г(𝑛) Г(𝑛)

is the digamma function [6]

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Substituting (32) into (27), we get 𝜃̂1 = Exp[𝜑(𝑛) − 𝑙𝑛(𝑇 − 𝑛𝑙𝑛𝛼)]

(33) Now, using (9) to estimate Reliability function we reach to: 𝑛 𝛼 ̂ (t)BJ = Exp [ R ln ( )] (14) 𝑇 − 𝑛𝑙𝑛𝛼 𝑡 ̂ (t)BJ is equivalent to the Maximum Likelihood We can notice that R Estimator for R(t). (ii) Posterior Distribution Using Exponential Prior Distribution. Assuming that 𝜃 has informative prior as Exponential prior, which takes the following form: 1 −𝜃 𝑔2 (𝜃) = 𝑒 𝜆 , 𝜃 ,𝜆 > 0 𝜆 So, the posterior distribution for the parameter 𝜃 given the data (𝑥1 , 𝑥2 , … 𝑥𝑛 ) is: 𝑛 𝜋𝑖=1 𝑓(𝑥𝑖 |𝜃)𝑔2 (𝜃) ℎ2 (𝜃|𝑥1 , 𝑥2 , … 𝑥𝑛 ) = ∞ 𝑛 ∫0 𝜋𝑖=1 𝑓(𝑥𝑖 |𝜃)𝑔2 (𝜃)𝑑𝜃 Then the posterior distribution became as follows: ℎ2 (𝜃|𝑥1 , 𝑥2 , … 𝑥𝑛 ) 1 𝑛+1 𝑛 −𝜃[𝑇−𝑛𝑙𝑛𝛼+𝜆1] [𝑇 − 𝑛𝑙𝑛𝛼 + ] 𝜃 𝑒 𝜆 = (15) Г(𝑛 + 1) This posterior density is recognized as the density of the Gamma distribution where: 1

𝜃~𝐺𝑎𝑚𝑚𝑎 (𝑛 + 1, + ∑n𝑖=1 𝑙𝑛 −𝑛𝑙𝑛𝛼 ), with: 𝜆

𝑛+1

𝐸(𝜃) =

, 𝑣𝑒𝑟(𝜃) 1 n + ∑𝑖=1 𝑙𝑛 xi − 𝑛𝑙𝑛𝛼 𝜆 𝑛+1 = 2 1 ( + ∑n𝑖=1 𝑙𝑛 xi − 𝑛𝑙𝑛𝛼) 𝜆 The Bayes estimator under Squared-Log error loss function will be: ∞

𝜃̂2 = Exp [∫ 𝑙𝑛𝜃 ℎ2 (𝜃|𝑡)𝑑𝜃] 0

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1 𝑛+1 𝑛 −𝜃[𝑇−𝑛𝑙𝑛𝛼+𝜆1] [𝑇 − 𝑛𝑙𝑛𝛼 + ] 𝜃 𝑒 𝜆 = ∫ 𝑙𝑛𝜃 𝑑𝜃 Г(𝑛 + 1) ∞

(16)

0

1 𝑦 = 𝜃 [𝑇 − 𝑛𝑙𝑛𝛼 + ] 𝜆 𝑦 1 ⇒ 𝜃= , 𝑑𝜃 = 𝑑𝑦 1 1 𝑇 − 𝑛𝑙𝑛𝛼 + 𝑇 − 𝑛𝑙𝑛𝛼 + 𝜆 𝜆 Substituting into (16), we have: 𝐸(𝑙𝑛𝜃|𝑥) 1 𝑛 ∞ (𝑇 − 𝑛𝑙𝑛𝛼 + ) 𝑦 𝑦 𝜆 ∫ ln( = )( )𝑛 𝑒 −𝑦 𝑑𝑦 1 1 Г(𝑛 + 1) 𝑇 − 𝑛𝑙𝑛𝛼 + 𝑇 − 𝑛𝑙𝑛𝛼 + 0 𝜆 𝜆 By simplification, we get: 1 𝐸(𝑙𝑛𝜃|𝑥) = 𝜑(𝑛 + 1) + ln(𝑇 − 𝑛𝑙𝑛𝛼 + ) 𝜆 Hence, 1 𝜃̂ = 𝐸𝑥𝑃 [𝜑(𝑛 + 1) + ln(𝑇 − 𝑛𝑙𝑛𝛼 + )] (17) 𝜆 Now, the corresponding Bayes estimator for R(𝑡) with posterior distribution (15) come out as: 𝛼 (𝑛 + 1)ln( ) 𝑡 ] 𝑅̂2 (𝑡) = 𝐸𝑥𝑃 [ (18) 1 𝑇 − 𝑛𝑙𝑛𝛼 + 𝜆 SIMULATION RESULTS In our simulation study, we generated I = 2500 samples of size n = 20, 50, and100 from Pareto distribution to represent small, moderate and large sample size with the several values of shape parameter, θ =0.5, 1.5 and 2.5 , the scale parameter 𝛼 = 1, 1.4 and taking t = 1.5, 3. We chose two values of 𝜆 for the Exponential prior (𝜆 =0.5, 3). In this section, Monte-Carlo simulation study is performed to compare the methods of estimation using mean square Errors (MSE’s) and integral mean squares error (IMSE), where 𝐿𝑒𝑡:

∑Ii=1(θ̂i − θ) MSE(θ̂) = I

2

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Bayes Estimators for the Reliability Function of Pareto Type I Distribution Under Squared-Log Error Loss Function Huda and Najam

The integrated mean squared error (IMSE) is an important global measure and it more accurate than MSE which is defined as distance between the estimate value of the reliability fonction and actual value of reliability fonction given by équation ̂ (t)) = 1 ∑Ii=1 [ 1 ∑nt (R ̂ i (t j ) − R(t j ))2 ] IMSE(R j=1 I

nt

nt

̂ (t)) = IMSE(R

1 ̂ i (t j )) ∑ MSE(R nt j=1

Where 𝑖 = 1, 2, … , 𝐿, 𝑛𝑡 the random limits of 𝑡𝑖 . In this paper, we use t = 1.5, 1.8, 2.1, 2.4, 2.7, 3 The results were summarized and tabulated in the following tables for each estimator and for all sample sizes. Table 1: IMSE's of the Different Estimators for Pareto Distribution Where 𝜃 = 0.5, 𝑅(𝑡)𝛼=1 = 0.57745967, 𝑅(𝑡)𝛼=1.4 = 0.68313 n Estimator

50

100

‫ = ג‬0.5

𝛼=1 0.0037297 0.0040447 0.0036041

𝛼 = 1.4 0.0018656 0.0021093 0.0018706

𝛼=1 0.0014338 0.0014703 0.0014089

𝛼 = 1.4 0.0007181 0.0007484 0.0007166

𝛼=1 0.0007068 0.0007168 0.0007021

𝛼 = 1.4 0.0003543 0.0003487 0.0003547

‫=ג‬3

0.0044940

0.0023735

0.0015395

0.0007883

0.0007346

0.0003723

BE(Sq. Log) ‫ = ג‬0.5

MinMSE

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BJ(Sq. Log)

MinMSE BJ(Sq. Log) BE (Sq. Log)

20

Best Estimator

Table 2: IMSE's of the Different Estimators for Pareto Distribution Where 𝜃 = 1.5, 𝑅(𝑡)𝛼=1 = 0.57745967, 𝑅(𝑡)𝛼=1.4 = 0.68313 n Estimator MinMSE BJ(Sq. Log) BE (Sq. Log)

‫ = ג‬0.5

‫=ג‬3 Best Estimator

20

50

100

𝛼=1 0.0069365 0.0060542 0.0052546

𝛼 = 1.4 0.0054207 0.0052179 0.0040936

𝛼=1 0.0025560 0.0024023 0.0022760

𝛼 = 1.4 0.0002048 0.0002003 0.0001823

𝛼=1 0.0012402 0.0012040 0.0011691

𝛼 = 1.4 0.0010033 0.0009937 0.0009456

0.0058954

0.0051766

0.0023749

0.0001998

0.0011977

0.0009932

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

BE(Sq. Log) ‫ = ג‬0.5

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Table 3: IMSE's of the Different Estimators for Pareto Distribution Where 𝜃 = 2.5, 𝑅(𝑡)𝛼=1 = 0.57745967, 𝑅(𝑡)𝛼=1.4 = 0.68313 n

20

Estimator

100

‫ = ג‬0.5

𝛼=1 0.0050510 0.0038279 0.0053541

𝛼 = 1.4 0.0056479 0.0048701 0.0056636

𝛼=1 0.00175183 0.0015517 0.0018501

𝛼 = 1.4 0.0020622 0.0019273 0.0020975

𝛼=1 0.0008306 0.0007823 0.0008580

𝛼 = 1.4 0.0009972 0.0009652 0.00100677

‫=ג‬3

0.0035196

0.0045048

0.0014976

0.0018669

0.0007684

0.0009501

BE(Sq. Log) ‫=ג‬3

BE(Sq. Log) ‫=ג‬3

BE(Sq. Log) ‫=ג‬3

BE(Sq. Log) ‫=ג‬3

BE(Sq. Log) ‫=ג‬3

BE(Sq. Log) ‫=ג‬3

MinMSE BJ(Sq. Log) BE (Sq. Log)

50

Best Estimator

DISCUSSION In general, the performance of Bayes estimator for reliability function of Pareto distribution under Squared-Log error loss function with exponential distribution was better in performance than using Jeffery prior (which equivalent to the Maximum likelihood estimator) for all sample sizes and with different values for θ. On the other hand, when θ < 2.5 the results show clearly, that, the increasing of the scale parameter α will decrease the values of IMSE for all estimators and with all cases. Finally, the simulation results show that, with large values of shape parameter of Pareto distribution (θ ≥ 2.5), the performance of Bayes estimator under Squared-Log error loss function using exponential prior with large value of ‫=ג( ג‬3) is more appropriate than using exponential prior with small value of ‫=ג( ג‬0.5), and vice versa.

REFERENCES [1] Al-Kutubi H. S., and Ibrahim, N. A. (2009): "Bayes'estimator for Exponential distribution with extension of Jeffrey prior information", Malaysian Journal of mathematical Sciences; 3(2), 297-313. [2] Nadarajah, S. & Kots, S.( 2003 ): Reliability in Pareto models, Metron, International, Journal of statistic, vol. LXI, No.2, pp.191204. [3] Podder, C.K. ( 2004 ): Comparison of Two Risk Functions Using Pareto distribution,pak. J. States. Vol. 20(3), pp. 369 – 378. [4] S. Dey, “Bayesian Estimation of the Parameter of the Generalized Exponential Distribution under Different Loss Functions”, Pak. J. Stat.Oper.Res., vol.6, no.2, pp. 163-174, 2010.

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[5] Setiya, P. & Kumar, V. " BAYESIAN ESTIMATION IN PARETO TYPE-I MODEL", Journal of Reliability and Statistical Studies; ISSN (Print): 0974-8024, (Online):2229-5666, Vol. 6, Issue 2 (2013): 139-150. [6] Shortle, J & Fischer, M. ( 2005 ): Using the Pareto distribution in queuing modeling, Submitted Journal of probability and Satistical Science, pp. 1 – 4. [7] S. K. Singh, U. Singh and D. Kumar, “Bayesian Estimation of the Exponentiated Gamma Parameter and Reliability Function under Asymmetric Loss Function”. REVSTAT – Statistical Journal, vol. 9, no.3, 247–260, 2011.

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