# Bayes Estimation of a Two-Parameter Geometric Distribution under Multiply Type II Censoring

Authors: J. B. Shah; M. N. Patel
Journal: International Journal of Quality, Statistics, and Reliability
 (2011)

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## Abstract

We derive Bayes estimators of reliability and the parameters of a two- parameter geometric distribution under the general entropy loss, minimum expected loss and linex loss,  functions for a noninformative as well as  beta prior from multiply Type II censored data. We have studied the robustness of the estimators using simulation and we observed that the Bayes estimators of reliability and the parameters of a two-parameter geometric distribution under all the above loss functions appear to be robust with respect to the correct choice of the hyperparametersa(b) and a wrong choice of the prior parameters b(a) of the beta prior.


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## Body

## 1. Introduction

In life testing experiments, a lot of work has been done under the continuous lifetime models. Sometimes it is neither possible nor convenient to measure the life length of an item continuously until its failure. Also failure time data is sometimes discrete either through the grouping of continuous data due to imprecise measurement or because time itself is discrete. In such circumstances one measures the life of a device on a discrete scale and considers the number of successful cycles, trials, or operations before failure. Therefore, the number of successful trials before failure is more pertinent than the time of continuous period. The one-parameter geometric distribution has an important position in discrete lifetime models. The geometric distribution can be used as a discrete failure to investigate the ability of electronic tubes to withstand successive voltage overloads and performance of electric switches, which are repeatedly turned on and off. Many authors like Yaqub and Khan [1], Patel and Gajjar [2], N. W. Patel and M. N. Patel [3] have contributed to the methodology and estimation of the parameter of the geometric distribution.Here we assume that the lifetime of certain items has a two-parameter geometric distribution with probability mass function (pmf), cumulative distribution function (cdf), and reliability, respectively, as(1)p(x)=P(X=x)=pqx-M,FX(x)=1-qx-M+1,RX(x)=qx-M+1;x=M,M+1,M+2,…,∞;M∈N={1,2,3,…},0<p<1,q=1-p.
Here p, and M are unknown parameters.This model is employed in situations where it is believed that death or failure cannot occur before certain cyclesM, where M is the warranty time or threshold parameter and M+(q/p) is the expected life of the item. One should be cautious when the model includes a threshold parameter M, since the data usually does not provide enough information about M and the inclusion of M could cause rather special statistical problems. If the threshold parameter is known, then there is no difficulty. But if it is unknown then it must be estimated. The models with threshold parameter are difficult to fit with maximum likelihood because for such models likelihood may be unbounded and so maximum likelihood does not work.In life testing experiments, the experimenter may not always be in a position to observe the lifetimes of all the items put on test. This may be because of time limitations and/or other restrictions on data collection. In this situations, Type I or Type II censoring schemes are used. Another way to get censored data is to censor firstr, middle t and largest s observations thus only available ordered observations are xr+1,xr+2,…,xr+k and xr+k+t+1,xr+k+t+2,…,xN-S. The scheme is known as the multiply Type II censoring scheme. This censoring scheme is useful in follow-up studies in epidemiology, reliability and endocrinology. Multiply Type II censoring has been considered by many authors like Balakrishnan [4], Fei and Kong [5], Kong and Fei [6] for continuous lifetime models.In this paper, we have obtained Bayes estimators of parametersM,p, and reliability R of the life model defined in (1) when the data available are of the multiply Type II censored. Here we have considered three types of loss functions, namely, general entropy loss function (GELF), minimum expected loss function (MELF), and linex loss function (LLF). A comparison has been made using Bayes risk among the Bayes estimators. Using simulation study robustness of Bayes estimators has been studied.


## 2. Likelihood: Prior, Posterior, and Marginal

For discrete failure mode, the likelihood function under the multiply Type II censoring scheme described in previous section can be written as(2)L(θ∣x)=N!r!t!s![FX(xr)]r[FX(xr+k+t)-FX(xr+k)]t×[1-FX(xN-S)]S∐i=r+1r+kp(xi)∐i=r+k+t+1N-Sp(xi).
Using (1) and (2), the likelihood becomes proportional to(3)L(θ∣x)αpn(1-qV1)r(1-qV2)tqV3,
where(4)n=N-r-t-s,V1=xr-M+1,V2=xr+k+t-xr+k,V3=∑i=r+1r+kxi+∑i=r+k+t+1N-Sxi+txr+k+sxN-S+t+s-M(N-r).
Here we consider beta prior density for p of the form(5)g1(p)α(1-p)a-1pb-1,0<p<1.
If the prior information about p is given in terms of the prior mean μ and variances σ2, respectively, then the prior parameters can be obtained by solving(6)μ=ba+b,σ2=ab(a+b)2(a+b+1).
For a=b=1, the above marginal prior reduces to diffuse (or natural) prior given by  g1(p)=1,0<p<1.Also the Jeffreys’ invariant noninformative prior(7)g1(p)α(1-p)-1p-1/2,0<p<1,
can be deduced from the above marginal prior with parameters a=0 and b=0.5.As in the stand by Broemeling and Tsurumi [7], we suppose the marginal prior distribution of M to be discrete uniform, that is,(8)g2(M)=1L,M=1,2,3,…,L(L∈N).
Using (5) and (8), the joint prior distribution of p and M is (9)g(p,M)α(1-p)a-1pb-1,0<p<1.The joint posterior density of p and M say π(p,M∣x̲), using (3) and (9), is (10)π(p,M∣x̲)=pn+b-1(1-qV1)r(1-qV2)tqV3+a-1Ψ(V1,V2,V3,M,n,a,b),0<p<1,
where                                                                             (11)Ψ(V1,V2,V3,M,n,a,b)=∑j1=0r∑j2=0t∑M=1L(-1)j1+j2(rj1)(tj2)×B(n+b,V1j1+V2j2+V3+a),
where  B(m,n)=∫01xm-1(1-x)n-1dx.Marginal posterior distributions ofM and p are obtained as(12)π(M∣x̲)=∑j1=0r∑j2=0t(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)Ψ(V1,V2,V3,M,n,a,b),M=1,2,3,…,L,π(p∣x̲)=∑j1=0r∑j2=0t∑M=1L(-1)j1+j2(rj1)(tj2)pn+b-1qV1j1+V2j2+V3+a-1Ψ(V1,V2,V3,M,n,a,b),0<p<1.


## 3. Bayes Estimator

The Bayesian concepts and methods are found useful in Econometrics, Sociology, Engineering, Life Testing and Reliability estimation, Quality Control and others. The Bayes method avoids the possibility of an essentially non-negative parameter having a negative estimator which could occur when estimation is done using sampling framework. For the given sample and the corresponding likelihood function, Bayes estimator will be unique and there will be no confusion about the choice of the estimator. In estimating certain numbers of parameters in the presence of nuisance parameters (threshold parameters), there would be considerable difficulties in non-Bayesian case whereas in the Bayesian setup inference can be made very simple [8].The Bayesian method usually requires less sample data to achieve the same quality of inference as compared to the method based on sampling theory. This is an especially important consideration in those areas of application where sample data may be either expensive or difficult to obtain, for example as in reliability. Over the years, reliability estimation methods based on sampling theory have been found to be extremely useful for a wide variety of problems. However, there are many instances in which the classical methods have been found to be less satisfactory. As reliability testing is directly related to cost and time, the use of sampling theory would be extremely limited. Sampling theory methods are especially inappropriate for reliability analysis based on scarce data.Most of the engineering designs are evolutionary rather than revolutionary processes in which current equipment is modified to suit new requirements. In such instances it would appear that the known facts regarding reliability of the current hardware could possibly be utilized in an attempt to improve the quality of the reliability estimates of the new design. The sampling theory methods are inappropriate for incorporating such related information.In reliability estimation, the Bayesian analysis has been found useful due to two facts: one is the increased quality of the inferences; provided the prior information accurately reflects the true variation in the parameter(s); the other is the reduction in testing requirements (like test time, sample size, etc.) that often occur in Bayesian reliability demonstration test programs. The Bayesian method produces believable results which go a long way in convincing the engineer of the utility of Bayesian methods.


## 4. Bayes Estimators under Minimum Expected Loss Function       (MELF)

In Bayesian analysis, widely used loss function is a quadratic loss function given by(13)L1(θ,d)=w(d-θ)2.
If  w=1, it reduces to squared error loss function (SELF) and for w=θ-2, it becomes (14)L1(θ,d)=θ-2(d-θ)2
known as MELF introduced by Rao Tummala and Sathe [9] in their study.The Bayes estimators ofθ under MELF are given by (15)θ̂M=E(θ-1)E(θ-2).
For the threshold parameter M, which is a positive integer quantity M=1,2,3,…,L(L∈N), the loss function is defined as L1(M,M̂M)=(M-M̂M)2, where M̂M = 1,2,3,…,L(L∈N) is the smallest integer greater than the analytical solution.The Bayes estimators ofM,p and R(t0) under MELF are(16)M̂M=∑j1=0r∑j2=0t∑M=1LM-1(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)∑j1=0r∑j2=0t∑M=1LM-2(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a),p̂M=Ψ(V1,V2,V3,M,n,a,b-1)Ψ(V1,V2,V3,M,n,a,b-2),R̂M(t0)=Ψ(V1,V2,V3,M,n,a-(t0-M+1),b)Ψ(V1,V2,V3,M,n,a-2(t0-M+1),b).


## 5. Bayes Estimators under General Entropy Loss Function      (GELF)

Sometimes, the use of symmetric loss function, namely, SELF, was found to be inappropriate in a situation where an overestimation of the reliability function is usually much more serious than an underestimation. Here we consider the asymmetric loss function, namely, GELF, proposed by Calabria and Pulcini [10], is (17)L2(θ,d)=(dθ)α1-α1ln⁡(dθ)-1,α1≠0,
whereas for the threshold parameter M, the loss function is defined as (18)L2(M,M̂G)=(M̂GM)α1-α1ln⁡(M̂GM)-1,
where α1≠0, M=1,2,3,…,L, and M̂G = 1,2,3,…,L(L∈N). Here M̂G is the smallest integer greater than the analytical solution. The sign of the shape parameter α1 reflects the deviation of the asymmetry, α1 > 0 if overestimation is more serious than underestimation, and vice versa. The magnitude of α1 reflects the degree of asymmetry. The Bayes estimator of θ under the GELF given in (17) is given by(19)θ̂G=[Eπ(θ-α1)]-1/α1.
The Bayes estimators of M,p, and R(t0) under GELF are, respectively,(20)M̂G=[∑j1=0r∑j2=0t∑M=1LM-α1(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)Ψ(V1,V2,V3,M,n,a,b)]-1/α1,(21)p̂G=[Ψ(V1,V2,V3,M,n,a,b-α1)Ψ(V1,V2,V3,M,n,a,b)]-1/α1,(22)R̂G(t0)=[Ψ(V1,V2,V3,M,n,a-α1(t0-M+1),b)Ψ(V1,V2,V3,M,n,a,b)]-1/α1.
The Bayes estimators of M,p, and R(t0) under SELF can be obtained by substituting α1=-1 in the results (20) to (22), respectively.


## 6. Bayes Estimators under Linex Loss Function (LLF)

Another asymmetric loss function given by Varian [11], known as LLF, is defined as (23)L3(θ,d)=exp⁡[α2(d-θ)]-α2(d-θ)-1(α2≠o),
whereas for the threshold parameter M,  the loss function is defined as(24)L3(M,M̂L)=exp⁡[α2(M̂L-M)]-α2(M̂L-M)-1(α2≠0),
where M=1,2,3,…,L and M̂G = 1,2,3,…,L(L∈N). Here M̂L is the smallest integer greater than the analytical solution. The sign and magnitude of the shape parameter α2 reflect the deviation and degree of asymmetry, respectively. The Bayes estimator of θ under LLF given in (23) is given by(25)θ̂L=-1α2ln⁡Eπ[exp⁡(-α2θ)].
Using marginal posterior distributions (12) in (25) and formula 3.478 (3), page 364 of Gradshteyn and Ryzhik [12], the Bayes estimators of M,p, and R(t0), respectively, can be obtained as                                                                        (26)M̂L=-1α2ln⁡[∑j1=0r∑j2=0t∑M=1Le-α2M(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)Ψ(V1,V2,V3,M,n,a,b)],p̂L=-1α2[ln⁡∑j1=0r∑j2=0t∑M=1L{(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)F11(n+b;V1j1+V2j2+V3+a+b+n;-α2)}-ln⁡Ψ(V1,V2,V3,M,n,a,b)],R̂L(t0)=-1α2[ln⁡∑j1=0r∑j2=0t∑M=1L{(-1)j1+j2(rj1)(tj2)B(n+b,V1j1+V2j2+V3+a)F(t0-M+1)(t0-M+1)(V1j1+V2j2+V3+at0-M+1,V1j1+V2j2+V3+a+1t0-M+1,…,V1j1+V2j2+V3+a+t0-Mt0-M+1;V1j1+V2j2+V3+n+a+bt0-M+1,V1j1+V2j2+V3+n+a+b+1t0-M+1,…,V1j1+V2j2+V3+n+a+b+t0-Mt0-M+1,-α2)}-ln⁡Ψ(V1,V2,V3,M,n,a,b)],where pFq(α1,α2,…,αp;β1,β2,…,βq;z) is called a generalized hypergeometric series which is defined as follows (see Gradshteyn and Ryzhik (see [12, page 1000,  formula 9.14 (1)]:(27)Fpq(α1,α2,…,αp;β1,β2,…,βq;z)=∑k=0∞(α1)k(α2)k⋯(αp)k(β1)k(β2)k⋯(βq)k(z)kk!,
where (γi)k=(γi)(γi+1)(γi+2)⋯(γi+k-1).The underconsidered multiply Type II censoring scheme reduces to right Type II, left Type II, and doubly Type II censoring schemes whenr=0 and t=0,t=0, and s=0 and t=0, respectively.


## 7. Numerical Example

We have generated 30 random observations withM=3,L=7,p=0.3 from the distribution defined in (1) having reliability R(t0)=0.2401 at t0=6. Suppose that information about first 3, middle 3, and largest 3 observations are censored, then the only available observations are given in Table 1.Table 1
Generated observations from geometric distributions.
I123456789101112131415xi———333333333344i161718192021222324252627282930xi444———666666———In Table1,  N=30,r=t=s=3,k=15. Prior parameters a and b themselves were random observations from beta distribution with means μ=0.3 and variance σ2=0.0233, resulting in a=7 and b=3. We have calculated Bayes estimators of M,p and R(t0) at t0 = 6 under MELF, SELF, GELF, and LLF. For a comparative purpose, point estimates under the diffuse prior (a=1,b=1) and Jeffreys’ prior (a=0,b=0.5) have also been calculated.The results are shown in Table2.Table 2
The values of bayes estimators ofM,p, and R(t0) at t0=6.
PriorLoss functionShape parameterM̂p̂R̂(t0)MELF—20.36290.1184GELF−350.38680.1730GELF−250.38290.1643SELF−150.37900.1555GELF130.37110.1370GELF230.36690.1274InformativeGELF320.36280.1175LLF−370.38350.1590LLF−270.38200.1581LLF−160.38060.1568LLF130.37760.1541LLF220.37610.1531LLF320.37470.1530DiffuseMELF—20.37600.1028GELF−350.40200.1613GELF−250.39790.1520SELF−150.39370.1425GELF130.38500.1226GELF230.38050.1122GELF320.37590.1016LLF−370.39870.1461LLF−270.39700.1451LLF−160.39540.1439LLF130.39200.1411LLF220.39040.1401LLF320.38870.1397Jeffreys’MELF—20.37230.1043GELF−350.39920.1650GELF−250.39500.1554SELF−150.39060.1455GELF130.38160.1249GELF230.37690.1141GELF320.37220.1031LLF−370.40140.1440LLF−270.39960.1429LLF−160.39790.1417LLF130.39460.1389LLF220.39290.1379LLF320.39120.1374From Table2, we observed that the Bayes estimator of M moves away from the true value of M under |α1|≥2 for GELF and |α2|≥2 for LLF, but the Bayes estimators of the other parameters are not affected, so much, under these choices. So, in subsequent Sections 8 and 9, we have considered the values of α1 only −2, −1, 1, and 2 whereas for α2 only  −2, −1, 1, and 2 for the comparison.


## 8. Sensitivity of Bayes Estimators

In this section, we have studied the sensitivity of the Bayes estimators with respect to changes in the parameters of the beta prior. We have computed Bayes estimators ofM,p and R(t0) at t0=6 in case of MELF, SELF, GELF, and LLF considering different sets of values of (a,b) for the data given in Table 1. The results are shown in Tables 3–6.Table 3
The Bayes estimators and their risks under SELF and MELF.
abSELFMELFM̂Sp̂SR̂S(t0)M̂Mp̂MR̂M(t0)150.36290.172020.34590.1325250.37110.163420.35450.12527350.37910.155520.36290.1184550.39450.140920.37910.1061750.40910.127920.39450.0954950.42310.116520.40910.0860150.41040.127720.39370.0909350.39940.137020.38290.0999550.38900.146220.37260.10927350.37910.155520.36290.1184850.37430.160020.35820.1231950.36970.164620.35370.1277Table 4
The Bayes estimators and their risks under GELF.
abα1=-2α1=1α1=2M̂Gp̂GR̂G(t0)M̂Gp̂GR̂G(t0)M̂Gp̂GR̂G(t0)150.36700.181330.35450.152430.35010.1421250.37510.172630.36290.144430.35870.13457350.38300.164330.37110.137030.36700.1274550.39820.149330.38690.123530.38300.1145750.41260.135930.40190.111630.39820.1032950.42650.123930.41620.101130.41260.0932150.41440.136730.40220.109230.39790.0996350.40330.145930.39120.118430.38700.1088550.39290.155230.38090.127730.37670.11817350.38300.164330.37110.137030.36700.1274850.37820.168930.36640.141630.36230.1320950.37350.173430.36180.146230.35770.1367Table 5
The Bayes estimators and their risks under LLF.
abα2=-1α2=-2M̂Lp̂LR̂L(t0)M̂Lp̂LR̂L(t0)160.36440.173670.36590.1749260.37260.165070.37410.16627360.38060.156870.38200.1581560.39590.142170.39740.1431760.41060.129070.41200.1299960.42450.117470.42600.1182160.41200.128970.41370.1300360.40090.138270.40260.1394560.39050.147670.39200.14877360.38060.156870.38200.1581860.37580.161570.37720.1627960.37110.166170.37260.1673Table 6
The Bayes estimators and their risks under LLF.
abα2=1α2=2M̂Lp̂LR̂L(t0)M̂Lp̂LR̂L(t0)130.36140.170420.35990.1694230.36960.162020.36810.16107330.37760.154120.37610.1531530.39300.139720.39160.1388730.40720.126920.40620.1261930.42170.115620.42030.1148130.40880.126620.40710.1256330.39780.135820.39620.1348530.38740.145020.38590.14407330.37760.154120.37610.1531830.37290.158720.37140.1577930.36830.163220.36690.1623Following Calabria and Pulcini [13], we also assume the prior information to be correct if the true value of p is close to prior mean μ and is assumed to be wrong if p is far from μ.We observed that the Bayes estimators ofM,p, and R(t0) under MELF and SELF appear to be robust with respect to the correct choice of prior parameters a (b) and a wrong choice of the prior parameters b(a). This can be seen from Table 3. These conclusions are also valid in case of GELF and LLF, as observed in Tables 4, 5, and 6.


## 9. Simulation Results

In this section, we have generated 500 different random samples withM=3,N=30,L=7,t0=6,p=0.3,r=t=s=3,k=15 from the distribution defined in (1) to justify the results obtained in Section 8 and obtained the Bayes estimators and their risk = Ed[L(d,θ)]under SELF, GELF, LLF and MELF for different choices of (a,b). The results are summarized in Tables 7, 8, 9, and 10.Table 7
The Bayes estimators and their risks under SELF and MELF.*
abSELFMELFM̂Sp̂SR̂S(t0)M̂Mp̂MR̂M(t0)140.29610.257220.28100.22240.074690.011050.030110.111110.023360.07138240.30340.247120.28870.21290.074110.010300.032070.111110.021400.076047340.31060.237520.29610.20400.081280.010150.035910.111110.020730.08321540.32440.219620.31060.18750.075280.011350.049310.111110.022900.10346740.33770.203520.32440.17270.083620.014110.070500.111110.029290.12949940.35050.188820.33770.15940.079960.017990.099690.111110.039380.15932140.33160.212520.31670.17810.080980.014330.070980.111110.030060.13048340.32430.220920.30950.18680.079810.012460.055560.111110.025580.11215540.31730.229320.30270.19540.082450.011080.044140.111110.022530.096407340.31060.237520.29610.20400.081280.010150.035910.111110.020730.08321840.30730.241620.29290.20830.083470.009850.032790.111110.020240.07757940.030420.245620.28980.21250.082590.009670.030260.111110.020000.07256*In each cell the first value gives Bayes estimate and second its risk.Table 8
The bayes estimators and their risks under GELF.*
abα1=-2α1=1α1=2M̂Gp̂GR̂G(t0)M̂Gp̂GR̂G(t0)M̂Gp̂GR̂G(t0)150.29980.265430.28870.240030.28480.23100.381650.043520.114240.0000.011420.033710.0000.046480.14289250.30700.255130.29610.230230.29240.22140.381650.040710.120740.0000.010540.036190.0000.042720.153547350.31410.245430.30340.220930.29980.21230.381650.040290.135130.0000.010310.040290.0000.041540.17015550.32780.227330.31760.203730.31410.19550.381650.045130.189510.0000.011470.052910.0000.046080.21865750.34090.210830.33120.188230.32780.18030.381650.055650.281540.0000.014450.070710.0000.058580.28386950.35360.195930.34420.174230.34100.16660.381650.070150.415910.0000.018950.092960.0000.077790.36221150.33520.220730.32420.195430.32040.18650.381650.056140.285250.0000.014770.071380.0000.059990.28830350.32790.229030.31700.204030.31320.19520.381650.049330.216580.0000.012710.058560.0000.051300.24075550.32080.237330.31010.212530.30630.20380.381650.043970.168240.0000.011240.048290.0000.045240.201617350.31410.245430.30340.220930.29980.21230.381650.040290.135130.0000.010310.040290.0000.041540.17015850.31080.249430.30020.225130.29660.21650.381650.039070.123100.0000.010030.037060.0000.040490.15711950.30760.253430.29710.229330.29350.22070.381650.038250.113610.0000.009870.034290.0000.039940.14575*In each cell the first value gives Bayes estimate and second its risk.Table 9
The bayes estimators and their risks under LLF.*
abα2=-1α2=-2M̂Lp̂LR̂L(t0)M̂Lp̂LR̂L(t0)160.29720.259170.29830.25982.04980.000940.001857.00030.003770.00717260.30450.248970.30560.24972.04980.000960.001677.00030.003870.006537360.31170.239270.31280.24002.04980.001040.000167.00030.004170.00628560.32550.221270.32670.22212.04980.001320.001707.00030.005300.00675760.33890.204970.34000.20582.04980.001770.002067.00030.007060.00828960.35160.190170.35270.19092.04980.002360.002627.00030.009350.01061160.33290.214270.33410.21512.04980.001770.002107.00030.007050.00842360.32550.222670.32670.22352.04980.001460.001867.00030.005840.00741560.31840.230970.31960.23182.04980.001220.001707.00030.004890.006707360.31170.239270.31280.24002.04980.001040.000167.00030.004170.00628860.30840.243270.30950.24412.04980.000970.001577.00030.003890.00617960.30520.247370.30630.24812.04980.000910.001567.00030.003660.00612*In each cell the first value gives Bayes estimate and second its risk.Table 10
The bayes estimators and their risks under LLF.
abα2=1α2=2M̂Lp̂LR̂L(t0)M̂Lp̂LR̂L(t0)130.29500.255520.29400.25720.0000.000930.001861.13530.003690.00845230.30230.245520.30130.24670.0000.000950.001681.13530.003780.007427330.30950.235920.30840.23680.0000.001020.001601.13530.004070.00690530.32330.218220.32230.21680.0000.001310.001691.13530.005240.00698730.33670.202220.33560.20220.0000.001780.002041.13530.007110.00817930.34940.187620.34830.18740.0000.002390.002561.13530.009630.01014130.33040.211020.32920.21120.0000.001770.002081.13530.007110.00850330.32310.219420.32200.21980.0000.001450.001851.13530.005800.00769530.31620.227720.31500.22830.0000.001210.001691.13530.004800.007167330.30950.235920.30840.23680.0000.001020.001601.13530.004070.00690830.30630.240020.30520.24100.0000.000960.001581.13530.003800.00688930.30310.244020.29500.25530.0000.000900.001571.13530.003470.00780*In each cell the first value gives Bayes estimate and second its risk.


## 10. Conclusions

From Table7, we observed that the Bayes estimators of the parameter perform well (minimum Bayes risk) under MELF compared to SELF for any choice of hyper parameters a and b.From Tables8–10, we find that under both the asymmetric loss functions, namely,  GELF and LLF, Bayes risks become smaller when parameters (α1 and α2) tend to zero,  that is, Bayes estimators perform well under these asymmetric loss functions.From Tables7–10 following Calabria and Pulcini [13], we also observed that the Bayes estimators of the parameters are robust with respect to the correct choice of prior parameters a(b) and a wrong choice of the prior parameters b(a) which we have concluded in Section 8.



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*Source: 618347-2011-05-11.xml*