Bayes Pre-Test Shrinkage Estimation of Rayleigh ...Bayes Pre-Test Shrinkage Estimation of Rayleigh...

IJAAMMInt. J. Adv. Appl. Math. and Mech. 7(4) (2020) 72 – 90 (ISSN: 2347-2529)

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International Journal of Advances in Applied Mathematics and Mechanics

Bayes Pre-Test Shrinkage Estimation of Rayleigh Distribution underDifferent Loss Functions

Research Article

Alaa Khlaif Jiheel∗, Ahmed Baqer Jaafar Al-Qatifi

Department of Mathematics,College of Education for Pure Sciences, University of Thi-Qar, Nasiriyah, Iraq

Received 26 March 2020; accepted (in revised version) 22 May 2020

Abstract: In this article, we suggest and study properties Bayes pre-test Shrinkage estimators of scale parameter for Rayleighdistribution. The equation of risk function and relative risk with respect classical estimator for the proposed estima-tors under squared error loss function (SELF) and Linex loss function (LLF) are derived. and the numerical resultsshow that performance our estimators than classical Bayes estimator.

Keywords: Rayleigh distribution • Bayes estimator • shrinkage estimator • mean squared error • squared error loss function• Linex loss function • relative risk

© 2020 The Author(s). This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/).

1. Introduction

Rayleigh distribution was introduced in the literature by Lord Rayleigh ,It represents a special case for the distri-bution of Weibull . it has been widely used in reliability theory and survival analysis because of its flexibility andsimplicity . [Dey (2011)]The characteristic of the Rayleigh distribution is that its failure rate is a linear function of time. The probability densityfunction of one parameter Rayleigh distribution is given . [Sinha , Howlader (1983)]

f (x;θ) = 2x

θexp

−x2

θ ; x > 0 , θ > 0. (1)

and the cumulative density function (cdf) has the form

F (x;θ) = 1−exp−

x2

θ ; x > 0 , θ > 0. (2)

The estimation of parameters is an important subject in statistical inference because it gives a full definition ofdistributions contain an unknown parameters. If we don’t have any information about the unknown parameters, weuse classic methods like maximum likelihood method (MLE) , the moment method (MM) , the ordinary least squaremethod (OLS) ,etsBut if the prior information is a prior distribution of the unknown parameter, we use the Bayes estimation.Sometimes the prior information is an initial value using based on past experiences.therefore, we should use it in the

∗ Corresponding author.E-mail address(es): [email protected] (Alaa Khlaif Jiheel), [email protected] (Ahmed Baqer Jaafar Al-Qatifi).

72

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Alaa Khlaif Jiheel, Ahmed Baqer Jaafar Al-Qatifi / Int. J. Adv. Appl. Math. and Mech. 7(4) (2020) 72 – 90 73

our estimator , The shrinkage estimator is a good way to give this aim , it was defined by Thompson (1968) based on alinear composition between the classical estimate and the guess value θ0 .defined as

kθ̂+ (1−k)θ0 (3)

where k is a shrinkage factor such that 0 < k < 1 .In many situations may not be sure that the true value θ close to the initial value θ0 ,therefore; the researchers sug-gested preliminary test procedure depending on testing hypothesis for the null hypothesis H0 : θ = θ0 against thealternative hypothesis H1 : θ 6= θ0

If H0 accepted then the estimator becomes

kθ̂+ (1−k)θ0

otherwise, then the classical estimator is our estimator .Thus, the pre-test estemster define

θ̂ =

kθ̂+ (1−k)θ0, if H0 accepted ,

θ̂, otherwise.(4)

The pretest define above studied by many researchers to estimate unknown parameters or parametric functionsfor different life distributions and different types of samples in the life testing e.g.Pandey(1985) , Singh ( 1996),Parkash and et al (2008),Al-Hemyari and et al (2009), Srivastava and Shah (2010) ,Sanubhogue and et al (2012), Nasiri and et al (2018) , Al-Joboori and Raeeda (2014) and Nasiri and Nooghabi (2018) .

2. Bayes Estimation of Rayleigh Distribution

Let X1, X2, ., ., ., Xn be a random sample size n from Rayleigh distribution . Assume that θ has Improper priordensity

g (θ) = exp−

a

θ

θc θ > 0

then it easy to show the posterior density function of θ given

∏(θ/x1, x2, ., ., ., xn) = (T +a)c+n−1 exp− T+a

θ

θc+nΓ(c +n −1)

then by making the derivations to extract the Bayes estimator of(θ) in this article, we use a squared error loss function(SELF) so the Beyes estimator given.

θ̂B = E(θ) =∫θθ

∏(x/θ)dθ

we get

θ̂B =∑n

i=1 x2i +a

(c +n −2)(5)

where c=2 and a=0 then we get the MLE of θAssume that T =∑n

i=1 x2i and let z = c +n −2 we get

θ̂B = T +a

z(6)

Now, we easy to prove the random variable T has Gamma distribution with parameter n, θ i.e

f (T ;θ) = T n−1 exp− Tθ

Γ(n)θn (7)

So;2T

θhas Chi-square distribution with degree freedom 2n

from (5) then T = θ̂B z −a

74 Bayes Pre-Test Shrinkage Estimation of Rayleigh Distribution under Different Loss Functions

dTd θ̂B

= z =| J |therefor; by using a transformation method we can show the estimator θ̂B has density function as

f(B ;θ) =

z(θ̂B z)n−1 exp− θ̂B z−a

θ

Γ(n)θn i f B > a

z

0 a.w

3. Proposed Pre-test Shrinkage Estimator

Assume the guess value θ0 is available then we must use it in our estimator, and depending above densityfunction of θ̂B we proposed the following pre-test estimators .The first proposed estimator (θ̂BS1) is defind

θ̂BS1 =

k1θ̂B + (1−k1)θ0, if H0 accepted ,

θ̂B , otherwise.(8)

where k1 is shrinkage factor it is a constant such that 0 < k1 < 1and R is a pretest region for testing the null hypothesis H0 : θ = θ0 .against the alternative hypothesis H1 : θ 6= θ0 with significance of level αThe second proposed estimator(θ̂BS1) is defind

θ̂BS2 =

k2θ̂B + (1−k2)θ0, if H0 accepted ,

θ̂B , otherwise.(9)

where k2 is the shrinkage factor depending statistical testing the null hypothesis H0 : θ = θ0 . against the alternativehypothesis H1 : θ 6= θ0 with significance of level α .

by (7) , then2T

θhas χ2 distribution with 2n .

at θ = θ0 ; then accepting region H0 is

θ0

2χ2

α2 ,2n < T < θ0

2χ2

1− α2 ,2n

Let r1 = θ0

2χ2

α2 ,2n

and r2 = θ0

2χ2

1− α2 ,2n

now;r1 < T < r2

then0 < T − r1 < r2

therefore;

0 < T − r1

r2< 1

then

k2 = T − r1

r2

Now; we can show the boundary the region R define as follow

χ2α2 ,2n

2z+ a

z< θ̂B <

χ21− α

2 ,2n

2z+ a

z

R = [χ2

α2 ,2n

2z+ a

z,χ2

1− α2 ,2n

2z+ a

z]


4. Risk function of Proposed Pre-test Shrinkage Estimator under square and linex losses func-tions

In this section, we shall drive the risk function of the proposed estimator under different symmetry squareerror loss function (SELF) and asymmetry linex losse function(LLF).

4.1. Risk function of (θ̂BS1) under SELF

The risk function of estimator (θ̂BS1) under square error loss function can define as follows

R(θ̂BS1/SELF ) = E(θ̂BS1 −θ)2

R(θ̂BS1/SELF ) = E(θ̂2BS1)−2θE(θ̂BS1)+θ2 (10)

Now

E(θ̂BS1) =∫

R(k1θ̂B + (1−k1)θ0) f (θ̂B )d θ̂B +

∫R̄

(θ̂B ) f (θ̂B )d θ̂B

=∫

R

Z (k1θ̂B + (1−k1)θ0)

Γ(n)θn (Z θ̂B −a)n−1 exp−Z θ̂B −a

θ d θ̂B

+∫ ∞

0

Z θ̂B

Γ(n)θn(Z θ̂B −a)n−1 exp

−Z θ̂B −aθ d θ̂B

−∫

R

Z θ̂B

Γ(n)θn(Z θ̂B −a)n−1 exp

−Z θ̂B −aθ d θ̂B

By using the transformation y = Z θ̂B −a

θ

then B = yθ+a

Zand d θ̂B = θ

Zd y

We get

E(θ̂BS1) = 1

Γ(n)

∫ b∗

a∗[

k1

Z(θyn +ayn−1)+ (1−k1)θ0 yn−1]exp−y d y

+ 1

ZΓ(n)(θΓ(n +1)+aΓ(n))− 1

ZΓ(n)

∫ b∗

a∗(θyn +ayn−1)exp−y d y

After calculating the integrals and simplifications we get

E(θ̂BS1) =k1[θ

ZJ1(a∗,b∗)+ a

ZJ0(a∗,b∗)−θ0 J0(a∗,b∗)]

− θ

zJ1(a∗,b∗)+ (θ0 − a

z)J0(a∗,b∗)+ 1

z(θn +a) (11)

where

Jm(a∗,b∗) = 1

Γ(n)

∫ b∗

a∗ym yn−1 exp−y d y at m = 0,1, ., ., . (12)

and

a∗ = λ

2χ2

α2 ,2n b∗ = λ

2χ2

1− α2 ,2n

where λ= θ0

θχ2

α2 ,2n

, χ21− α

2 ,2nis respectively lower and upper (

α

2) the percentile value chi-square distribution with 2n

degrees of freedom.for accpte the hypothesis H0 : θ = θ0 against H0 : θ 6= θ0

Now we calculate

E(θ̂BS1)2 =∫

R

Z (k1θ̂B + (1−k1)θ0)2


θ d θ̂B

+∫

R̄

Z θ̂2B


θd θ̂B


By using the same transformation and previous assumptions used in E(θ̂BS1) we get

E(θ̂BS1)2 =k21[θ2

Z 2 J2(a∗,b∗)+ 2aθ

Z 2 J1(a∗,b∗)+ a2

Z 2 J0(a∗,b∗)

− 2θ0θ

ZJ1(a∗,b∗)− 2aθ0

ZJ0(a∗,b∗)+θ2

0 J0(a∗,b∗)]

+2k1[θ0θ

ZJ1(a∗,b∗)+ aθ0

ZJ0(a∗,b∗)−θ2

0 J0(a∗,b∗)]

+ 1

Z 2 (θ2n(n +1)+2aθn +a2)

− 1

Z 2 (θ2 J2(a∗,b∗)+2aθJ1(a∗,b∗)

+a2 J0(a∗,b∗))+θ20 J0(a∗,b∗) (13)

By useing (11) ,(12) and(14) then

R(θ̂BS1/SELF ) =θ2k21[

1

Z 2 J2(a∗,b∗)+ 2aλ

Z 2θ0J1(a∗,b∗)+ a2λ2

Z 2θ20

J0(a∗,b∗)

− 2λ

ZJ1(a∗,b∗)− 2aλ2

Zθ0J0(a∗,b∗)+λ2 J0(a∗,b∗)] subsect i on

+2k1[1

Z(λ−1)J1(a∗,b∗)+λ(λ−1)(

a

Zθ0−1)J0(a∗,b∗)]

− 1

Z 2 J2(a∗,b∗)− (2aλ

Z 2θ0− 2

Z)J1(a∗,b∗)

− ((a2λ2

Z 2θ0− 2aλ

Zθ0)+λ(λ−2))J0(a∗,b∗)

+ 1

Z 2 (n(n +1)+ 2anλ

θ0+ a2λ2

θ20

)− 2

Z(n + aλ

θ0)+1} (14)

4.2. Risk function of (θ̂BS2) under SELF

The Risk function of the estimator θ̂BS2 under SELF is given as follows

R(θ̂BS2/SELF ) =E(θ̂BS2 −θ)2

=∫

R(k2(θ̂B −θ0)− (θ−θ0))2 f (θ̂B )d θ̂B +

∫R̄

(θ̂B −θ)2 f (θ̂B )d θ̂B

=∫

R[(

T − r1

r2)2(θ̂B −θ0)2 −2(

T − r1

r2)(θ−θ0)(θ̂B −θ0)

+ (θ−θ0)2] f (θ̂B )d θ̂B +∫

R̄[(θ̂B )2 −2θθ̂B +θ2] f (θ̂B )d θ̂B

by using T = zθ̂B +a

R(θ̂BS2/SELF ) = 1

r 22

[∫

R(zθ̂B +a)2((θ̂B )2 −2θ0θ̂B +θ2

0) f (θ̂B )d θ̂B

−2r1

∫R

(zθ̂B +a)((θ̂B )2 −2θ0θ̂B +θ20) f (θ̂B )d θ̂B

+ r 21

∫R

((θ̂B )2 −2θ0θ̂B +θ20) f (θ̂B ) d θ̂B ]

− 2(θ−θ0)

r2[∫

R((zθ̂B −a)(θ̂B −θ0) f (θ̂B )d θ̂B

− r1

∫Rθ̂B f (θ̂B )d θ̂B + r1θ0) f (θ̂B )d θ̂B ]

+∫

R(θ−θ0)2 f (θ̂B )d θ̂B +

∫R̄

[(θ̂B )2 −2θθ̂B +θ2] f (θ̂B )d θ̂B


In same way used to drive the risk function of the first estimator θ̂BS1 under SELF we get

R(θ̂BS2/LSEF ) =θ2[1

r 22 z2

(θ2

λ2 J3(a∗,b∗)+4aθ0

λJ3(a∗,b∗)+6a2 J2(a∗,b∗)+ 4a3λ

θ0J1(a∗,b∗)

+ a4λ2

θ20

J0(a∗,b∗))− 2

r 22 z2

(θ0z +a + r1)(θ0

λJ3(a∗,b∗)+3a J2(a∗,b∗)

+ 3a2λ

θ0J1(a∗,b∗)+ a3λ2

θ20

J0(a∗,b∗))+ 1

r 22 z2

(z2θ20 +4zaθ0 +a2 +4zr1θ0

+ r1(r1 +2a)−2zr2(θ0

λ−θ0)− r 2

2 )(J2(a∗,b∗)+ 2aλ

θ0J1(a∗,b∗)+ a2λ2

θ20

J0(a∗,b∗)

− 2

zr 22

(zaθ0λ+a2λ+ zr1θ0λ+λr1(r1 +2a)

− r2(1−λ)(zθ0 +a)− r2r1(1−λ)− r 22 )(J1(a∗,b∗)+ aλ

θ0J0(a∗,b∗))

+ 1

r 22

(a2λ2 + r1(r1 +2a)λ2 −2ar2λ(1−λ)

−2r1r2λ(1−λ)−1)+ (1−λ)2)J0(a∗,b∗)+1

+ 1

z2 (n(n +1)+ 2anλ

θ0+ a2λ2

θ20

)− 2

z(n + aλ

θ0)] (15)

4.3. Risk function of (θ̂BS1) under LLF

In many real-life situations, overestimate or underestimate more seriously like reliability estimation the over-estimation more seriously than underestimation in these cases, the asymmetry loss more useful than symmetrysquared error loss function,the LINEX loss function is an asymmetric loss function, which was introduced by Klebanov(1972) and used by Varian (1975) in the context of real estate assessment. Zellner (1986) used it for the estimation ofa scalar parameter and prediction of a scalar random variable. This function is rising almost twice on one side fromzero and almost linear to the other side.It is define as

L(4) = b(expa4−a 4−1)

where 4= θ̂−θ’ b’ is the scale parameter and ’a’ is the shape parameter. The sign and value of ’a’ respectively represent the directionand degree of asymmetry. The positive value of an is used when overestimation is more serious than underestimationand a negative value is used for the other case . [Shanubhogue, Ashok and Jiheel(2013)] .


Now; we will find risk function of θ̂BS1 under LINEX loss function.

R(θ̂BS1/LLF ) =E(L(θ̂BS1/L4))

=E [b(expa1(θ̂BS1−θ)−a1(θ̂BS1 −θ)−1)]

=b∫

R(expa1(k1θ̂B+(1−k1)θ0−θ) f (θ̂B )d θ̂B

−a1

∫R

(k1θ̂B + (1−k1)θ0 −θ)−1) f (θ̂B )d θ̂B

+∫

R̄(expa1(θ̂B−θ)−a1(θ̂B −θ)−1) f (θ̂B )d θ̂B

=b[z expa1((1−k1)θ0−θ)

Γ(n)θn

∫R

expa1(k1θ̂B )(zθ̂B −a)n−1 exp−

zθ̂B −a

θ d θ̂B

− za1k1

Γ(n)θn

∫Rθ̂B (zθ̂B −a)n−1 exp

−zθ̂B −a

θ d θ̂B

− z(a1((1−k)θ0 −θ)+1)

Γ(n)θn

∫R

(zθ̂B −a)n−1 exp−

zθ̂B −a

θ d θ̂B

+ z exp−a1θ

Γ(n)θn

∫R̄

expa1θ̂B (zθ̂B −a)n−1 exp−

zθ̂B −a

θ d θ̂B

− za1

Γ(n)θn

∫R̄θ̂B (zθ̂B −a)n−1 exp

−zθ̂B −a

θ d θ̂B

+ z(a1θ−1)

Γ(n)θn

∫R̄

(zθ̂B −a)n−1 exp−

zθ̂B −a

θ d θ̂B ]

By using the transformation y = zθ̂B −a

θand x = y(1− a1θ

z )

assume x1 = y(1− a1kθz ) =⇒ x = y(

z −a1kθ

z) =⇒ y = x(

z

z −a1kθ)

d y = (z

z −a1kθ)d x and θ = θ0

λ

and by (13) at m=0,1,5,6

R(θ̂BS1/LLF ) =b expa1((1−k1)θ0− θ0λ+ k1 a

z )(zλ

zλ−a1k1θ0)n j6(a∗∗∗,b∗∗∗)

− a1k1

z(θ0

λj1((a∗,b∗)+a j0(a∗,b∗))

− (a1((1−k1)θ0 − θ0

λ)+1) j0(a∗,b∗)

+exp−a1(θ0λ− a

z )(zλ

zλ−a1θ0)n(1− j5(a∗∗,b∗∗))

− a1

z(θ0n

λ+a)+ a1

z(θ0

λj1(a∗,b∗)

+a j0(a∗,b∗))+ (a1θ0

λ−1)(1− j0(a∗,b∗)) (16)

where

1− a∗ = λ

2χ2

α2 ,2n

b∗ = λ

2χ2

1− α2 ,2n

2− a∗∗ = 1

2χ2

α2 ,2n

(λ− a1θ0

z) b∗∗ = 1

2χ2

1− α2 ,2n

(λ− a1θ0

z))

3− a∗∗∗ = 1

2χ2

α2 ,2n

(λ− a1θ0k

z) b∗∗∗ = 1

2χ2

1− α2 ,2n

(λ− a1θ0k

z))

5. Relative Risk

Now; we need to derive the risk function of Beyas estimator θ̂B under SELF and LLF to get relative risk.Firstly; we drive the risk function of θ̂B under SELF .


to find the risk function of θ̂B under squared error loss function as follows

E(θ̂B ) =∫ ∞

a/zθ̂B f (θ̂B )d θ̂B =

∫ ∞

a/zθ̂B

z(zθ̂B −a)n−1 exp− zθ̂B −aθ

Γ(n)θn d θ̂B

By using transformation y = zθ̂B −a

θwe get

E(θ̂B ) = θn +a

z(17)

By using the same transformation and previous assumptions used in E(θ̂B ) we get

E(θ̂B )2 =∫ ∞

0θ̂2

B f (θ̂B )d θ̂B =∫ ∞

0θ̂2

Bz(zθ̂B −a)n−1 exp− zθ̂B −a

θ

Γ(n)θn d θ̂B

we get

E(θ̂B )2 = θ2n(n +1)+2θan +a2

z2 (18)

by using (18) and (19)then

R(θ̂B /SELF ) = θ2[n(n +1)

z2 + 2anλ

z2θ0+ a2λ2

z2θ20

− 2n

z− 2aλ

zθ0+1] (19)

Secondly; the risk function of θ̂B under LLF .Now find risk function of the estimator θ̂B under LINEX loss function as follows

L(θ̂B /4) =b(expa1(θ̂B−θ)−a1(θ̂B −θ)−1)

Then

R(θ̂B /LLF ) =E [b(expa1(θ̂B−θ)−a1(θ̂B −θ)−1)]

=∫ ∞

0[b(expa1(θ̂B−θ)−a1(θ̂B −θ)−1)] f (θ̂B ) d θ̂B

b[z exp−a1θ

Γ(n)θn

∫ ∞

0expa1θ̂B (zθ̂B −a)n−1 exp

−zθ̂B −a

θ d θ̂B d θ̂B

− a1z

Γ(n)θn

∫ ∞

0θ̂B (zθ̂B −a)n−1 exp

−zθ̂B −a

θ d θ̂B +aθ−1]

By transformation y = zθ̂B −a

θand x = y(1− a1θ

z )

R(θ̂B /LLF ) =b[exp−a1(θ− a

z )

Γ(n)(

z

z −a1θ)n−1

∫ ∞

0(x)n−1 exp−x (

z

z −a1θ)d x

− a1

zΓ(n)

∫ ∞

0(ynθ+ayn−1)exp−y d y +aθ−1]

=b[exp−a1(θ− a

z )

Γ(n)(

z

z −a1θ)n

∫ ∞

0(x)n−1 exp−x d x

− a1

zΓ(n)

∫ ∞

0(ynθ+ayn−1)exp−y d y +aθ−1]

=b[ (z

z −a1θ)n exp−a1(θ− a

z )−a1

z(nθ+a)+a1θ−1]

then

R(θ̂B /LEF ) = b[(zλ

zλ−a1θ0)n exp−a1(

θ0λ− a

z )−a1

z(

nθ0

λ+a)+ a1θ0

λ−1] (20)

Therefore; we can extract Relative Risks of proposed pre-test shrinkage estimator θ̂BS1 , θ̂BS2 with respect toclassical estimator θ̂B denoted by R1.R(θ̂BS1/SELF ), R2.R(θ̂BS2/SELF ) respectively under square error loss function


and θ̂BS1 under linex loss function denoted by R3.R(θ̂BS1/LLF ) as follows.

R1.R(θ̂BS1/SELF ) = R(θ̂B /SELF )

R(θ̂BS1/SELF )(21)

R2.R(θ̂BS2/SELF ) = R(θ̂B /SELF )

R(θ̂BS2/SELF )(22)

R3.R(θ̂BS1/LLF ) = R(θ̂B /LLF )

R(θ̂BS1/LLF )(23)

We notice that equations relative risk to estimators θ̂BS1 and θ̂BS2 under loss squared error function and θ̂BS1 underlinex loss function It contains constants .To study these equations numerically, we assumed the following values forthese constants .n=10,15,20 λ= 0.4,0.8,1,1.2,1.6 α= 0.01,0.02 k1 = 0.1,0.2c = 1,2,3 a = 0.5,1,2 a1 =−1,0.5,1,2 θ0 = 1

6. Numerical Results

i. The proposed estimators θ̂BS1 and θ̂BS2 given high relative risk under squared error loss function and θ̂BS1 underlinex loss function with respect to classical estimator θ̂B whenλ equal to one i.e the true value equal to the initialvalue .

ii. The relative risk of the estimators θ̂BS1 and θ̂BS2 under squared error loss function and the estimator θ̂BS1 underLinex loss function is increasing with ’a’ , see figures (1),(5) and (6) .

iii. The relative risk of the estimators θ̂BS1 and θ̂BS2 under squared error loss function and the estimator θ̂BS1 underLinex loss function at a1 =−1 it is decreasing with ’n’ , as shown in figures (3) and (7). While the relative risk ofthe estimator θ̂BS1 under Linex loss function at a1 = 0.5,1,2 it is increasing with ’n’, see figures (9) , (11) and (13).

iv. The relative risk of the estimators θ̂BS1 and θ̂BS2 under squared error loss function and the estimator θ̂BS1 underLinex loss function at a1 = −1,0.5,1 it has a high value when the value of c=1 , see figures (2) ,(8),(10) and (12),but, The relative risk of the estimator θ̂BS1 under Linex loss function at a1 = 2 it has a high value when the valueof c=3 as shown in figure (15) .

v. The relative risk of the estimator θ̂BS1 under Linex loss function is decreasing with ’a1’ . i.e it has a high value whenthe value of a1 =−1 at 0.82 <λ< 1.15 as shown in figure (14) .

vi. The relative risk of the first estimator θ̂BS1 is better than the second estimator θ̂BS2 under squared error loss func-tion , for all the given values and constants as shown in figure (4) .

vii. The relative risk of the proposed estimator θ̂BS1 under squared error loss function better than θ̂BS1 under Linexloss function at 0.8 < λ < 1.15 ,for all a1 except for at a1 = −1 is θ̂BS1 under Linex loss function better than theestimator θ̂BS1 under squared error loss function as shown in figure (16) .

vi. The relative risk of The first estimator θ̂BS1 under Linex loss function for all a1 better than θ̂BS2 uunder square errorloss function at 0.8 <λ< 1.15 see figure (17) .


0

2

4

6

8

10

12

14

16

0.4 0.8 1 1.2 1.6λ

R1, a=0.5

R1, a=1

R1, a=2

R2, a=0.5

R2, a=1

R2, a=2

Fig. 1. R1.R(θ̂BS1/SELF ) and k=0.1 α=0.01 c=1 n=10

0

2

4

6

8

10

12

0.4 0.8 1 1.2 1.6λ

R1, c=1

R1, c=2

R1, c=3

R2, c=1

R2, c=2

R2, c=3

Fig. 2. R1.R(θ̂BS1/SELF ) and k=0.1 α=0.01 a=0.5 c=1


0

2

4

6

8

10

12

0.4 0.8 1 1.2 1.6λ

R1, n=10

R1, n=15

R1, n=20

R2, n=10

R2, n=15

R2, n=20

Fig. 3. R1.R(θ̂BS1/SELF ) and k=0.1 α=0.01 a=0.5 n=10

0

2

4

6

8

10

12

0.4 0.8 1 1.2 1.6

λ

R1, k=0.1 α=0.01

R1, k=0.1 α=0.02

R1, k=0.2 α=0.01

R1, k=0.2 α=0.02

R2, k1 α=0.01

R2, k=2, α=0.02



0

2

4

6

8

10

12

14

16

18

0.4 0.8 1 1.2 1.6λ

a1=-1 , a=0.5

a1=-1 , a=1

a1=-1 , a=2

a1=0.5 , a=0.5

a1=0.5 , a=1

a1=0.5 , a=2


0

2

4

6

8

10

12

0.4 0.8 1 1.2 1.6λ

a1=1 , a=0.5

a1=1 , a=1

a1=1 , a=2

a1=2 , a=0.5

a1=2 , a=1

a1=2 , a=2



0

2

4

6

8

10

12

14

0.4 0.8 1 1.2 1.6

λ

n=10

n=15

n=20


0

2

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10

12

14

0.4 0.8 1 1.2 1.6

λ

c=1

c=2

c=3



0

2

4

6

8

10

12

0.4 0.8 1 1.2 1.6

λ

n=10

n=15

n=20


0

2

4

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8

10

12

0.4 0.8 1 1.2 1.6

λ

c=1

c=2

c=3



0

2

4

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10

12

0.4 0.8 1 1.2 1.6

λ

n=10

n=15

n=20


0

1

2

3

4

5

6

7

8

9

10

0.4 0.8 1 1.2 1.6

λ

c=1

c=2

c=3



0

1

2

3

4

5

6

7

8

9

0.4 0.8 1 1.2 1.6

λ

n=10

n=15

n=20


0

2

4

6

8

10

12

14

0.4 0.8 1 1.2 1.6

λ

a1=-1

a1=0.5

a1=1

a1=2



0

1

2

3

4

5

6

7

0.4 0.8 1 1.2 1.6

λ

c=1

c=2

c=3


0

2

4

6

8

10

12

14

0.4 0.8 1 1.2 1.6

λ

R1

R3 a1=-1

R3 a1=0.5

R3 a1=1

R3 a1=2



0

2

4

6

8

10

12

14

0.4 0.8 1 1.2 1.6

λ

R2

R3 a1=-1

R3 a1=0.5

R3 a1=1

R3 a1=2



Acknowledgements

The author(s) would like to thank some institutions for support and so on.

References

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[9] Ferreira, Johan T. ,Objective Bayesian estimators for the censored Rayleigh model using different loss functions,Suid-Afrikaans Tydskrif vir Natuurwetenskap en Tegnologie/South African Journal of Science and Technology33(1) (2014)2-bladsye.

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