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7
BAYESIAN AND E-BAYESIAN ESTIMATION OF EXPONENTIATED INVERSE RAYLEIGH DISTRIBUTION
USING CONJUGATE PRIOR
RAMESH KUMAR1, HEMANI SHARMA*2, RAHUL GUPTA2, ABLEEN KAUR3
Department of statistics, University of Jammu, Jammu, J&K, India 180006 rk1825308@gmail.com1, hemanisharma124@gmail.com*2, rahulgupta68@gmail.com2
,ableenkaur23@gmail.com3
Abstract
This study explores the application of Bayesian and E-Bayesian techniques to estimate the scale parameter of the Exponentiated Inverse Rayleigh distribution. Bayesian estimates for the parameter are derived using an informative Gamma prior and evaluated under three distinct loss functions: De-Groot, Squared Error, and Al-Bayyati loss functions. Various Properties of the E-Bayesian estimators under different loss functions have also been studied. To compare the effectiveness of E-Bayesian estimates against the Bayesian counterpart, a simulation study is conducted using MatLab. The various derived estimators were compared in terms of their Mean Squared Error. The results of a simulation study reveal that E-Bayesian estimates exhibit a smaller Mean Squared Error in comparison to Bayesian estimates, thereby demonstrating their enhanced efficiency. Among the E-Bayesian estimates, the third one stands out as the most effective. Moreover, the analysis highlights that the Squared Error loss function outperforms the Al-Bayyati and De-Groot loss functions, exhibiting a smaller MSE. Furthermore, the efficacy of these estimators is demonstrated through an analysis of a real-life dataset.
Keywords: Al-Bayyati loss function, De-Groot loss function, Exponentiated inverse Rayleigh distribution, Gamma prior, Squared error loss function.
1. Introduction
The Exponentiated Inverse Rayleigh distribution (EIRD) finds extensive utility in life testing and reliability studies, playing a crucial role in domains like electronic component longevity and wind speed analysis. Its significance also extends to physics and signal processing, facilitating investigations into radiations, sounds, and light phenomena. This versatility prompts statisticians to frequently employ the EIRD across diverse datasets.
Rehman and Dar [1] conducted a comprehensive examination of the Exponentiated Inverse Rayleigh distribution, delving into its mathematical properties and harnessing Bayesian estimation techniques for parameter estimation. The probability density function (PDF) and cumulative distribution function (CDF) of the EIRD, characterized by scale parameter 9 and shape parameter a, are as follows:
2a0e x2
/(x,0,a) =--- ;x>0,a ,>0 (1)
F(x,0,a) = e*2 ;x>0,a ,0 > 0 (2)
ae
R. KUMAR, H. SHARMA, A. KAUR, R. GUPTA RT&A, No 3 (79) BAYESIAN AND E-BAYESIAN ESTIMATION..._Volume 19, September 2024
Numerous authors have explored the Inverse Rayleigh distribution (IRD) from various angles. Voda
[2] delved into essential properties such as Maximum Likelihood Estimation (MLE), confidence
intervals, and hypothesis tests. Siddiqui [3] focused on the diverse practical applications of the
Inverse Rayleigh Distribution. Soliman et al. [4] utilized squared error and zero-one loss functions
to devise Bayesian estimators for IRD, centered around lower record values. Reshi et al. [5] tackled
parameter estimation for the Generalized Inverse Rayleigh distribution.
Dey [6] derived Bayes estimators for IRD parameters using distinct loss functions and a noninformative prior. Sindhua et al. [7] explored Bayesian estimators and associated risks for IRD parameters, emphasizing left-censored data and showcasing the efficacy of the gamma prior under Quasi-Quadratic loss functions. Okasha [8] explored E-Bayesian estimation for the Lomax distribution with type-II censored data.
This paper's objective is to conduct a statistical comparison between Bayesian estimators and Expected Bayesian estimators for the Exponentiated Inverse Rayleigh distribution's scale parameter. The analysis involves the utilization of gamma priors and different loss functions. The ensuing layout of the paper is outlined as follows: Section 2 outlines the derivation of the likelihood function, prior distribution, and posterior distribution. Section 3 presents Bayesian estimators for the EIRD scale parameter using Al-Bayyati, Squared Error, and De-Groot loss functions. In Section 4, E-Bayesian estimates are derived and their properties are examined. Section 5 is dedicated to a simulation study comparing Bayes and E-Bayes estimates. Real data analysis is tackled in Section 6, while Section 7 concludes by summarizing the findings.
2. Likelihood function, Prior and Posterior Distribution
2.1 Likelihood function
Let x — x-±, %2,..., xn be a random sample of size n drawn from EIRD. Then the likelihood function is given by
n
L(x,e) — — e-a9^=ixi2 (3)
i=i 1
In the context of Bayesian estimation, the selection of an appropriate prior holds paramount importance in parameter estimation. When a substantial understanding of the parameter(s) is available, the inclination is towards informative priors; however, when such knowledge is lacking, non-informative priors may be more appropriate. In this study, we opt for an informative prior, specifically the Gamma Prior, to derive the corresponding posterior distribution.
2.2 Prior distribution
The gamma distribution is employed as a conjugate prior distribution for the parameter 9. The subsequent Probability Density Function (PDF) is formulated using the shape parameter 'c' and the scale parameter 'r'.
r° dc-1e-er
h(e\c,r)—-—- ; c,r > 0 ,e>0 (4)
2.3 Posterior distribution
The posterior distribution for the parameter 6 using (3) and (4), is given as
L(x,e)*h(6) 9(i,e) — f™L(x,8)*h(8)d8
R. KUMAR, H. SHARMA, A. KAUR, R. GUPTA RT&A, No 3 (79) BAYESIAN AND E-BAYESIAN ESTIMATION._Volume 19, September 2024
2nan0nnn=iJ_e-«0zi=1*r a e-
__^_
As a result, the posterior distribution of is equal to
e(n+c)-1e-e{a2i=ixr2 + r}
= J0Me(n+c)-1e-e{a2F=iXr2+r}de ^n+cg(n+c)-1e-se ( „ .
g(x,0) =- + -, 0 > 0 where 5 = {a^ x-2 + rj (5)
3. Bayesian Estimation
In this section, we find the Bayes estimate of scale parameter of EIRD under three different loss functions as:
3.1 Under the Al-Bayyati loss function
Al-Bayyati [9] proposed a loss function, defined as ¿(0 ,0) = 0d(0 - 0)2; where 0 is the estimate of 0. By using the Al-Bayyati loss function, the bayes estimator is given as
£{¿(0 ,0)} = J ¿(0 ,0)*^(x,0)d0
o
q
= J 0d(0 — 0)
o
^n+c^ (n+c)-1g-
dtQ — a\2 *_ ¿Q
r(n + c)
^32 j. ^2 _ oaa\2 _
= J 0d(02 + 02 — 200)2
r(n + c)
o
^n+cg(n+c+d)-1g-se ^n+cg(n+c+d+2)-1g-se ^n+e^(n+e+d + 1)-1g-S6
-w-;-+ I -w-;-d0 — 20 I -—---d0
r(n + c) J r(n + c) J r(n + c)
By solving the above integral, finally we get
, ^ _ T(n + c + d) T(n + c + d + 2) _ T(n + c + d + 1)
5dr(n + c) 5d+2r(n + c) 5d+1r(n + c)
And consequently, the Bayes estimator is
0B, = ^ (6)
3.2 Under the Squared error loss function
The Squared Error Loss Function [10] is defined as follows: ¿(0 ,0) = (0 — 0)2
By using the Squared error loss function, the bayes estimator is given as
£■{¿(0 ,®)} = J ¿(^ ,0)*^(x,0)d0
o
q
r ^n+c^ (n+c)-1(
= I (0 — 0)2 *-—--
J y 1 T(n + c)
d0
QO
o
o
o
o
= j(e2 + e2- 2B9)2 *
$n+cQ (n+c)-1q-S9
dS
r(n + c)
By solving the above integral, finally we get And consequently, the Bayes estimator as
SBS = ^ (7)
3. 3 Under the De-Groot loss function
The De-Groot loss function [11] is defined as follows: _ , (8 — 8)2
By using the De-Groot loss function, the bayes estimator is given as
m
E{l(§ >6)} = f L(8 ,e)*g(x,e)de
)
(8 — 8)2 Sn+cd(n+c)-1e-se
0
m
= I
d8
0
m
r(n + c)
£n+CQ(n + C)-1g-S9
r(n + c)
= 1I (®2 + 02 - 2®6)2 * --, ^-de
0
By solving the above integral, finally we get
r[](n , mn + C + V 2r(n + C + D
t{L(tl, )} + §2 S2r(n + c) 8 ST(n + c) And consequently, the Bayes estimator as
eBD - (8)
4. E-Bayesian Estimation
According to Han [12], the prior parameters 'c' and 'r' should be chosen so that the prior given in (4) is a decreasing function of 8.
d rc „ n
— h(8\c,r) -—8c-2e-er{(c — 1)—r8}, dd rc
As a result, our prior distribution (4) becomes a decreasing function of 8, for 0<c<1 and r>0. The E-Bayesian estimate of is calculated as follows:
Qeb = I I Qbe * n(6, c, r) * drdc
00
The intervals of integration for the first and second integrals correspond to the domains of the hyperparameters 'c' and 'r,' respectively, ensuring that our prior density function exhibits a decreasing trend with respect to 8. The Bayesian estimate of 8, denoted as 8EB, is calculated utilizing three distinct loss functions.
Subsequently, for the E-Bayesian estimates of 8, we deliberate on the choice of prior distributions for the hyperparameters 'c' and 'r.' These distributions serve primarily to explore the influence of different prior choices on the E-Bayesian estimations of 8. The hyperparameters 'c' and 'r' are governed by the following distributions 2(t — r)
ni(8,c,r) = ; 0<c<l; 0<r<t (9)
0
*
2
rc2(0,c,r)=- ; 0 < c < 1; 0<r<t (10)
2r
n:3(0,c,r)= — ; 0 < c < 1; 0<r<t (11)
t2
Now follows the E- Bayesian estimates of the scale parameter of EIRD under proposed loss functions 4.1 E-Bayesian estimation of 6 under the Al-Bayyati loss function
E-Bayesian estimate of the parameter 0 based on % (0, c, r), is provided by
i t
®eba1 = J J 0B,4 * %(0,c,r)drdc
0 0
=?{i(n+e+d)dc*f(tij:)*dr }
On solving the above equation, we get
2n + 2d + 1 r /t+P\ )
^ =-p-* {(t + P) * log (—) - f} (12)
E-Bayesian estimate of the parameter 0 based on rc2(0, c, r), is provided by
i t
^EBA-t = J J V* ^2 (0,c, r)drdc
00
+ c + ^ 1 -} * — * drdc
o o 5 } t
2n+2d+1
where P = (a ^f=1 Xj 2) and J (n + c + d)dc =
Hence, on solving we get
2n + 2d + 1 r /t+P\")
-2F—*M-T-)j (13)
E-Bayesian estimate of the parameter 0 based on tc3(0, c, r), and is given by
1 t
¿W43 = J J ^3 (0,c, r)drdc
oo
f1 f'rn + c + d} 2r
= J U^^* t2*drdC
Hence on solving, we get
(2n + 2d + 1) r ( P \ -)
^ = --p-^ * {P * log (t+P) + tj (14)
4.2 E-Bayesian estimation of 6 under the Squared error loss function
E-Bayesian estimate of the parameter 0 under based on %(0, c,r), and is provided by
1 t
= J J ¿'ss * %(0,c,r)drdc
oo 1 t
= JJ(^l+£)* ^r)j*drdC
oo
= ||Jo(n + c)dc*Jo(i^:+r)*dr ^ Where ^ = xi-2 + rj P = xt-2
On solving the above integrals, finally we get 2n + l ( ft + P
{(t+P),log(i+-)-tj (15)
E-Bayesian estimate of the parameter в based on п2(в, c, r), and is provided by
i t
Qebs2 = J J eBs* n2(e,c,r)drdc
0 0
^Ш* rdrdc
1
= l{j0(n + c)dc*J0Q*dr
where S = x-2 + rj and P = x-2
On solving the above integrals, finally we get
W^MTT)} (16)
E-Bayesian estimate of the parameter в based on п3(в, c, r), and is provided by
i t
@ebs3 = J J ®BS* пз (Q,c, r)drdc
00
Г1 Г1т + с) 2r
= Jo АПГГ F*drdC
on solving the above intervals, finally we get f2n + 1\ r f p
JEBS3
H^VM^gt+p^ }t (17)
4.3 E-Bayesian estimation of в under the De-Groot loss function
E-Bayesian estimate of the parameter в based on n1 (в, c, r), and is provided by
i t
Sebd1 = J J 6bd* ni (d,c, r)drdc
fn + c + 1\ r2(t-r)ï
00 i t
00
--JJm* FP-j-dc
\P + rJ
Where S = x-2 + rj and P = x-2
On solving the above integrals, we get
6EBDl=^*[(t+P)*logffl-t} (18)
E-Bayesian estimate of the parameter в based on л2(в, c, r), and is provided by
i t
ê>ebd2 = J J ôbd* n2(d,c,r)drdc
00
= /o1/ot[^+Ti}* T*drdc = l{J (n + c)dc* J (i)*dr
,o KSJ
= \{J\n + c + 1)dc*fo{1±;)*dr
= {a ^ ^ Xj 2 + r} and P = a ^ ^ Xj 2
*il°g("+r~)l (19)
Where S
¿=1
â (2n + 3) r /t+P 6bbd2 = 2t * {l°g (_F
E-Bayesian estimate of the parameter в based on я3(б, c, r), and is provided by
бяВСз = /o1/0t^BD * ^3(ö,c,r)drdc
+ с+Л 2r -} * —
S J t2
с 1 rt ("п+с+Л 2r , ,
= Jo Jol-S-j* ^*drdc
4{/01(п+с+Шс*Г(^тНг
= tL{^*{t-P*i°g(t+f) j} On solving the above intervals, finally we get
0ВВО3 = F^ * {P * l°g (t+Pp) +t jj (20)
4.4 Properties of E-Bayesian estimates under Different Loss Functions
In this section, we will discuss the relationship amongst the different E-Bayesian estimators obtained under the Al-Bayyati loss function i.e, бВВЛ1,бввл2,0ввл3 0 = 1,2,3)
Theorem 4.1 E-Bayesian estimators obtained under the Al-Bayyati loss function will follow the following results:
(i) < бввл2 < ^BB^
(ii) lim (0BBj4l) = lim (0ВВл2 ) = lim (0ВВЛз )
Proof (i) From (12) and (13), we get
Öbb.1 - ^ = * {(t + P) * l°g (+P) - t j - ^ * {l°g (+P)j
(2n + 2d + 1) ( ( twP h 2
= (-i-W + PXÏ + ÏH (21)
From (13) and (14), we get
Ö.B.2 - ÖBB.3 = ^^^ * {l°g (^j - ^^^ *{t-P*l°g (+P)j
(2n + 2d + 1) f / twP 1\
2d +1) ( ( t\ (P 1\ -)
-t—{l°g(1 + P)(T + ü)-1} (22)
Since l°g(1 + â = {Î-2& + 3t33--^j
p) (.p
_ (2n + 2d + 1)[ t2 t3
t (12P2 6P3
0ВВЛ! - бввл2 > 0
hence
^BB^ > (23)
Similarly,
~ ~ _ (2n + 2d + 1) r t2 t3
— = t (12P2 + 6P3 '
<9bb^2 — <9bb^3 > 0, and hence
#BB,42 > #BB,43 (24)
Combining (23) and (24), we get
$BB,43 < 0£B,42 < ^BB^
Proof (ii): From (21) and (22), we get
(2n + 2d + 1) ( ( t\ (P 1\ -)
— = ^¿2 — ^3--1- {log (1 + p)(7 + 2) — 1j
After taking the limit, we get
~ . (2n + 2d + 1)( t2 t3 lim( dEBAl — eEBA2) = lim---+
1 2J P^m t 112P2 6P3
~ , (2n + 2d + 1)( t2 t3 lim (9EBa2 — QebaJ = lim---{ —2 + ■
2 3' P^m t [12P2 6P3
On solving the above , we have
¡lm (®EBA1 - ®EBA2) = Jim {£>EBA2 - t>EBA3) = 0 P^m 1 2J P^m 2 3J
Hence
lim ( BesaJ - lim (Geba2) - lim (Seba3)
Now we will discuss the relationship amongst the different E-Bayesian estimators obtained under the Square Error loss function i.e, (¡ebs1,^^ebs2,@ebs3 (i - 1,2,3)
Theorem 4.2 E-Bayesian estimators obtained under the Squared Error loss function will follow the following results:
(i) 9ebs3 < @EBS2 < @EBS1
(ii) lim (dEBS1) - lim (9EBs2 ) - lim (9EBS3 )
Proof (i): From (15) and (16), we get
2n + 1 ( ) 2n + 1 ( ¡t + P\-) 9ebs1 - 9ebs2 *((t+p)* log (—) -tj--— * (log (-)j
(2n + 1) ^ ft+P\/P
2)
— Vebs2 t^*((L+p) *log^-^-) — <-j- -(2n + 1) ( /t+P\/P 1\ -) ^ — ^ - —* (l°g {-1T) {-t+1) — 1\
Qebs, - Qebs2 > 0
(25)
j1 ~CBJ2
S2
0ebs1 > Qebs2 (26)
Similarly, from (16) and (17), we get
(2n + 1) ( ft+P\) (2n + 1) ( ft+P\
( ft+P\) (2n + 1) ( it+P\)
1EBS2—VEBS3 ---* 1log(—5—)j--72- ""
2t P 2 P
(2n + 1)( / t\ (P 1\ 1
e,B52-^B53 ^^-+^(log(1 + p)(T + 1)-1j (27)
9ebs2 - 9ebs3 > 0
¿2
@EBS2 > @EBS3 (28)
Combining (26) and (28), we get
@EBS3 < @EBS2 < @EBS1
Proof (ii): From (25) and (27), we get
(2n + 1)( / t\/P1\ -) . eEBS1 - 0ebs2 - —— (tog (1 + p)(t + 2)-1j- QEBS2 - e.
After taking the limit, we get
~ - x (2n + 1)( t2 t3 )
Pimm(e^1- - Pimm-^-^
and
Hz^ - 6^3) = pm-2T1{lS2+:P;—■■■■-}- 0
On solving, we have
Jim (Qebsi - Qebsj - lim (?ebs2 - Qebs3) - 0
P^m 1 2J P^m 2 3J
hence
lim (Bses^ - \im(9EBS2) - \im(dEBS1)
Now we will discuss the relationship amongst the different E-Bayesian estimators obtained under the De-Groot loss function i.e, 8EBSi, 9EBS2,9EBS3 (i - 1,2,3)
Theorem 4.3 E-Bayesian estimators obtained under the Square Error loss function will follow the following results: (S)@ebd3 < @ebd2 < @ebd1
(ii) \im ((>EBD1) - lim (dEBD2) - \im (eEBD3)
EBS3
Proof (i): From (18) and (19), we get
^ - ^ ^{(t+P)* log K (t)}
9
(2n + 3W /t+PwP 1\ 1
#BBD2 > 0
#BBD1 > #BBD2 (30)
Similarly, from eq. (19) and (20), we get
(2n + 3) r ft + P\) (2n + 3W A + P\
(2n + 3W /t+P\l (2n + 3W /t+P\-)
0BB.2 - ^ = * H(—)} - *{t-P*log (—)}
(2n + 3) r ( t\ (P 1\ -)
^ - ^ = (1 + p)(T + 2)- 1} (31)
$bbd2 $bbd3 > 0
#BBD2 > ^BBD3 (32)
Combining (30) and (32), we get
0£BD3 < 0£BD2 < $£BD1
Proof (ii): From (29) and (31), we get
(2n + 3) r / t\ (P 1\ -) ^ - 0bbd2 = t {log (1 + p)(7 + 2)-!} = -
After taking the limit , we get
^ ~ , (2n + 3)( t2 t3 )
Hmfa^i- = —t— {T2P2 + 6P3 — -j = 0 And
- \ (2n + 3)( t2 t3
Hm^ - ^ = lim —+ 6P3---j = 0
On solving, we have
Hm(0BBDi - 0BBDJ = Hm(0BBD2 - 0BBDJ = 0 limi^i) = lim(^BBD2) = lim^s^
Hence the proof is complete.
The part (a) of theorem 4.(i), 4.(ii) and 4.(iii) shows that with different priors (9)-(11) of the parameters c and r, the associated E-Bayesian estimate , 0BBD;, and 0BBSi; (i=1,2,3) are different. The property (b) of the theorems shows that , c; (i=1,2,3) are asymptotically equivalent to each other as ^f=1x-2 i to, that means (i=1,2,3) are all close to each other when ^f=1x-2 is
sufficiently large and 0BBD;, and 0BBSi; (i = 1,2,3) are also close to each other.
5. Simulation Study
In order to compare the performance of Bayesian and E-Bayesian techniques of estimation, a simulation study was conducted using MatLab. We chose a sample of size of n=20, 50, 70, 100, 120 to represent small, medium and large data set. The following steps were conducted:
1. The shape (a) and scale (0) parameters has been fixed at 0.5 and 0.25 respectively.
2. For given value of t, we generate c and r from uniform and gamma distribution respectively.
3. For given value of a and B, we generate a random sample of different sizes from Exponentiated inverse Rayleigh distribution (EIRD) using the quantile function.
4. The above steps are iterated 1000 times to find the MSE of Bayesian and E-Bayesian estimates of scale parameter using different loss functions.
5. The MSE for Bayesian and E-Bayesian estimates under different loss function are shown in table 1.
6. The MSE of 8 for Bayesian and E-Bayesian estimation under different loss functions are also illustrated in Figure 1, 2 and 3.
Table 1: Mean Squared Error (MSE) of 8 under different loss functions for a = 0.25,8 = 0.5 , t = 1.5, d = 3.
n ^EB^ ^BS ^EBSi ®EBS2 eEBS3 ^BD ^EBDi ®ebd2 ®EBD3
20 0.1357 0.1315 0.1304 0.1294 0.1029 0.0994 0.0987 0.0979 0.1134 0.1096 0.1087 0.1079
50 0.0958 0.0948 0.0945 0.0942 0.0854 0.0844 0.0842 0.0839 0.0888 0.0878 0.0876 0.0873
70 0.0735 0.0727 0.0725 0.0724 0.0677 0.0668 0.0667 0.0666 0.0696 0.0688 0.0686 0.0685
100 0.0604 0.0599 0.0599 0.0598 0.0569 0.0565 0.0564 0.0563 0.0581 0.0576 0.0576 0.0575
120 0.0546 0.0543 0.0543 0.0542 0.0519 0.0517 0.0517 0.0516 0.0528 0.0526 0.0525 0.0525
0.14
0.05-'-1-1-1-1-1-1-'-1-
20 30 40 50 60 70 80 90 100 110 120
n
Figure 1: MSE of Bayesian and E-Bayesian estimates of 8 under Al-Bayyati loss function.
0.12
0.05 -'-'-'-1-'-'-'-'-1-
20 30 40 50 60 70 80 90 100 110 120
n
Figure 3: MSE of Bayesian and E-Bayesian estimates of в under De-Groot loss function.
6. Real Data Analysis The dataset was sourced from [13], comprising monthly actual tax revenues in Egypt spanning fro m January 2006 to November 2010. The data, expressed in 1000 million Egyptian pounds, are as foll ows: 5.9, 20.4, 14.9, 16.2, 17.2, 7.8, 6.1, 9.2, 10.2, 9.6, 13.3, 8.5, 21.6, 18.5, 5.1, 6.7, 17, 8.6, 9.7, 39.2, 35.7, 15.7, 9.7, 10, 4.1, 36, 8.5, 8, 9.2, 26.2, 21.9, 16.7, 21.3, 35.4, 14.3, 8.5, 10.6, 19.1, 20.5, 7.1, 7.7, 18.1, 16.5, 1 1.9, 7, 8.6, 12.5, 10.3, 11.2, 6.1, 8.4, 11, 11.6, 11.9, 5.2, 6.8, 8.9, 7.1, 10.8. The Kolmogorov-Smirnov (K-S ) statistic's value is 0.082194, with an associated p-value of 0.8203. This suggests that the Exponentia ted Inverse Rayleigh Distribution (EIRD) is the best fit for this dataset. Based on this data, the Maxi mum Likelihood estimates yield § = 8.9362 and a= 9.8028. Figure 4 and 5 displays the histogram an d the estimated Cumulative Distribution Function (CDF) of the EIRD for the dataset, while the esti
Figure 4: Histogram and the fitted density for the monthly actual taxes revenue.
Empirical CDF
r
jy
f
■ Empirical cdf -EIRDcdf
0 5 10 15 20 25 30 35 40
X
Figure 5: Plot for the ECDF of the EIRD model.
Table 2: Bayesian and E-Bayesian estimates of 8 based on real dataset.
^BS ®EBS1 ®EBSZ eEBS3 ^BD ®EBD1 ®EBDZ ®EBD3
94.257 129.236 95.558 71.446 89.769 123.032 90.971 68.017 91.265 125.100 92.500 69.160
7. Conclusion
This paper focuses on employing Bayesian and E-Bayesian methods to estimate the scale parameter of the Exponentiated Inverse Rayleigh distribution (EIRD) through the use of diverse loss functions. Additionally, certain properties of the E-Bayesian estimates are explored. Notably, the results of a simulation study reveal that E-Bayesian estimates exhibit a smaller Mean Squared Error (MSE) in comparison to Bayesian estimates, thereby demonstrating their enhanced efficiency. Among the E-Bayesian estimates, the third one stands out as the most effective.
Moreover, the analysis highlights that the Squared Error loss function outperforms the Al-Bayyati and De-Groot loss functions, exhibiting a smaller MSE. The conclusions drawn from the simulation study are further substantiated by validating the findings through a real-life dataset.
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