Bayesian inference for Parameter and Reliability ...

Australian Journal of Basic and Applied Sciences, 10(16) November 2016, Pages: 241-248

AUSTRALIAN JOURNAL OF BASIC AND APPLIED SCIENCES ISSN:1991-8178 EISSN: 2309-8414 Journal home page: www.ajbasweb.com

Bayesian inference for Parameter and Reliability function of Inverse Rayleigh Distribution Under Modified Squared Error Loss Function 1

Huda A. Rasheed and 2 Raghda, kh. Aref

1

AL-Mustansiriyah University Collage of Science, Dept. of Math. AL-Mustansiriyah University Collage of Science, Dept. of Math.

2

Address For Correspondence: Huda, A. Rasheed, AL-Mustansiriyah University Collage of Science, Dept. of Math.

ARTICLE INFO Article history: Received 11 September 2016 Accepted 10 November 2016 Published 28 November 2016 Keywords: Inverse Rayleigh, MLE, Bayes estimator, Modified squared error loss function, Jefferys prior and Gamma prior, Mean squared error, Integrated mean squared errors.

ABSTRACT In this study, obtained some Bayes estimators based on Modified squared error loss function as well as Maximum likelihood estimator for scale parameter and reliability function of Inverse Rayleigh distribution. In order to get better understanding of our Bayesian analysis, we consider non-informative prior for the scale parameter using Jefferys prior information as well as informative prior density represented by Gamma distribution. Based on Monte-Carlo simulation study, the behavior of Bayes estimates of the scale parameter of inverse Rayleigh distribution have been compared depending on the mean squared errors (MSE’s), while the estimates of the reliability function have been compared depending on the Integrated mean squared errors (IMSE’s). In the current study, we observed that, the performance of Bayes estimator for the scale parameter and reliability function under Modified squared error loss function with Gamma prior is better than the corresponding estimators with Jefferys prior, for all cases.

INTRODUCTION The inverse Rayleigh distribution is introduced by Voda. (1972). He studies some properties of the MLE of the scale parameter of inverse Rayleigh distribution which is also being used in lifetime experiments. Gharraph (1993) derived five measures of location for the Inverse Rayleigh distribution. These measures are the mean, harmonic mean, geometric mean, mode, and the median. He, also, estimated the unknown parameter using different methods of estimation. A comparison of these estimates was discussed numerically in term of their bias and root mean square error. In (2010) Soliman and other researchers studied the estimation and prediction from Inverse Rayleigh distribution based on lower record values, Bayes estimators have been developed under squared error and zero one-loss functions. In (2012) Dey discussed the Bayesian estimation of the parameter and reliability function of an Inverse Rayleigh distribution using different loss function represented by Square error, LINEX loss function. In (2013) Tabassum and others studied the Bayes estimation of the parameters of the Inverse Rayleigh distribution for left censored data under Symmetric and asymmetric loss functions. In (2014) Muhammad Shuaib Khan obtained the Modified Inverse Rayleigh Distribution as a special case of Inverse Weibull, which is extension to it. In (2015) Guobing discussed Bayes estimation for Inverse Rayleigh model under different loss functions represented by squared error loss, LINEX loss and entropy loss functions. In (2015) Rasheed and others compared between some classical estimators with the Bayes estimators of one parameter Inverse Rayleigh distribution under Generalized squared error loss function. In (2016) Rasheed and Aref obtained and discussed the Bayesian approach for estimating the scale parameter of Inverse Rayleigh distribution under different loss function. Finally, in (2016) Rasheed and Aref obtain Reliability Estimation in Inverse Rayleigh Distribution using Precautionary Loss Function. Open Access Journal Published BY AENSI Publication © 2016 AENSI Publisher All rights reserved This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

To Cite This Article: Huda A. Rasheed and Raghda, kh. Aref., Bayesian inference for Parameter and Reliability function of Inverse Rayleigh Distribution Under Modified Squared Error Loss Function. Aust. J. Basic & Appl. Sci., 10(16): 241-248, 2016

242

Huda A. Rasheed and Raghda, kh. Aref, 2016 Australian Journal of Basic and Applied Sciences, 10(16) November 2016, Pages: 241-248

1.

One parameter Inverse Rayleigh distribution: The probability density function (pdf) of the Inverse Rayleigh distribution with scale parameter 𝜃 is define as follows: (Shawky, A.I. and M.M. Badr, 2012) 2𝜃 𝜃 𝑓(𝑥; 𝜃) = 3 𝑒𝑥𝑝 (− 2 ) x > 0 , 𝜃 > 0 (1) 𝑥

𝑥

The corresponding cumulative distribution function (C.D.F), is given by: 𝜃

𝐹(𝑥; 𝜃) = 𝑒𝑥𝑝 (− 2) ; 𝑥 > 0,𝜃 > 0 𝑥 The Reliability function of inverse Rayleigh distribution is given by: 𝑅(x; 𝜃) = 1 − 𝐹(x; 𝜃) 𝜃 = 1 − 𝑒𝑥𝑝 (− 2 ) 𝑥 > 0 ,𝜃 > 0 𝑥 The Hazard function of Inverse Rayleigh distribution is given by: 𝑓(𝑥; 𝜃) ℎ(x; 𝜃) = 𝑅(𝑥; 𝜃) After substitution (1) and (3) into ℎ(x; 𝜃), yields: ℎ(x; 𝜃) =

2𝜃 𝜃 𝑒𝑥𝑝(− 2 ) 𝑥3 𝑥 𝜃 1−𝑒𝑥𝑝(− 2 ) 𝑥

=

2𝜃

(2)

(3)

(4)

𝜃 𝑥 3 (𝑒𝑥𝑝( 2 )−1) 𝑥

2. Estimation of parameter and the Reliability function In this section, we estimate the scale parameter 𝜃 and the Reliability function R(t) using Maximum Likelihood Estimator in addition of some Bayesian estimators, as follows 1. Maximum likelihood estimator: The Maximum likelihood estimator of the scale parameter 𝜃 and the reliability function 𝑅(𝑡) of inverse Rayleigh distribution have been derived as one of the classical estimators. The maximum likelihood estimator 𝜃̂𝑀𝐿 of the parameter  that maximizes the likelihood function is defined as: 1 1 𝐿(𝑥1 , 𝑥2 , … , 𝑥𝑛 ; 𝜃) = 2𝑛 𝜃 𝑛 ∏𝑛𝑖=1 3 exp (−𝜃 ∑𝑛𝑖=1 2) (5) 𝑥𝑖

𝑥𝑖

Taking the partial derivatives for the natural log-likelihood function, with respect to θ and then, equating to zero we have: 𝑛 𝜕 ln 𝐿(𝑥𝑖 ; θ) 𝑛 1 = −∑ 2 =0 𝜕θ 𝜃 𝑥𝑖 𝑖=1 Hence, the MLE of θ denoted by θ̂ ML is: 𝜃̂𝑀𝐿 =

𝑛 1

∑𝑛 𝑖=1𝑥 2

=

𝑛

(6)

𝑇

𝑖

Where T = ∑𝑛𝑖=1

1 𝑥𝑖 2

Since the Maximum likelihood estimator is invariant and one to one mapping (Singh, S.K., et al., 2011), hence the Maximum likelihood estimator of reliability function will be: 𝜃̂𝑀𝐿 𝑅̂𝑀𝐿 (t) = 1 − 𝑒𝑥𝑝 (− 2 ) (7) 𝑡 2. Bayes estimators: We provide Bayesian estimation method including informative and non-informative priors, under Modified squared error loss function to estimate scale parameter and reliability function of Inverse Rayleigh distribution. 2.1. Jefferys prior information: Assume that θ has a non-informative prior density defined as using Jeffreys prior information g1 (θ) which is given by (Rasheed, H. A. and Khalifa, Z. N. 2016) 𝑔1 ∝ √𝐼(𝜃) Where 𝐼(𝜃) represented Fisher information, defined as, follows: 𝜕2 ln 𝑓(𝑥;𝜃)

𝐼(𝜃) = −𝑛𝐸 [

𝜕𝜃 2

]

243


Therefore, 𝑔1 (𝜃) = 𝑏√−𝑛𝐸 (

𝜕2 ln 𝑓(𝑥;𝜃)

)

𝜕𝜃 2

, b is a constant

(8)

Now, taking the second partial derivative of log 𝑓(𝑥; 𝜃) with respect to 𝜃, gives 𝜕 2 ln 𝑓(𝑥𝑖 ; 𝜃) 1 = − 2 𝜕 𝜃2 𝜃 𝜕 2 ln 𝑓 (𝑥𝑖 ; 𝜃) 1 )= − 2 𝜕 𝜃2 𝜃 After substitution into (8), we get 𝑏 𝑔1 (𝜃) = √𝑛 , 𝜃>0 𝜃 The posterior density function is defined as: g(θ)L(θ; x1 , … , xn ) h(θ|x1 , … , xn ) = ∞ ∫0 g(θ)L(θ; x1 , … , xn ) dθ Hence, 𝐸 (

(9)

𝑏 1 1 𝑛 2𝑛 𝜃 𝑛 ∏𝑛𝑖=1 3 exp (−𝜃 ∑𝑛𝑖=1 2 ) 𝜃 √ 𝑥𝑖 𝑥𝑖 h1 (θ|x1 , … , xn ) = ∞𝑏 1 1 ∫0 𝜃 √𝑛 2𝑛 𝜃 𝑛 ∏𝑛𝑖=1 3 exp (−𝜃 ∑𝑛𝑖=1 2 ) 𝑑𝜃 𝑥𝑖 𝑥𝑖 h1 (θ|x1 , … , xn ) =

𝜃 𝑛−1 𝑒 −𝜃𝑇

∞ ∫0 𝜃 𝑛−1 𝑒 −𝜃𝑇 𝑑𝜃

, 𝑇 = ∑𝑛𝑖=1

1 𝑥𝑖 2

,

𝜃>0

Hence, the posterior density function of θ with Jeffreys prior can be written as 𝑇 𝑛 𝜃 𝑛−1 𝑒 −𝜃𝑇 h1 (θ|x1 , … , xn ) = (10) Γn It is clear, h1 (θ|x1 , … , xn ) is recognized as the density of the Gamma distribution, i.e., θ|x~𝐺𝑎𝑚𝑚𝑎(𝑛, 𝑇), with 𝑛 𝑛 𝐸(𝜃) = , 𝑉𝑎𝑟(𝜃) = 2 𝑇 𝑇 2.2. Gamma prior distribution: Assuming that θ has informative prior as Gamma prior which takes the following form 𝑔2 (𝜃) =

𝛽 𝛼 𝜃 𝛼−1 𝑒 −𝜃𝛽 Γ𝛼

;

𝜃>0

,𝛼 > 0,𝛽 > 0

(11)

Where, 𝛽, 𝛼 are the shape and the scale parameters respectively. From Bayesian theorem the posterior density function of  denoted by ℎ2 (𝜃|𝑥) can be obtained as ℎ2 (𝜃|𝑥) =

𝑔2 (𝜃)𝐿(𝜃; 𝑥1 𝑥2 … … . . 𝑥𝑛 ) ∞ ∫0 𝑔2 (𝜃)𝐿(𝜃; 𝑥1 𝑥2 … … . . 𝑥𝑛 )𝑑𝜃

Now, combining (5) and (11), gives ℎ2 (𝜃|𝑥) =

𝜃 𝛼−1+𝑛 𝑒 −𝜃(𝑇+𝛽)

∞ ∫0 𝜃 𝛼−1+𝑛 𝑒 −𝜃(𝑇+𝛽) 𝑑𝜃

So, the posterior density function of θ with Gamma prior is: ℎ2 (𝜃|𝑥) =

𝑃𝛼+𝑛 𝜃 𝛼−1+𝑛 𝑒 −𝜃𝑃 𝛤(𝛼 + 𝑛)

,

𝜃>0

Where, 𝑃 = 𝑇 + 𝛽 𝛼+𝑛 𝛼+𝑛 Notice that: θ|x~Gamma(𝛼 + 𝑛 , 𝑃), with, E(𝜃) = , 𝑉𝑎𝑟 (𝜃) = 2 𝑃 𝑃 2.3.Bayes estimator for θ under Modified squared error loss function: The modified squared error loss function can be defined as follows (Al-Baldawi, T. H., 2013)

(12)

244


2 L(θ̂ , θ) = 𝜃 𝑟 (θ̂ − θ) The Risk function under the Modified squared error loss function which is denoted by R MS (θ̂, θ) is

R MS (θ̂ , θ) = E[L(θ̂ , θ)] ∞

2

𝑅𝑀𝑆 (𝜃̂ , 𝜃) = ∫ 𝜃 𝑟 (θ̂ − θ) h(θ|x)dθ

(13)

0 ̂2

𝑅𝑀𝑆 (𝜃̂ , 𝜃) = 𝜃 E(𝜃 𝑟 |x) − 2𝜃̂E(𝜃 𝑟+1 |x) + E(θr+2 |x) Taking the partial derivative for R MS (θ̂ , θ) with respect to θ̂ and setting it equal to zero, gives the Bayes estimator relative to Modified square error loss function which is denoted by 𝜃̂𝑀𝑆 as 𝜃̂𝑀𝑠 = 𝑟+1 E(𝜃 |x) (14) 𝑟 E(𝜃 |x) (𝒊) 𝑾ith Jefferys prior information: According to the posterior density function (10), the Bayes estimator for θ under Modified squared error loss function can be derived as follows: ∞

𝑚

𝐸(𝜃 |𝑥) = ∫ 𝜃 𝑚 h1 (θ|x)dθ 0 𝑚

∞ 𝜃 𝑚+𝑛−1 𝑒 −𝜃𝑇 𝑇 𝑛

𝐸(𝜃 |𝑥) = ∫0 𝐸(𝜃 𝑚 |𝑥) = 𝑚

E(𝜃 |𝑥) =

𝑑𝜃

Γ𝑛 ∞ 𝜃 𝑚+𝑛−1 𝑒 −𝜃𝑇 𝑇 𝑛+𝑚 ∫ Γ𝑛 𝑇 𝑚 0 Γ(𝑛+𝑚) Γ(𝑛+𝑚)

Γ(𝑛+𝑚)

𝑑𝜃 (15)

Γ𝑛 𝑇 𝑚

We get the Bayes estimator for the scale parameter of inverse Rayleigh distribution under Modified square loss function with Jeffreys prior denoted by θ̂ MSJ1 , θ̂ MSJ2 with r = 1, 3 respectively, are: E(𝜃1+1 |x) 𝑛+1 (16) θ̂ MSJ1 = = 1 𝑇 E(𝜃 |x) E(𝜃1+3 |x) 𝑛+3 (17) θ̂ MSJ2 = = 𝑇 E(𝜃 3 |x) (𝖎𝖎) With Gamma prior information: According to the posterior density function (12), the Bayes estimator of θ of Inverse Rayleigh distribution under Modified squared error loss function we substitute two values of r, r = 1, 3 respectively into (14) as follows: ∞

𝑚

𝐸(𝜃 |𝑥) = ∫ 𝜃 𝑚 h2 (θ|x)dθ 0 ∞

= ∫0 𝜃 𝑚 𝐸(𝜃 𝑚 |𝑥)

𝑃𝛼+𝑛 𝜃 𝛼+𝑛−1 𝑒 −𝜃𝑃

𝑑𝜃

𝛤(𝛼+𝑛) 𝛤(𝛼+𝑛+𝑚) ∞ 𝜃 𝑚+𝛼+𝑛−1 𝑒 −𝜃𝑃 𝑃𝛼+𝑛+𝑚 = ∫ 𝛤(𝛼+𝑛) 𝑃𝑚 0 𝛤(𝛼+𝑛+𝑚) 𝛤(𝛼+𝑛+𝑚)

E(𝜃 𝑚 |𝑥) =

𝛤(𝛼+𝑛) 𝑃𝑚

𝑑𝜃 (18)

the Bayes estimator for the scale parameter of inverse Rayleigh distribution under Modified square loss function with Gamma prior denoted by θ̂ MSG1 , θ̂ MSG2 with r = 1, 3 respectively, are: E(𝜃1+1 |x) 𝛼 + 𝑛 + 1 (19) θ̂ MSG1 = = 𝑃 E(𝜃|x) E(𝜃1+3 |x) 𝛼 + 𝑛 + 3 (20) θ̂ MSG2 = = 𝑃 E(𝜃 3 |x) 2.4.Bayes estimator for R(t) Under Modified squared error loss function: We can find the Bayes estimator for the reliability function R(t) by using the probability density function for 𝜃. According to (14), the Bayes estimator for R(t) under Modified squared error loss function, will be :

245


𝑅̂ (𝑡)𝑀𝑆 =

𝐸((𝑅(𝑡))𝑟+1 |t)

(21) 𝐸((𝑅(𝑡))𝑟 |t) (𝖎) Bayes estimator for R(t) based on Jeffreys prior information: To derive the Bayes estimator for the 𝑅(𝑡) under Modified squared error loss function (MSLF) with Jeffreys prior denoted by 𝑅̂ (𝑡)𝑀𝑆𝐽 , we substitute two values of r, r = 1, 3 respectively into (21) which required to obtain 𝐸(𝑅(𝑡)|𝑡) , 𝐸((𝑅(𝑡))2 |𝑡), 𝐸((𝑅(𝑡))3 |𝑡) 𝑎𝑛𝑑 𝐸((𝑅(𝑡))4 |𝑡) as follows: ∞

(22)

𝐸(𝑅(𝑡)|𝑡) = ∫ 𝑅(𝑡)ℎ1 (𝜃|𝑡)𝑑𝜃 0

𝜃 ) 𝑡2 ∞ 𝜃 𝑇 𝑛 𝜃 𝑛−1 𝑒 −𝜃𝑇 𝐸(𝑅(𝑡)|𝑡) = ∫0 (1 − exp (− 2) ) 𝑑𝜃

Since R(t) = 1 − exp (−

𝑇𝑡 2 𝐸(𝑅(𝑡)|𝑡) = 1 − ( 2 ) 𝑇𝑡 + 1

𝑛

Γ𝑛

𝑡

(23)

By the same way, we can find 𝐸((𝑅(𝑡))2 |𝑡), 𝐸((𝑅(𝑡))3 |𝑡) 𝑎𝑛𝑑 𝐸((𝑅(𝑡))4 |𝑡) so, 𝑛

𝐸((𝑅(𝑡))2 |𝑡) = 1 − 2 (

𝑇𝑡 2 𝑇𝑡 2 ) + ( ) 𝑇𝑡 2 + 1 𝑇𝑡 2 + 2 n

3

E ((R(t)) |t) = 2 − 3 (

Tt 2 Tt 2 ) + ( ) t2T + 1 t2 T + 2 𝑛

𝑛

(24)

n

(25) 𝑛

𝑇𝑡 2 𝑇𝑡 2 ) + 2 ( ) (26) 𝑇𝑡 2 + 1 𝑇𝑡 2 + 2 Hence, from (24), (23) we get the Bayes estimator for the R(t) of inverse Rayleigh distribution under ̂ (t)MSJ1 as follows Modified square error loss function with Jefferys prior with r = 1, which is denoted by R n n Tt 2 Tt 2 1− 2( 2 ) +( 2 ) Tt + 1 Tt +2 ̂ (t)MSJ1 = R (27) n Tt 2 1−( 2 ) Tt + 1 Now, from (26) and (25) we obtain the Bayes estimator for the 𝑅(𝑡) of inverse Rayleigh, distribution under Modified squared error loss function with Jefferys prior with r = 3, which is denoted by 𝑅̂(𝑡)𝑀𝑆𝐽2 𝑛 𝑛 𝑇𝑡 2 𝑇𝑡 2 2 − 4( 2 ) + 2( 2 ) 𝑇𝑡 + 1 𝑇𝑡 + 2 𝑅̂ (𝑡)𝑀𝑆𝐽2 = (28) 𝑛 𝑛 𝑇𝑡 2 𝑇𝑡 2 2 − 3( 2 ) +( 2 ) 𝑇𝑡 + 1 𝑇𝑡 + 2 𝐸((𝑅(𝑡))4 |𝑡) = 2 − 4 (

(𝖎𝖎) Bayes estimator for R(t) based on Gamma prior information: To derive the Bayes estimator for the 𝑅(𝑡) under Modified squared error loss function (MSLF) with Gamma prior, that is denoted by 𝑅̂(𝑡)𝑀𝑆𝐺 , we'll derive 𝐸(𝑅(𝑡)|𝑡) , 𝐸((𝑅(𝑡))2 |𝑡), 𝐸((𝑅(𝑡))3 |𝑡) 𝑎𝑛𝑑 𝐸((𝑅(𝑡))4 |𝑡) as follows: ∞

(29)

𝐸(𝑅(𝑡)|𝑡) = ∫ 𝑅(𝑡)ℎ2 (𝜃|𝑡)𝑑𝜃 0

𝜃 ) 𝑡2 ∞ 𝜃 𝐸(𝑅(𝑡)|𝑡) = ∫0 (1 − exp (− 2) )

since R(t) = 1 − exp (−

2

𝑃𝛼+𝑛 𝜃 𝛼−1+𝑛 𝑒 −𝜃𝑃

𝑡 𝛼+𝑛

𝛤(𝛼+𝑛)

𝑑𝜃

𝑃𝑡 ) 𝑃𝑡 2 + 1 By same way we can find 𝐸((𝑅(𝑡))2 |𝑡), 𝐸((𝑅(𝑡))3 |𝑡) 𝑎𝑛𝑑 𝐸((𝑅(𝑡))4 |𝑡). Hence,

𝐸(𝑅(𝑡)|𝑡) = 1 − (

𝑃𝑡 2 𝐸 ((𝑅(𝑡)) |𝑡) = 1 − 2 ( 2 ) 𝑃𝑡 + 1 2

𝛼+𝑛

𝑃𝑡 2 + ( 2 ) 𝑃𝑡 + 2

(30)

𝛼+𝑛

(31)

246

Huda A. Rasheed and Raghda, kh. Aref, 2016 Australian Journal of Basic and Applied Sciences, 10(16) November 2016, Pages: 241-248 𝛼+𝑛

𝛼+𝑛

𝑃𝑡 2 𝑃𝑡 2 ) + ( ) 𝑡 2𝑃 + 1 𝑡 2𝑃 + 2 𝛼+𝑛 𝛼+𝑛 𝑃𝑡 2 𝑃𝑡 2 𝐸((𝑅(𝑡))4 |𝑡) = 2 − 4 ( 2 ) + 2( 2 ) 𝑃𝑡 + 1 𝑃𝑡 + 2 𝐸((𝑅(𝑡))3 |𝑡) = 2 − 3 (

(32) (33)

From (31), (30) we can get the Bayes estimator for the R(t) using Modified squared error loss function ̂ (t)MSG1 as follows based on Gamma prior with r = 1, which is denoted by R 𝛼+𝑛 𝛼+𝑛 2 2 𝑃𝑡 𝑃𝑡 1 − 2( 2 ) +( 2 ) 𝑃𝑡 + 1 𝑃𝑡 +2 (34) 𝑅̂ (𝑡)𝑀𝑆𝐺1 = 𝛼+𝑛 𝑃𝑡 2 1−( 2 ) 𝑃𝑡 + 1 Now, from (33), (32) we get the Bayes estimator for the R(t) under Modified squared error loss function ̂ (t)MSG2 based on Gamma prior with r = 3, that is denoted by R α+n α+n 2 2 Pt Pt 2 −4( 2 ) + 2( 2 ) Pt + 1 Pt +2 ̂ (t)MSG2 = R (35) α+n α+n Pt 2 Pt 2 2 − 3( 2 ) +( 2 ) Pt + 1 Pt + 2 4. Simulation Study: In our simulation study, the process have been repeated 5000 times (L=5000). We generated samples of sizes n = 10, 25, 50, and 100 from Inverse Rayleigh distribution with 𝜃 = 0.5, 1.5 and 3. The values of the parameters Gamma prior are chosen to be 𝛽 = 1.2,3 , 𝛼 = 0.3, 0.8. The expected values and mean squared errors (MSE's) for all estimates of the parameter 𝜃 are obtained, where: 2 ∑𝐿𝑖=1(𝜃̂𝑖 − 𝜃) 𝑀𝑆𝐸(𝜃) = ; 𝑖 = 1, 2 , 3, … , 𝐿 𝐿 and integral mean squares error (IMSE) for all estimates of the reliability function of Inverse Rayleigh distribution which is defined as distance between the estimate value of the reliability function and actual value of reliability function that is given as follows: 𝐼𝑀𝑆𝐸 (𝑅̂(𝑡)) =

𝑡 ∑𝑛𝑗=1 𝑀𝑆𝐸(𝑅̂𝑖 (𝑡𝑗 ))

𝑛𝑡 𝐿

𝑛𝑡

𝑖=1

𝑗=1

, i = 1, 2, … , L , nt the random limits of 𝑡𝑖

1 1 2 𝐼𝑀𝑆𝐸(𝑅̂ (𝑡)) = ∑ [ ∑(𝑅̂𝑖 (𝑡𝑗 ) − 𝑅(𝑡𝑗 )) ] 𝐿 𝑛𝑡 The results were summarized and tabulated in the following tables for each estimator and for all sample sizes as follows: Table 1: Expected Values and MSE’s of the Different Estimates for the Inverse Rayleigh Distribution when θ=0.5 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior Estimates r=1 r=3 n MLE Criteria r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 EXP 0.55330 0.60863 0.71929 0.58207 0.60782 0.52841 0.55179 0.68509 0.71084 0.62193 0.64532 10 MSE 0.04207 0.05926 0.11438 0.04294 0.05111 0.02452 0.02854 0.08442 0.09846 0.04772 0.05648 EXP 0.52179 0.54266 0.58440 0.53493 0.54510 0.51529 0.52508 0.57561 0.58578 0.55447 0.56427 25 MSE 0.01282 0.01517 0.02260 0.01349 0.01477 0.01074 0.01154 0.01992 0.02207 0.01513 0.01673 EXP 0.51021 0.52042 0.54083 0.51701 0.52205 0.50762 0.51256 0.53717 0.54221 0.52741 0.53235 50 MSE 0.00568 0.00622 0.00793 0.00587 0.00617 0.00523 0.00543 0.00740 0.00791 0.00634 0.00674 EXP 0.50536 0.51041 0.52052 0.50881 0.51132 0.50420 0.50669 0.51886 0.52137 0.51416 0.51665 100 MSE 0.00265 0.00278 0.00320 0.00270 0.00278 0.00255 0.00260 0.00309 0.00321 0.00283 0.00293 Table 2: Expected Values and MSE’s of the Different Estimates for the Inverse Rayleigh Distribution when θ=1.5 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior Estimates r=1 r=3 n MLE Criteria r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 EXP 1.65991 1.82589 2.15788 1.53852 1.60660 1.22092 1.27494 1.87891 1.87891 1.43701 1.49104 10 MSE 0.37858 0.53335 1.02940 0.18921 0.21607 0.14937 0.12860 0.42355 0.42355 0.10299 0.10669 EXP 1.56536 1.62797 1.75320 1.52721 1.55624 1.37818 1.40438 1.64335 1.67238 1.48298 1.50918 25 MSE 0.11534 0.13651 0.20343 0.09064 0.09652 0.07380 0.07037 0.12464 0.13752 0.06856 0.07079 EXP 1.53064 1.56125 1.62247 1.51368 1.52844 1.43598 1.44998 1.57270 1.58745 1.49197 1.50596 50 MSE 0.05110 0.05594 0.07137 0.04561 0.04712 0.04074 0.03986 0.05432 0.05761 0.03962 0.04034 EXP 1.51607 1.53123 1.56156 1.50807 1.51551 1.46834 1.47559 1.53784 1.49733 1.54529 1.50458 100 MSE 0.02385 0.02505 0.02882 0.02256 0.02296 0.02121 0.02100 0.02483 0.02102 0.02567 0.02123

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Table 3: Expected Values and MSE’s of the Different Estimates for the Inverse Rayleigh Distribution when θ=3 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior Estimates r=1 r=3 n MLE Criteria r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 EXP 3.31981 3.65179 4.31575 2.62050 2.73645 1.82819 1.90909 3.08430 3.20026 2.15177 0.61056 10 MSE 1.51433 2.13342 4.11759 0.52651 0.48655 1.46172 1.28670 0.53698 0.61056 0.84222 2.23266 EXP 3.13071 3.25594 3.50640 2.84880 2.90296 2.37333 2.41845 3.06543 3.11960 2.55381 0.33802 25 MSE 0.46136 0.54603 0.81374 0.29282 0.28973 0.52111 0.47152 0.31685 0.33802 0.34774 2.59893 EXP 3.06128 3.12250 3.24495 2.92193 2.95041 2.64613 2.67192 3.03585 3.06432 2.74929 0.17707 50 MSE 0.20442 0.22377 0.28546 0.16333 0.16278 0.23032 0.21479 0.17102 0.17707 0.17630 2.77509 EXP 3.03214 3.06246 3.12311 2.96269 2.97732 2.81326 2.82713 3.02119 3.03581 2.86880 0.08920 100 MSE 0.09542 0.10018 0.11529 0.08513 0.08508 0.10285 0.09853 0.08752 0.08920 0.08790 2.88268 Table 4: IMSE's of the Different Estimates for R(t) of Inverse Rayleigh Distribution where 𝜃 = 0.5, R(t) = 0.054041 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior r=1 r=3 n MLE r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 10 0.00177 0.00219 0.01711 0.00165 0.00194 0.00102 0.00116 0.01488 0.01679 0.01090 0.01245 25 0.00059 0.00066 0.01137 0.00059 0.00064 0.00049 0.00051 0.01081 0.01149 0.00949 0.01011 50 0.00027 0.00028 0.00952 0.00027 0.00028 0.00024 0.00025 0.00930 0.00962 0.00870 0.00901 100 0.00013 0.00013 0.00871 0.00013 0.00013 0.00012 0.00012 0.00860 0.00876 0.00832 0.00847 Table 5: IMSE's of the Different Estimates for R(t) of Inverse Rayleigh Distribution where 𝜃 = 1.5,R(t) = 0.1535183 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior r=1 r=3 n MLE r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 10 0.00710 0.00750 0.03970 0.00391 0.00405 0.00497 0.00415 0.02420 0.02767 0.00970 0.01173 25 0.00265 0.00275 0.02942 0.00208 0.00213 0.00214 0.00198 0.02385 0.02539 0.01626 0.01750 50 0.00125 0.00127 0.02561 0.00111 0.00112 0.00111 0.00107 0.02299 0.02379 0.01892 0.01962 100 0.00060 0.00061 0.02386 0.00056 0.00057 0.00056 0.00055 0.02259 0.02299 0.02047 0.02085 Table 6: IMSE's of the Different Estimates for R(t) of Inverse Rayleigh Distribution where 𝜃 = 3, R(t) = 0.2834688 Under Modified Squared Error Loss Function Jeffreys prior Gamma prior r=1 r=3 n MLE r =1 r =3 β = 1.2 β=3 β = 1.2 β=3 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 α =0.3 α =0.8 10 0.01068 0.01019 0.03888 0.00853 0.00714 0.02683 0.02341 0.01334 0.01586 0.00254 0.00262 25 0.00434 0.00426 0.03074 0.00375 0.00349 0.00841 0.00757 0.01939 0.02082 0.00816 0.00904 50 0.00209 0.00208 0.02742 0.00194 0.00187 0.00334 0.00310 0.02161 0.02241 0.01421 0.01486 100 0.00102 0.00102 0.02587 0.00098 0.00096 0.00136 0.00129 0.02292 0.02335 0.01866 0.01905

Discussion: 1. The results of the simulation study for estimating the scale parameter (𝜃) of Inverse Rayleigh distribution show that:  From table (1), when the 𝜃=0.5, the performance of Bayes estimator under Modified squared error loss function with (𝛽=3, 𝛼=0.3 and r=1) is the best estimator comparing to the other estimators for all sample size.  From table (2), when the 𝜃=1.5, it is a clear that, the performance of Bayes estimator under Modified squared error loss function with (𝛽=3, 𝛼=0.3 and r=2) is the best estimator comparing to the other estimators for all sample size except the sample (100).  From table (3), we observed that, the performance of Bayes estimator under Modified squared error loss function with (𝛽=1.2, 𝛼=0.8 and r=1) is the best estimator comparing to the other estimators for all sample size. 2. The results of the simulation study for estimating the reliability function R(t) of Inverse Rayleigh distribution show that  From table (4), notice that, the performance of Bayes estimator under Modified squared error loss function with (𝛽=3, 𝛼=0.3 and r=1) is the best estimator comparing to the other estimators for all sample size.  From table (5), we observed that, the performance of Bayes estimator under Modified squared error loss function with (𝛽=3, 𝛼=0.3 and r=1) is the best estimator comparing to the other estimator for all sample size except the sample (10).  From table (6), it is a clear that, the performance of Bayes estimator under Modified squared error loss function (MSELF) with (𝛽=1.2, 𝛼=0.8 and r=1) is the best estimator comparing to the other estimator for all sample size except the sample (10). In general, we conclude that, in situation involving estimation of scale parameter(𝜃) and reliability function R(t) of Inverse Rayleigh distribution under Modified squared error loss function using Gamma prior for all samples sizes. REFERENCES Al-Baldawi, T., 2013, "Comparison of Maximum Likelihood and some Bayes Estimators for Maxwell Distribution based on Non-informative Priors", Baghdad Science Journal, 10(2): 480-488. Dey, S., 2012. "Bayesian estimation of the parameter and reliability function of an inverse Rayleigh distribution", Malaysian Journal of Mathematical Sciences, 6(1): 113-124.

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