# Bayesian Estimation for Burr Distribution Type III Based on Trimmed Samples

Authors: A. M. Abd-Elfattah; A. H. Alharbey
Journal: ISRN Applied Mathematics
 (2012)

---

## Abstract

Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the parameters of Burr distribution type III based on a trimmed samples and prior information will be considered. In this paper, we study the estimation of unknown parameters based on doubly censored type II. The problem discussed using maximum likelihood method and Bayesian approach to estimate the shape parameters of Burr type III distribution. The numerical illustration requires solving nonlinear equations, therefore, MathCAD (2001) statistical package used to asses these effects numerically.


---

## Body

## 1. Introduction

Burr [1] introduced a family of twelve cumulative distribution functions for modeling lifetime data. The two important members of the family are Burr types III and XII. The two important distributions, Burr type III and Burr type XII, are interrelated through simple transformation. Burr type III distribution allows for a wider region for the skewness and kurtosis plane, which covers several distributions including the log-logistic, and the Weibull and Burr type XII distributions.However, outliers may occur in the data set. Trimmed samples are widely employed in several areas of statistical practice, especially when some sample values at either or both extremes might have been contaminated. The problem of estimating the parameters of Burr distribution type III based on a trimmed sample and prior information is considered.Many authors discuss different methods of estimation for Burr type XII distribution. Al-Hussaini and Jaheen [2] and Al-Hussaini et al. [3] used different techniques for obtaining Bayes estimates of the shape parameters c and k, reliability and failure rate functions based on type II censored samples. Al-Hussaini et al. [4] obtained the maximum likelihood, uniformly minimum variance unbiased, Bayes and empirical Bayes estimators for the parameter k and reliability function when c is known. Wingo [5] developing the theory for the ML point estimation of the parameters of the Burr distribution when Type II singly censored sample is at hand. Ali-Mousa and Jaheen [6] obtained interval estimates of the parameter k and reliability function when c is known, using a Bayesian approach based on Type II censored data. Ali-Mousa [7] obtained empirical Bayes estimation of the parameter k and the reliability function based on accelerated Type II censored data. Gupta et al. [8] discusses analysis of failure time data by Burr distribution. Wang et al. [9] obtained the maximum likelihood estimation of the Burr XII distributions parameter with censored and uncensored data. Hossain and Nath [10] deal with unweighted least squares estimation of the parameters. They compared the results with the maximum likelihood and maximum product of spacing methods. Ali-Mousa and Jaheen [11] obtained the maximum likelihood and Bayes estimates for two parameters c,  k and the reliability function r(t) of the Burr Type XII distribution based on progressive type II censored samples.The objective of this paper is to obtain the estimators of the unknown shape parameters of Burr type III based on doubly censored type II. The problem discussed using maximum likelihood method and Bayesian approach to estimate the shape parameters of Burr type III distribution requires solving nonlinear equations, therefore, numerical study is carried out to asses these effects using MathCAD (2001) statistical package.


## 2. Burr Type III Distribution

The Burr family of distributions has, in recent years, assumed an important position in the field of life testing because of its uses to fit business failure data, also it includes the well-known exponential and Weibull distributions as special cases. Burr [1] has suggested this family of distributions by solving the following differential equation:
(2.1)dydx=y(1-y)g(x,y),y=F(x),
where g(x,y)>0,  0≤y≤1,  x in the range over which the solution is being satisfied. By using different forms of g(x,y) and Pearson systems, Burr obtained twelve distribution functions which listed in Burr [1].Burr family of distributions includes twelve types of cumulative distribution functions, which yield a variety of density shapes. Many standard theoretical distributions, including the Weibull, exponential logistic, generalized logistic, Gompertz, normal, extreme value, and uniform distributions are special cases or limiting cases of Burr family of distributions. The simple closed form of these distributions has been applied to studies in simulation. Burr [1] developed the family of Burr distributions as an outgrowth into methods for fitting cumulative frequency functions rather than probability density functions to frequency data. Some of the forms of Burr distributions are related by simple transformation For example, the Burr type III distribution can be obtained from Burr type II distribution by replacing X with ln(X), see Johnson et al. [12]. Similarly, Burr type XII distribution can be derived from Burr type III distribution by replacing X with (1/X), see Burr and Cislak [13]. The Burr type I family is more commonly known as the uniform distribution. The Burr III, IV, V, IX, and XII families have a variety of density shapes. Types III and XII are the simplest functionally and therefore, the two distributions are the most desirable for statistical modeling. The Burr type III distribution is much richer than Burr type XII distribution.The twelve distributions of Burr are listed in Burr [1] and Johnson et al. [12].Properties of Type III Burr(c,k)
The distribution of Burr(c,k) type III is
(2.2)F(x)=(1+x-c)-k,x>0,
where the parameters c>0 and k>0 are the shape parameters of the distribution. Its density function is
(2.3)f(x)=ckx-(c+1)(1+x-c)-(k+1).
The rth moment
(2.4)E(xr)=μr′=ck∫0∞xrx-(c+1)(1+x-c)k+1dx=kβ(1-rc,k+rc),r<c,
where β(a,b) is the standard beta function.The expectation of the distribution is obtained as follows.Ifr=1 in (2.4) we have:
(2.5)μ1′=kβ(1-1c,k+1c)=λ1Γ(k),c>1,
where λi=Γ(1-i/c)Γ(k+i/c),  i=1,2,…,r.The variance(2.6)var(x)=Γ(k)λ2-λ12Γ2(k),c>2.The mode(2.7)x={c(k+1)c+1-1}1/c,ck≥1,c>0,k>0.The median(2.8)x=1(21/k-1)1/c.


## 3. Bayesian and Non-Bayesian Estimation Methods

In this section we will obtain the estimation of Burr parameters based on trimmed samples using Bayesian and non Bayesian methods of estimation.


### 3.1. Non-Bayesian Estimation for Burr Parameters Based on Trimmed Sample

In this section, the estimation problem of Burr type III with two parameters (c,k) under type II double censored data (trimmed samples) are obtained. Some data may not be observed, a known number of observation in an ordered sample are missing at both ends in failure censored experiments, the observations are the smallest r and the largest rs are random then the data collected will be x(r+1)≤x(r+2)≤⋯≤x(n-s) and the likelihood function in double censored type II takes the following form:
(3.1)L(x,θ)=n!(r-1)!(n-s)!∏i=rsf(x(i),θ)[F(x(r),θ)]r-1[1-F(x(s),θ)]n-s,L(x,c,k)=n!(r-1)!(n-s)![1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s*∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)].
It is usually easier to use maximize the natural logarithm of the likelihood function rather than the likelihood function itself. Therefore, the logarithm of the likelihood function is
(3.2)lnL=lnw+(r-1)ln[1-(1+x(r)-c)-k]-(n-s)kln(1+x(s)-c)-∑i=rs[(c+1)lnx(i)+(k+1)ln(1+x(i)-c)]+(s-r+1)ln(ck),
where w=n!/((r-1)!(n-s)!). The maximum likelihood estimators c^,k^ of c,k are the solutions of the system of equations obtained by letting the first partial derivatives of the total log likelihood with respect to c, k be zero. The systems of equations are as follows:
(3.3)∂lnL∂k=(r-1)ln(1+x(r)-c)[1+x(r)-c]-k1-(1+x(r)-c)-k-(n-s)ln(1+x(s)-c)+(s-r+1)k-∑i=rsln(1+x(i)-c),(3.4)∂lnL∂c=-(r-1)k(1+x(r)-c)-k-1x(r)-cln(x(r))1-(1+x(r)-c)-k+(n-s)kx(s)-cln(x(s))1+x(s)-c+(s-r+1)c-∑i=rsln(x(i))-(k+1)∑i=rsx(i)-cln(x(i))1+x-c(i).
From (3.3) the maximum likelihood estimator of k is expressed by
(3.5)(n-s)ln(1+x(s)-c^)+∑i=1sln(1+x(i)-c^)=(s-r+1)k^+(r-1)ln(1+x(r)-c^)[1+x(r)-c^]k^-1.
From (3.4) the maximum likelihood estimator of c is expressed by
(3.6)(n-s)kx(s)-c^ln(x(s))1+x(s)-c^+(k+1)∑i=rsx(i)-c^ln(x(i))1+x(i)-c^=(r-1)k(1+x(r)-c^)-k-1x(r)-c^ln(x(r))1-(1+x(r)-c^)-k+(s-r+1)c^+∑i=rsln(x(i)).
Since the closed form solution to nonlinear equations (3.5) and (3.6) is very hard to obtain, Newton-Raphson method is applied for solving the nonlinear equations simultaneously to obtain c^,k^.


#### 3.1.1. Asymptotic Variances Covariance Matrix

The asymptotic variances covariance matrix of the parameters (c and k) is obtained by inverting the Fisher information matrix
(3.7)Iij=E[-∂2L∂θi∂θj],i,j=1,2,
where θ1,θ2=c or k. Hence the approximate variance-covariance matrix is given by
(3.8)[∂2lnL∂c2|c^,k^∂2lnL∂c∂k|c^,k^∂2lnL∂k∂c|c^,k^∂2lnL∂k2|c^,k^]-1=[Var(c^)Cov(c^,k^)Cov(k^,c^)Var(k^)].
The elements of matrix I are the negative of second derivatives of the natural logarithm of likelihood function defined in (3.2).The elements of matrixI are given as follows:
(3.9)∂2lnL∂k2=-(s-r+1)k2-(r-1)[ln(1+x(r)-c)]2(1+x(r)-c)-k[1-(1+x(r)-c)-k]2,(3.10)∂2lnL∂c2=(r-1)kln(x(r))[𝒜x(r)cln(x(r))-x(r)-c(k+1)(1+x(r)-c)kx(r)-cln(x(r))-x(r)-cln(x(r))][(1+x(r)c)k+1-(1+x(r)c)]2-(n-s)klnx(s)x(s)-clnx(s)(1+x(s)c)2-(s-r+1)c2+(k+1)∑i=rsln(x(i))2(1+x(i)-c)2,(3.11)∂2lnL∂c∂k=-∑i=rsx(i)-cln(x(i))1+x(i)-c+(n-s)x(s)-cln(x(s))1+x(s)-c-(r-1)x(r)-clnx(r)1+x(r)-c+(r-1)(1-(1+x(r)-c)-k)x(r)-cln(x(r))(1+x(r)-c)((1-(1+x(r)-c)-k))2-[ln(1+x(r)-c)]k(1+x(r)-c)-k-1+x(r)-clnx(r),
where 𝒜 denotes [(1+x(r)-c)k+1-(1+x(r)-c)]. The maximum likelihood estimators c^, k^ have asymptotic variance-covariance matrix  defined by inverting the Fisher information matrix.


### 3.2. Numerical Illustration

In estimation problem, it is required to study the properties of the derived expressions for the estimators theoretically or analytically. Sometimes it seems very difficult to study the properties of the estimators theoretically because of the complicated formula of the estimators. Consequently, a simulation study will be set up, treating separately the sampling distribution of the estimators. Simulation studies have been performed using MathCAD (2001) for illustrating the theoretical results of the estimation problem. The simulation procedures will described below.Step 1.
Generate a random sample of size 10, 20, 40, 60, 80, and 100 from Burr type III distribution. The generation of Burr type III distribution is very simple ifU has a uniform (0,1) random number, then Y=[U(-1/k)-1](-1/c) follows a Burr type III distribution. In double censored type II, the true parameters selected values are (c=1.2,k=1.5), (c=1.2,k=2), (c=2.5,k=1.5), and (c=2.5,k=2).Step 2.
Choose censored failure(r=0.6,s=6).Step 3.
For each sample and for the four sets of parameters distribution were estimated under doubly censored type II.Step 4.
Newten-Raphson method was used for solving the nonlinear equations forc and k defined in (3.4) and (3.6).Step 5.
Equations (3.9) and (3.10) were used to obtain the variance covariance matrix of (c^,k^).Results are tabulated in Table1.Table 1
The maximum likelihood estimator, the standard deviation, and covariance of the Burr distribution with two parameter under doubly censored sample when (a) (c=1.2, k=1.5), (b) (c=1.2, k=2), (c) (c=2.5, k=1.5), (d) (c=2.5, k=2).
(a)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
4.619
3.689
7.841
4.341
58.814
10
7
2
4.534
3.524
4.672
4.172
50.042
9
4.338
3.308
4.722
3.202
22.784
10
6.874
3.874
6.654
4.154
66.589
20
15
2
6.483
2.483
6.439
3.939
64.752
18
6.147
3.147
6.307
3.807
41.058
20
6.498
3.498
6.802
4.302
74.941
40
30
3
5.384
3.384
6.471
3.971
62.603
38
5.287
2.287
5.629
3.129
39.913
30
5.692
2.692
6.794
4.294
62.054
60
45
3
5.542
2.542
6.678
4.178
47.829
58
5.201
2.201
5.489
2.989
16.475
40
5.448
2.448
6.824
4.324
78.475
80
65
5
5.707
2.707
6.671
4.171
66.942
78
5.334
2.334
5.391
2.891
11.313
50
6.689
3.689
6.841
4.341
58.814
100
75
5
6.524
3.524
6.672
4.172
50.042
98
6.308
3.308
5.702
3.202
30.784
(b)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
6.689
3.689
6.841
4.341
58.814
10
7
2
6.524
3.524
6.672
4.172
50.042
9
6.308
3.308
5.702
3.202
30.784
10
6.874
3.874
6.654
4.154
66.589
20
15
2
6.483
3.483
6.439
3.939
64.752
18
6.147
3.147
6.307
3.807
41.058
20
6.498
3.498
6.802
4.302
74.941
40
30
3
6.384
3.384
6.471
3.971
62.603
38
5.287
2.287
5.629
3.129
39.913
30
5.692
2.692
6.794
4.294
62.054
60
45
3
5.542
2.542
6.678
4.178
47.829
58
5.201
2.201
5.489
2.989
16.475
40
5.448
2.448
6.824
4.324
78.475
80
65
5
5.707
2.707
6.671
4.171
66.942
78
5.334
2.334
5.391
2.891
11.313
50
6.689
3.689
6.841
4.341
58.814
100
75
5
6.524
3.524
6.672
4.172
50.042
98
6.308
3.308
5.702
3.202
30.784
(c)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
5.952
2.952
5.159
3.159
69.043
10
7
2
5.712
2.712
4.914
2.914
55.876
9
5.819
2.819
4.249
2.249
38.874
10
5.984
2.984
5.147
3.147
68.497
20
15
2
5.147
2.147
4.271
2.271
24.984
18
5.078
2.078
4.014
2.014
14.872
20
5.618
2.618
5.081
3.081
69.187
40
30
3
5.491
2.491
4.941
2.941
57.078
38
5.107
2.107
4.028
2.028
23.174
30
5.347
2.347
4.627
2.627
70.842
60
45
3
5.148
2.148
4.489
2.489
58.247
58
5.004
2.004
4.309
2.309
41.872
40
5.334
2.334
4.658
2.658
74.194
80
65
5
5.238
2.238
4.748
2.748
60.827
78
4.997
1.997
4.102
2.102
47.145
50
0.447
0.553
1.39
0.19
1.987
100
75
5
0.474
0.526
1.317
0.117
1.854
98
1.218
0.218
1.234
0.034
1.823
(d)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
1.395
0.395
2.717
0.717
42.175
10
7
2
1.501
0.501
2.585
0.585
35.214
9
1.014
0.014
2.461
0.461
18.258
10
1.284
0.284
2.648
0.648
40.954
20
15
2
1.481
0.481
2.441
0.441
33.254
18
1.008
0.008
2.314
0.314
16.418
20
1.658
0.658
2.995
0.995
68.284
40
30
3
1.018
0.018
2.147
0.147
50.258
38
1.001
0.001
1.654
0.346
52.547
30
1.228
0.228
2.954
0.954
68.952
60
45
3
1.109
0.109
2.756
0.756
53.218
58
1.019
0.019
2.341
0.341
40.021
40
1.721
0.721
2.481
0.481
77.248
80
65
5
1.856
0.856
2.441
0.441
80.514
78
1.007
0.007
1.761
0.239
54.951
50
1.395
0.395
2.717
0.717
42.175
100
75
5
1.501
0.501
2.585
0.585
35.214
98
1.014
0.014
2.461
0.461
18.258From Table1, we note that the standard deviation decreases when n is increasing. Also, we note that, when c and k increase the standard deviation of the estimators decrease. Similarly, the maximum likelihood estimator of the parameters has the same behaviors when the sample size becomes large and the properties of two parameters c and k at (1.2 and 1.5), respectively is better than the other values.


### 3.3. Bayesian Analysis for Burr Distribution Type III Based on Trimmed Samples

The Bayesian approach allows both sample and prior information to be incorporated into analysis, which will improve the quality of the inferences. In this section, Bayesian estimators and posterior variance of the shapes parameters are obtained in the case of double censored type II. The prior distribution could be “none informative” such as a flat, uniform distribution, assuming equal probability for the parameter value within a realistic range, and when the prior could be more informative such as a normal distribution or other possible distribution that represents an initial assessment of what is known about the parameter before the collection of data and the analysis. Moreover, a numerical examples are given for illustration study.


#### 3.3.1. Bayesian Estimation for Burr Distribution Type III Parameters Based on Trimmed Sample in Case of Noninformative Prior

The likelihood function takes the form in (3.1). Assumed that the parameters c, k have independent prior distribution and let the noninformative prior (NIP) for cand k are, respectively, given by:
(3.12)u1(c)αc-1c>0,u2(k)αk-1k>0.
Consequently, the joint (NIP) will be defined as follows:
(3.13)u(c,k)=(ck)-1c,k>0.
The joint posterior density functions of c, k under double censored sample type II will be
(3.14)g(c,k∣x)=u(c,k)L(x∣c,k)λc>0,k>0,g(c,k∣x)=(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]λ,
where λ is the normalized constant defined as follows:
(3.15)∬0∞(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Now, the marginal posterior of one parameter is obtained by integrating the joint posterior distribution with respect to the other parameter, hence the marginal posterior probability density function of c will be
(3.16)g1(c∣k,x)=c-1∫0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dkλ.
Similarly integrating the joint posterior (3.14) with respect to k. Then, the marginal posterior of k is
(3.17)g2(k∣c,x)=k-1∫0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c-1)(1+x(i)-c)-(k+1)]dcλ.It is well known that under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance, a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters(c,k) are expressed as follows:
(3.18)E(c∣k,x)=c~,c~=∬0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(c~∣k,x)=E(c~-c)2,var(c~∣k,x)=∬0∞(c~-c)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,E(k∣c,x)=k~,k~=∬0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(k~∣c,x)=E(k~-k)2,var(k~∣c,x)=∬0∞(k~-k)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Equations (3.18) are very difficult to be solved exactly so that an iterative procedure is needed to solve these equations numerically. Using the statistical package, MathCAD (2001), the posterior mean and posterior variance of the shapes parameter (c,k) will be obtained.The posterior mean and the posterior variance ofc, k are obtained numerically as described below.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain posterior variance of the shape parameter(c,k) using (3.18).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameters(c,k) for all sample size and for sets of parameter values were obtained.
Numerical results are summarized in Table2.Table 2
The posterior mean and posterior variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) in case of noninformative prior.
n
r
S
c
k
Posterior mean
Posterior variance
Posterior mean
Posterior variance
5
1.017
0.685
0.671
0.765
10
2
7
0.701
0.514
0.569
0.648
9
0.608
0.338
0.408
0.491
10
1.010
0.742
0.948
0.785
20
2
15
0.871
0.537
0.653
0.676
18
0.691
0.338
0.527
0.527
20
0.437
0.674
0.741
0.576
40
3
30
0.288
0.582
0.687
0.477
38
0.119
0.449
0.354
0.345
30
0.609
0.762
0.671
0.758
60
3
45
0.547
0.684
0.507
0.862
58
0.489
0.449
0.407
0.647
40
0.827
0.761
0.718
0.901
80
5
65
0.667
0.895
0.651
0.763
78
0.524
0.612
0.581
0.522
50
0.718
0.761
0.718
0.901
100
5
75
0.651
0.895
0.651
0.763
98
0.581
0.612
0.581
0.522


#### 3.3.2. Bayes Estimator under LINEX Loss Function

Under the assumption that minimal loss function occursθ~~=θ, the LINEX loss function for θ=θ(c,k) can be expressed as defined by Zellner [14], the Bayes estimator θ~~ of θ under LINEX loss function is
(3.19)θ~~=-1alog(Eθ(exp(-aθ))),θ~~-1aln∫e-aθf(θ∣x)dθ.Now, in (3.19) put θ=c, then the Bayes estimate c~~ of parameter c relative to the LINEX loss function is
(3.20)c~~=-1aln∫e-acg1(c∣k,x)dc.
Set θ=k in (3.19), then the Bayes estimate k~~ of parameter k relative to the LINEX loss function is
(3.21)k~~=-1aln∫e-akg2(k∣c,x)dk.
The equations (3.20) to (3.21) are very difficult to obtain their solutions exactly, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameters (c,k). The numerical procedures will describe as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain the posterior variance of the shape parameter(c,k) in (3.20) and (3.21).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table3.Table 3
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) under LINEX loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.698
0.396
2.596
1.631
10
2
7
1.353
0.205
1.700
0.810
9
1.189
0.292
1.663
0.726
10
1.468
0.267
2.552
1.819
20
2
15
1.238
0.166
2.032
1.619
18
1.237
0.178
1.773
1.152
20
1.382
0.294
2.343
1.198
40
3
30
1.092
0.200
1.411
0.848
38
1.022
0.196
0.936
0.847
30
1.075
0.197
1.543
1.237
60
3
45
1.013
0.174
1.018
1.062
58
0.988
0.170
0.935
1.138
40
1.275
0.162
1.007
1.188
80
5
65
0.986
0.182
0.340
1.193
78
0.906
0.151
1.762
2.269
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


#### 3.3.3. Bayes Estimator under General Entropy (GE) Loss Function

The Bayes estimatorθ~^ under GE loss function is
(3.22)θ~^=(Eθ(θ-v))-1/v,θ~^=[∫θ-vf(θ∣x)dθ]-1/vv>0.
Now, in (3.22) put θ~^=c, then the Bayes estimate c~^ of parameter c relative to the GE loss function is
(3.23)c~^=[∫c-vg1(c∣k,x)dc]-1/v.
In (3.22), set θ~^=k, then the Bayes estimate k~^ of parameter k relative to the GE loss function is
(3.24)k~^=[∫k-vg2(k∣c,x)dk]-1/v.The equations (3.23) to (3.24) cannot have exact solutions, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameter. The numerical procedures will be described as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solving the nonlinear Bayesian for posterior variance of the shape parameter(c,k) in (3.23) and (3.24).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table4.Table 4
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) general entropy loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.641
0.685
3.825
0.765
10
2
7
1.375
0.514
2.614
0.648
9
1.255
0.338
1.661
0.491
10
1.474
0.742
3.423
0.785
20
2
15
1.299
0.537
2.490
0.676
18
1.147
0.338
1.550
0.527
20
1.294
0.674
3.076
0.576
40
3
30
1.087
0.582
1.378
0.477
38
1.016
0.449
1.263
0.345
30
1.032
0.762
1.672
0.758
60
3
45
0.977
0.684
1.173
0.862
58
0.924
0.449
1.038
0.647
40
1.162
0.761
0.860
0.901
80
5
65
1.034
0.895
0.396
0.763
78
0.888
0.612
0.185
0.522
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


### 3.4. Bayesian Estimation for Burr Distribution Type III Parameters Based on Trimmed Sample in Case of Informative Prior (IP)

Assume that the parametersc, k have independent prior distribution and let the informative prior (IP) for the parameter k is given by the gamma density as follow:
(3.25)π1(k∣c)=cα+1Γα+1kαe-kck>0,c>0,α+1>0,
and the prior density function of c is the exponential density as follow:
(3.26)π2(c)=1βe-c/βc>0,β>0.
Then the joint prior density of c and k, given by
(3.27)π(c,k)=π1(k∣c)π2(c),π(c,k)=cα+1βΓα+1kαe-(k+1/β)cc>0,k>0.
The joint posterior density function of c, k using double censored sample type II will be
(3.28)f(c,k∣x)=π(c,k)L(x∣c,k)Ec>0,k>0=cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]EβΓα+1,
where E is the normalized constant equal to
(3.29)E=∬0∞cα+βΓα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Now, the marginal posterior of one parameter is obtained by integrating the joint posterior distribution with respect to the other parameter, hence the marginal posterior probability density function of c is
(3.30)f1(c∣k,x)=cα+1βΓα+1∫0∞kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dkλ.
Similarly integrating the joint posterior (3.28) with respect to k the marginal posterior of k is
(3.31)f2(k∣c,x)=kαβΓα+1∫0∞cα+1e-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcλ.Hence, under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters(c,k) are expressed as follows:
(3.32)E(c∣k,x)=c~˘,c~˘=1βΓα+1∬0∞cα+2kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(c~˘∣k,x)=E(c~˘-c)2,var(c~˘∣k,x)=1βΓα+1∬0∞(c~˘-c)2∬0∞cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,E(k∣c,x)=k~˘,k~˘=1βΓα+1∬0∞cα+1kα+1e-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(K~˘∣c,x)=E(K~˘-k)2,var(K~˘∣c,x)=1βΓα+1∬0∞(k~˘-k)2∬0∞cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Nonlinear equations (3.32) are very difficult to be solved exactly an iterative procedure is needed to solve these equations numerically. By using the statistical package, MathCAD (2001), the posterior mean and posterior variance of the shape parameters (c,k) will be obtained.The posterior mean and the posterior variance ofc, k are obtained numerically in the following procedures as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear Bayesian for posterior variance of the shape parameter(c,k) in (3.32).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table5.Table 5
The posterior mean and posterior variance of the Burr distribution with two parameter under doubly censored sample when (c=1.2, k=1.5) in case of informative prior.
n
r
s
c
k
Posterior mean
Posterior variance
Posteriormean
Posterior variance
5
0.495
0.204
1.017
0.719
10
2
7
0.384
0.214
0.701
0.645
9
0.258
0.217
0.608
0.201
10
0.234
0.391
1.010
0.604
20
2
15
0.128
8.215
0.871
0.556
18
0.087
0.319
0.691
0.493
20
0.482
0.339
0.437
0.911
40
3
30
0.352
0.337
0.288
0.425
38
0.241
0.190
0.119
0.307
30
0.852
0.199
0.609
0.827
60
3
45
0.729
0.160
0.547
0.647
58
0.635
0.139
0.489
0.558
40
0.957
1.171
0.827
0.728
80
5
65
0.721
0.206
0.667
0.661
78
0.408
0.078
0.524
0.337
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522It is noted that the posterior mean decreases whenn is increasing. Similarly the posterior variance of the parameters has the same behaviors when the sample size becomes large.


## 3.1. Non-Bayesian Estimation for Burr Parameters Based on Trimmed Sample

In this section, the estimation problem of Burr type III with two parameters (c,k) under type II double censored data (trimmed samples) are obtained. Some data may not be observed, a known number of observation in an ordered sample are missing at both ends in failure censored experiments, the observations are the smallest r and the largest rs are random then the data collected will be x(r+1)≤x(r+2)≤⋯≤x(n-s) and the likelihood function in double censored type II takes the following form:
(3.1)L(x,θ)=n!(r-1)!(n-s)!∏i=rsf(x(i),θ)[F(x(r),θ)]r-1[1-F(x(s),θ)]n-s,L(x,c,k)=n!(r-1)!(n-s)![1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s*∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)].
It is usually easier to use maximize the natural logarithm of the likelihood function rather than the likelihood function itself. Therefore, the logarithm of the likelihood function is
(3.2)lnL=lnw+(r-1)ln[1-(1+x(r)-c)-k]-(n-s)kln(1+x(s)-c)-∑i=rs[(c+1)lnx(i)+(k+1)ln(1+x(i)-c)]+(s-r+1)ln(ck),
where w=n!/((r-1)!(n-s)!). The maximum likelihood estimators c^,k^ of c,k are the solutions of the system of equations obtained by letting the first partial derivatives of the total log likelihood with respect to c, k be zero. The systems of equations are as follows:
(3.3)∂lnL∂k=(r-1)ln(1+x(r)-c)[1+x(r)-c]-k1-(1+x(r)-c)-k-(n-s)ln(1+x(s)-c)+(s-r+1)k-∑i=rsln(1+x(i)-c),(3.4)∂lnL∂c=-(r-1)k(1+x(r)-c)-k-1x(r)-cln(x(r))1-(1+x(r)-c)-k+(n-s)kx(s)-cln(x(s))1+x(s)-c+(s-r+1)c-∑i=rsln(x(i))-(k+1)∑i=rsx(i)-cln(x(i))1+x-c(i).
From (3.3) the maximum likelihood estimator of k is expressed by
(3.5)(n-s)ln(1+x(s)-c^)+∑i=1sln(1+x(i)-c^)=(s-r+1)k^+(r-1)ln(1+x(r)-c^)[1+x(r)-c^]k^-1.
From (3.4) the maximum likelihood estimator of c is expressed by
(3.6)(n-s)kx(s)-c^ln(x(s))1+x(s)-c^+(k+1)∑i=rsx(i)-c^ln(x(i))1+x(i)-c^=(r-1)k(1+x(r)-c^)-k-1x(r)-c^ln(x(r))1-(1+x(r)-c^)-k+(s-r+1)c^+∑i=rsln(x(i)).
Since the closed form solution to nonlinear equations (3.5) and (3.6) is very hard to obtain, Newton-Raphson method is applied for solving the nonlinear equations simultaneously to obtain c^,k^.


### 3.1.1. Asymptotic Variances Covariance Matrix

The asymptotic variances covariance matrix of the parameters (c and k) is obtained by inverting the Fisher information matrix
(3.7)Iij=E[-∂2L∂θi∂θj],i,j=1,2,
where θ1,θ2=c or k. Hence the approximate variance-covariance matrix is given by
(3.8)[∂2lnL∂c2|c^,k^∂2lnL∂c∂k|c^,k^∂2lnL∂k∂c|c^,k^∂2lnL∂k2|c^,k^]-1=[Var(c^)Cov(c^,k^)Cov(k^,c^)Var(k^)].
The elements of matrix I are the negative of second derivatives of the natural logarithm of likelihood function defined in (3.2).The elements of matrixI are given as follows:
(3.9)∂2lnL∂k2=-(s-r+1)k2-(r-1)[ln(1+x(r)-c)]2(1+x(r)-c)-k[1-(1+x(r)-c)-k]2,(3.10)∂2lnL∂c2=(r-1)kln(x(r))[𝒜x(r)cln(x(r))-x(r)-c(k+1)(1+x(r)-c)kx(r)-cln(x(r))-x(r)-cln(x(r))][(1+x(r)c)k+1-(1+x(r)c)]2-(n-s)klnx(s)x(s)-clnx(s)(1+x(s)c)2-(s-r+1)c2+(k+1)∑i=rsln(x(i))2(1+x(i)-c)2,(3.11)∂2lnL∂c∂k=-∑i=rsx(i)-cln(x(i))1+x(i)-c+(n-s)x(s)-cln(x(s))1+x(s)-c-(r-1)x(r)-clnx(r)1+x(r)-c+(r-1)(1-(1+x(r)-c)-k)x(r)-cln(x(r))(1+x(r)-c)((1-(1+x(r)-c)-k))2-[ln(1+x(r)-c)]k(1+x(r)-c)-k-1+x(r)-clnx(r),
where 𝒜 denotes [(1+x(r)-c)k+1-(1+x(r)-c)]. The maximum likelihood estimators c^, k^ have asymptotic variance-covariance matrix  defined by inverting the Fisher information matrix.


## 3.1.1. Asymptotic Variances Covariance Matrix

The asymptotic variances covariance matrix of the parameters (c and k) is obtained by inverting the Fisher information matrix
(3.7)Iij=E[-∂2L∂θi∂θj],i,j=1,2,
where θ1,θ2=c or k. Hence the approximate variance-covariance matrix is given by
(3.8)[∂2lnL∂c2|c^,k^∂2lnL∂c∂k|c^,k^∂2lnL∂k∂c|c^,k^∂2lnL∂k2|c^,k^]-1=[Var(c^)Cov(c^,k^)Cov(k^,c^)Var(k^)].
The elements of matrix I are the negative of second derivatives of the natural logarithm of likelihood function defined in (3.2).The elements of matrixI are given as follows:
(3.9)∂2lnL∂k2=-(s-r+1)k2-(r-1)[ln(1+x(r)-c)]2(1+x(r)-c)-k[1-(1+x(r)-c)-k]2,(3.10)∂2lnL∂c2=(r-1)kln(x(r))[𝒜x(r)cln(x(r))-x(r)-c(k+1)(1+x(r)-c)kx(r)-cln(x(r))-x(r)-cln(x(r))][(1+x(r)c)k+1-(1+x(r)c)]2-(n-s)klnx(s)x(s)-clnx(s)(1+x(s)c)2-(s-r+1)c2+(k+1)∑i=rsln(x(i))2(1+x(i)-c)2,(3.11)∂2lnL∂c∂k=-∑i=rsx(i)-cln(x(i))1+x(i)-c+(n-s)x(s)-cln(x(s))1+x(s)-c-(r-1)x(r)-clnx(r)1+x(r)-c+(r-1)(1-(1+x(r)-c)-k)x(r)-cln(x(r))(1+x(r)-c)((1-(1+x(r)-c)-k))2-[ln(1+x(r)-c)]k(1+x(r)-c)-k-1+x(r)-clnx(r),
where 𝒜 denotes [(1+x(r)-c)k+1-(1+x(r)-c)]. The maximum likelihood estimators c^, k^ have asymptotic variance-covariance matrix  defined by inverting the Fisher information matrix.


## 3.2. Numerical Illustration

In estimation problem, it is required to study the properties of the derived expressions for the estimators theoretically or analytically. Sometimes it seems very difficult to study the properties of the estimators theoretically because of the complicated formula of the estimators. Consequently, a simulation study will be set up, treating separately the sampling distribution of the estimators. Simulation studies have been performed using MathCAD (2001) for illustrating the theoretical results of the estimation problem. The simulation procedures will described below.Step 1.
Generate a random sample of size 10, 20, 40, 60, 80, and 100 from Burr type III distribution. The generation of Burr type III distribution is very simple ifU has a uniform (0,1) random number, then Y=[U(-1/k)-1](-1/c) follows a Burr type III distribution. In double censored type II, the true parameters selected values are (c=1.2,k=1.5), (c=1.2,k=2), (c=2.5,k=1.5), and (c=2.5,k=2).Step 2.
Choose censored failure(r=0.6,s=6).Step 3.
For each sample and for the four sets of parameters distribution were estimated under doubly censored type II.Step 4.
Newten-Raphson method was used for solving the nonlinear equations forc and k defined in (3.4) and (3.6).Step 5.
Equations (3.9) and (3.10) were used to obtain the variance covariance matrix of (c^,k^).Results are tabulated in Table1.Table 1
The maximum likelihood estimator, the standard deviation, and covariance of the Burr distribution with two parameter under doubly censored sample when (a) (c=1.2, k=1.5), (b) (c=1.2, k=2), (c) (c=2.5, k=1.5), (d) (c=2.5, k=2).
(a)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
4.619
3.689
7.841
4.341
58.814
10
7
2
4.534
3.524
4.672
4.172
50.042
9
4.338
3.308
4.722
3.202
22.784
10
6.874
3.874
6.654
4.154
66.589
20
15
2
6.483
2.483
6.439
3.939
64.752
18
6.147
3.147
6.307
3.807
41.058
20
6.498
3.498
6.802
4.302
74.941
40
30
3
5.384
3.384
6.471
3.971
62.603
38
5.287
2.287
5.629
3.129
39.913
30
5.692
2.692
6.794
4.294
62.054
60
45
3
5.542
2.542
6.678
4.178
47.829
58
5.201
2.201
5.489
2.989
16.475
40
5.448
2.448
6.824
4.324
78.475
80
65
5
5.707
2.707
6.671
4.171
66.942
78
5.334
2.334
5.391
2.891
11.313
50
6.689
3.689
6.841
4.341
58.814
100
75
5
6.524
3.524
6.672
4.172
50.042
98
6.308
3.308
5.702
3.202
30.784
(b)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
6.689
3.689
6.841
4.341
58.814
10
7
2
6.524
3.524
6.672
4.172
50.042
9
6.308
3.308
5.702
3.202
30.784
10
6.874
3.874
6.654
4.154
66.589
20
15
2
6.483
3.483
6.439
3.939
64.752
18
6.147
3.147
6.307
3.807
41.058
20
6.498
3.498
6.802
4.302
74.941
40
30
3
6.384
3.384
6.471
3.971
62.603
38
5.287
2.287
5.629
3.129
39.913
30
5.692
2.692
6.794
4.294
62.054
60
45
3
5.542
2.542
6.678
4.178
47.829
58
5.201
2.201
5.489
2.989
16.475
40
5.448
2.448
6.824
4.324
78.475
80
65
5
5.707
2.707
6.671
4.171
66.942
78
5.334
2.334
5.391
2.891
11.313
50
6.689
3.689
6.841
4.341
58.814
100
75
5
6.524
3.524
6.672
4.172
50.042
98
6.308
3.308
5.702
3.202
30.784
(c)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
5.952
2.952
5.159
3.159
69.043
10
7
2
5.712
2.712
4.914
2.914
55.876
9
5.819
2.819
4.249
2.249
38.874
10
5.984
2.984
5.147
3.147
68.497
20
15
2
5.147
2.147
4.271
2.271
24.984
18
5.078
2.078
4.014
2.014
14.872
20
5.618
2.618
5.081
3.081
69.187
40
30
3
5.491
2.491
4.941
2.941
57.078
38
5.107
2.107
4.028
2.028
23.174
30
5.347
2.347
4.627
2.627
70.842
60
45
3
5.148
2.148
4.489
2.489
58.247
58
5.004
2.004
4.309
2.309
41.872
40
5.334
2.334
4.658
2.658
74.194
80
65
5
5.238
2.238
4.748
2.748
60.827
78
4.997
1.997
4.102
2.102
47.145
50
0.447
0.553
1.39
0.19
1.987
100
75
5
0.474
0.526
1.317
0.117
1.854
98
1.218
0.218
1.234
0.034
1.823
(d)
n
s
r
c
^
k
^
Co
var  (c^,k^)
MLE
Standard deviation
MLE
Standard deviation
5
1.395
0.395
2.717
0.717
42.175
10
7
2
1.501
0.501
2.585
0.585
35.214
9
1.014
0.014
2.461
0.461
18.258
10
1.284
0.284
2.648
0.648
40.954
20
15
2
1.481
0.481
2.441
0.441
33.254
18
1.008
0.008
2.314
0.314
16.418
20
1.658
0.658
2.995
0.995
68.284
40
30
3
1.018
0.018
2.147
0.147
50.258
38
1.001
0.001
1.654
0.346
52.547
30
1.228
0.228
2.954
0.954
68.952
60
45
3
1.109
0.109
2.756
0.756
53.218
58
1.019
0.019
2.341
0.341
40.021
40
1.721
0.721
2.481
0.481
77.248
80
65
5
1.856
0.856
2.441
0.441
80.514
78
1.007
0.007
1.761
0.239
54.951
50
1.395
0.395
2.717
0.717
42.175
100
75
5
1.501
0.501
2.585
0.585
35.214
98
1.014
0.014
2.461
0.461
18.258From Table1, we note that the standard deviation decreases when n is increasing. Also, we note that, when c and k increase the standard deviation of the estimators decrease. Similarly, the maximum likelihood estimator of the parameters has the same behaviors when the sample size becomes large and the properties of two parameters c and k at (1.2 and 1.5), respectively is better than the other values.


## 3.3. Bayesian Analysis for Burr Distribution Type III Based on Trimmed Samples

The Bayesian approach allows both sample and prior information to be incorporated into analysis, which will improve the quality of the inferences. In this section, Bayesian estimators and posterior variance of the shapes parameters are obtained in the case of double censored type II. The prior distribution could be “none informative” such as a flat, uniform distribution, assuming equal probability for the parameter value within a realistic range, and when the prior could be more informative such as a normal distribution or other possible distribution that represents an initial assessment of what is known about the parameter before the collection of data and the analysis. Moreover, a numerical examples are given for illustration study.


### 3.3.1. Bayesian Estimation for Burr Distribution Type III Parameters Based on Trimmed Sample in Case of Noninformative Prior

The likelihood function takes the form in (3.1). Assumed that the parameters c, k have independent prior distribution and let the noninformative prior (NIP) for cand k are, respectively, given by:
(3.12)u1(c)αc-1c>0,u2(k)αk-1k>0.
Consequently, the joint (NIP) will be defined as follows:
(3.13)u(c,k)=(ck)-1c,k>0.
The joint posterior density functions of c, k under double censored sample type II will be
(3.14)g(c,k∣x)=u(c,k)L(x∣c,k)λc>0,k>0,g(c,k∣x)=(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]λ,
where λ is the normalized constant defined as follows:
(3.15)∬0∞(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Now, the marginal posterior of one parameter is obtained by integrating the joint posterior distribution with respect to the other parameter, hence the marginal posterior probability density function of c will be
(3.16)g1(c∣k,x)=c-1∫0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dkλ.
Similarly integrating the joint posterior (3.14) with respect to k. Then, the marginal posterior of k is
(3.17)g2(k∣c,x)=k-1∫0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c-1)(1+x(i)-c)-(k+1)]dcλ.It is well known that under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance, a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters(c,k) are expressed as follows:
(3.18)E(c∣k,x)=c~,c~=∬0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(c~∣k,x)=E(c~-c)2,var(c~∣k,x)=∬0∞(c~-c)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,E(k∣c,x)=k~,k~=∬0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(k~∣c,x)=E(k~-k)2,var(k~∣c,x)=∬0∞(k~-k)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Equations (3.18) are very difficult to be solved exactly so that an iterative procedure is needed to solve these equations numerically. Using the statistical package, MathCAD (2001), the posterior mean and posterior variance of the shapes parameter (c,k) will be obtained.The posterior mean and the posterior variance ofc, k are obtained numerically as described below.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain posterior variance of the shape parameter(c,k) using (3.18).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameters(c,k) for all sample size and for sets of parameter values were obtained.
Numerical results are summarized in Table2.Table 2
The posterior mean and posterior variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) in case of noninformative prior.
n
r
S
c
k
Posterior mean
Posterior variance
Posterior mean
Posterior variance
5
1.017
0.685
0.671
0.765
10
2
7
0.701
0.514
0.569
0.648
9
0.608
0.338
0.408
0.491
10
1.010
0.742
0.948
0.785
20
2
15
0.871
0.537
0.653
0.676
18
0.691
0.338
0.527
0.527
20
0.437
0.674
0.741
0.576
40
3
30
0.288
0.582
0.687
0.477
38
0.119
0.449
0.354
0.345
30
0.609
0.762
0.671
0.758
60
3
45
0.547
0.684
0.507
0.862
58
0.489
0.449
0.407
0.647
40
0.827
0.761
0.718
0.901
80
5
65
0.667
0.895
0.651
0.763
78
0.524
0.612
0.581
0.522
50
0.718
0.761
0.718
0.901
100
5
75
0.651
0.895
0.651
0.763
98
0.581
0.612
0.581
0.522


### 3.3.2. Bayes Estimator under LINEX Loss Function

Under the assumption that minimal loss function occursθ~~=θ, the LINEX loss function for θ=θ(c,k) can be expressed as defined by Zellner [14], the Bayes estimator θ~~ of θ under LINEX loss function is
(3.19)θ~~=-1alog(Eθ(exp(-aθ))),θ~~-1aln∫e-aθf(θ∣x)dθ.Now, in (3.19) put θ=c, then the Bayes estimate c~~ of parameter c relative to the LINEX loss function is
(3.20)c~~=-1aln∫e-acg1(c∣k,x)dc.
Set θ=k in (3.19), then the Bayes estimate k~~ of parameter k relative to the LINEX loss function is
(3.21)k~~=-1aln∫e-akg2(k∣c,x)dk.
The equations (3.20) to (3.21) are very difficult to obtain their solutions exactly, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameters (c,k). The numerical procedures will describe as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain the posterior variance of the shape parameter(c,k) in (3.20) and (3.21).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table3.Table 3
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) under LINEX loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.698
0.396
2.596
1.631
10
2
7
1.353
0.205
1.700
0.810
9
1.189
0.292
1.663
0.726
10
1.468
0.267
2.552
1.819
20
2
15
1.238
0.166
2.032
1.619
18
1.237
0.178
1.773
1.152
20
1.382
0.294
2.343
1.198
40
3
30
1.092
0.200
1.411
0.848
38
1.022
0.196
0.936
0.847
30
1.075
0.197
1.543
1.237
60
3
45
1.013
0.174
1.018
1.062
58
0.988
0.170
0.935
1.138
40
1.275
0.162
1.007
1.188
80
5
65
0.986
0.182
0.340
1.193
78
0.906
0.151
1.762
2.269
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


### 3.3.3. Bayes Estimator under General Entropy (GE) Loss Function

The Bayes estimatorθ~^ under GE loss function is
(3.22)θ~^=(Eθ(θ-v))-1/v,θ~^=[∫θ-vf(θ∣x)dθ]-1/vv>0.
Now, in (3.22) put θ~^=c, then the Bayes estimate c~^ of parameter c relative to the GE loss function is
(3.23)c~^=[∫c-vg1(c∣k,x)dc]-1/v.
In (3.22), set θ~^=k, then the Bayes estimate k~^ of parameter k relative to the GE loss function is
(3.24)k~^=[∫k-vg2(k∣c,x)dk]-1/v.The equations (3.23) to (3.24) cannot have exact solutions, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameter. The numerical procedures will be described as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solving the nonlinear Bayesian for posterior variance of the shape parameter(c,k) in (3.23) and (3.24).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table4.Table 4
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) general entropy loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.641
0.685
3.825
0.765
10
2
7
1.375
0.514
2.614
0.648
9
1.255
0.338
1.661
0.491
10
1.474
0.742
3.423
0.785
20
2
15
1.299
0.537
2.490
0.676
18
1.147
0.338
1.550
0.527
20
1.294
0.674
3.076
0.576
40
3
30
1.087
0.582
1.378
0.477
38
1.016
0.449
1.263
0.345
30
1.032
0.762
1.672
0.758
60
3
45
0.977
0.684
1.173
0.862
58
0.924
0.449
1.038
0.647
40
1.162
0.761
0.860
0.901
80
5
65
1.034
0.895
0.396
0.763
78
0.888
0.612
0.185
0.522
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


## 3.3.1. Bayesian Estimation for Burr Distribution Type III Parameters Based on Trimmed Sample in Case of Noninformative Prior

The likelihood function takes the form in (3.1). Assumed that the parameters c, k have independent prior distribution and let the noninformative prior (NIP) for cand k are, respectively, given by:
(3.12)u1(c)αc-1c>0,u2(k)αk-1k>0.
Consequently, the joint (NIP) will be defined as follows:
(3.13)u(c,k)=(ck)-1c,k>0.
The joint posterior density functions of c, k under double censored sample type II will be
(3.14)g(c,k∣x)=u(c,k)L(x∣c,k)λc>0,k>0,g(c,k∣x)=(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]λ,
where λ is the normalized constant defined as follows:
(3.15)∬0∞(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Now, the marginal posterior of one parameter is obtained by integrating the joint posterior distribution with respect to the other parameter, hence the marginal posterior probability density function of c will be
(3.16)g1(c∣k,x)=c-1∫0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dkλ.
Similarly integrating the joint posterior (3.14) with respect to k. Then, the marginal posterior of k is
(3.17)g2(k∣c,x)=k-1∫0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c-1)(1+x(i)-c)-(k+1)]dcλ.It is well known that under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance, a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters(c,k) are expressed as follows:
(3.18)E(c∣k,x)=c~,c~=∬0∞k-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(c~∣k,x)=E(c~-c)2,var(c~∣k,x)=∬0∞(c~-c)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,E(k∣c,x)=k~,k~=∬0∞c-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(k~∣c,x)=E(k~-k)2,var(k~∣c,x)=∬0∞(k~-k)2×(ck)-1[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Equations (3.18) are very difficult to be solved exactly so that an iterative procedure is needed to solve these equations numerically. Using the statistical package, MathCAD (2001), the posterior mean and posterior variance of the shapes parameter (c,k) will be obtained.The posterior mean and the posterior variance ofc, k are obtained numerically as described below.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain posterior variance of the shape parameter(c,k) using (3.18).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameters(c,k) for all sample size and for sets of parameter values were obtained.
Numerical results are summarized in Table2.Table 2
The posterior mean and posterior variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) in case of noninformative prior.
n
r
S
c
k
Posterior mean
Posterior variance
Posterior mean
Posterior variance
5
1.017
0.685
0.671
0.765
10
2
7
0.701
0.514
0.569
0.648
9
0.608
0.338
0.408
0.491
10
1.010
0.742
0.948
0.785
20
2
15
0.871
0.537
0.653
0.676
18
0.691
0.338
0.527
0.527
20
0.437
0.674
0.741
0.576
40
3
30
0.288
0.582
0.687
0.477
38
0.119
0.449
0.354
0.345
30
0.609
0.762
0.671
0.758
60
3
45
0.547
0.684
0.507
0.862
58
0.489
0.449
0.407
0.647
40
0.827
0.761
0.718
0.901
80
5
65
0.667
0.895
0.651
0.763
78
0.524
0.612
0.581
0.522
50
0.718
0.761
0.718
0.901
100
5
75
0.651
0.895
0.651
0.763
98
0.581
0.612
0.581
0.522


## 3.3.2. Bayes Estimator under LINEX Loss Function

Under the assumption that minimal loss function occursθ~~=θ, the LINEX loss function for θ=θ(c,k) can be expressed as defined by Zellner [14], the Bayes estimator θ~~ of θ under LINEX loss function is
(3.19)θ~~=-1alog(Eθ(exp(-aθ))),θ~~-1aln∫e-aθf(θ∣x)dθ.Now, in (3.19) put θ=c, then the Bayes estimate c~~ of parameter c relative to the LINEX loss function is
(3.20)c~~=-1aln∫e-acg1(c∣k,x)dc.
Set θ=k in (3.19), then the Bayes estimate k~~ of parameter k relative to the LINEX loss function is
(3.21)k~~=-1aln∫e-akg2(k∣c,x)dk.
The equations (3.20) to (3.21) are very difficult to obtain their solutions exactly, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameters (c,k). The numerical procedures will describe as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear equations to obtain the posterior variance of the shape parameter(c,k) in (3.20) and (3.21).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table3.Table 3
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) under LINEX loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.698
0.396
2.596
1.631
10
2
7
1.353
0.205
1.700
0.810
9
1.189
0.292
1.663
0.726
10
1.468
0.267
2.552
1.819
20
2
15
1.238
0.166
2.032
1.619
18
1.237
0.178
1.773
1.152
20
1.382
0.294
2.343
1.198
40
3
30
1.092
0.200
1.411
0.848
38
1.022
0.196
0.936
0.847
30
1.075
0.197
1.543
1.237
60
3
45
1.013
0.174
1.018
1.062
58
0.988
0.170
0.935
1.138
40
1.275
0.162
1.007
1.188
80
5
65
0.986
0.182
0.340
1.193
78
0.906
0.151
1.762
2.269
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


## 3.3.3. Bayes Estimator under General Entropy (GE) Loss Function

The Bayes estimatorθ~^ under GE loss function is
(3.22)θ~^=(Eθ(θ-v))-1/v,θ~^=[∫θ-vf(θ∣x)dθ]-1/vv>0.
Now, in (3.22) put θ~^=c, then the Bayes estimate c~^ of parameter c relative to the GE loss function is
(3.23)c~^=[∫c-vg1(c∣k,x)dc]-1/v.
In (3.22), set θ~^=k, then the Bayes estimate k~^ of parameter k relative to the GE loss function is
(3.24)k~^=[∫k-vg2(k∣c,x)dk]-1/v.The equations (3.23) to (3.24) cannot have exact solutions, an iterative procedure is needed to solve these equations numerically using MathCAD (2001) statistical package to obtain posterior variance of shapes parameter. The numerical procedures will be described as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solving the nonlinear Bayesian for posterior variance of the shape parameter(c,k) in (3.23) and (3.24).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table4.Table 4
The expectation mean and variance of the Burr distribution with two parameter distribution under doubly censored sample when (c=1.2, k=1.5) general entropy loss function.
n
r
S
c
k
Expectation mean
Variance
Expectation mean
Variance
5
1.641
0.685
3.825
0.765
10
2
7
1.375
0.514
2.614
0.648
9
1.255
0.338
1.661
0.491
10
1.474
0.742
3.423
0.785
20
2
15
1.299
0.537
2.490
0.676
18
1.147
0.338
1.550
0.527
20
1.294
0.674
3.076
0.576
40
3
30
1.087
0.582
1.378
0.477
38
1.016
0.449
1.263
0.345
30
1.032
0.762
1.672
0.758
60
3
45
0.977
0.684
1.173
0.862
58
0.924
0.449
1.038
0.647
40
1.162
0.761
0.860
0.901
80
5
65
1.034
0.895
0.396
0.763
78
0.888
0.612
0.185
0.522
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522


## 3.4. Bayesian Estimation for Burr Distribution Type III Parameters Based on Trimmed Sample in Case of Informative Prior (IP)

Assume that the parametersc, k have independent prior distribution and let the informative prior (IP) for the parameter k is given by the gamma density as follow:
(3.25)π1(k∣c)=cα+1Γα+1kαe-kck>0,c>0,α+1>0,
and the prior density function of c is the exponential density as follow:
(3.26)π2(c)=1βe-c/βc>0,β>0.
Then the joint prior density of c and k, given by
(3.27)π(c,k)=π1(k∣c)π2(c),π(c,k)=cα+1βΓα+1kαe-(k+1/β)cc>0,k>0.
The joint posterior density function of c, k using double censored sample type II will be
(3.28)f(c,k∣x)=π(c,k)L(x∣c,k)Ec>0,k>0=cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]EβΓα+1,
where E is the normalized constant equal to
(3.29)E=∬0∞cα+βΓα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Now, the marginal posterior of one parameter is obtained by integrating the joint posterior distribution with respect to the other parameter, hence the marginal posterior probability density function of c is
(3.30)f1(c∣k,x)=cα+1βΓα+1∫0∞kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dkλ.
Similarly integrating the joint posterior (3.28) with respect to k the marginal posterior of k is
(3.31)f2(k∣c,x)=kαβΓα+1∫0∞cα+1e-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcλ.Hence, under a squared error loss function, the Bayes estimator of the parameter will be its posterior expectation. To obtain the posterior mean and posterior variance a numerical integration is required. Then, the posterior mean and posterior variance of the shape parameters(c,k) are expressed as follows:
(3.32)E(c∣k,x)=c~˘,c~˘=1βΓα+1∬0∞cα+2kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(c~˘∣k,x)=E(c~˘-c)2,var(c~˘∣k,x)=1βΓα+1∬0∞(c~˘-c)2∬0∞cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,E(k∣c,x)=k~˘,k~˘=1βΓα+1∬0∞cα+1kα+1e-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk,var(K~˘∣c,x)=E(K~˘-k)2,var(K~˘∣c,x)=1βΓα+1∬0∞(k~˘-k)2∬0∞cα+1kαe-(k+1/β)c[1-(1+x(r)-c)-k]r-1[(1+x(s)-c)-k]n-s×∏i=rs[ckx(i)-(c+1)(1+x(i)-c)-(k+1)]dcdk.
Nonlinear equations (3.32) are very difficult to be solved exactly an iterative procedure is needed to solve these equations numerically. By using the statistical package, MathCAD (2001), the posterior mean and posterior variance of the shape parameters (c,k) will be obtained.The posterior mean and the posterior variance ofc, k are obtained numerically in the following procedures as follow.Repeat Steps1 and 2 in Section 3.2.Step 3.
Solve the nonlinear Bayesian for posterior variance of the shape parameter(c,k) in (3.32).Step 4.
The posterior mean and the posterior variance of the estimators for the shape parameter(c,k) for all sample size and for sets of parameters were obtained.Numerical results are summarized in Table5.Table 5
The posterior mean and posterior variance of the Burr distribution with two parameter under doubly censored sample when (c=1.2, k=1.5) in case of informative prior.
n
r
s
c
k
Posterior mean
Posterior variance
Posteriormean
Posterior variance
5
0.495
0.204
1.017
0.719
10
2
7
0.384
0.214
0.701
0.645
9
0.258
0.217
0.608
0.201
10
0.234
0.391
1.010
0.604
20
2
15
0.128
8.215
0.871
0.556
18
0.087
0.319
0.691
0.493
20
0.482
0.339
0.437
0.911
40
3
30
0.352
0.337
0.288
0.425
38
0.241
0.190
0.119
0.307
30
0.852
0.199
0.609
0.827
60
3
45
0.729
0.160
0.547
0.647
58
0.635
0.139
0.489
0.558
40
0.957
1.171
0.827
0.728
80
5
65
0.721
0.206
0.667
0.661
78
0.408
0.078
0.524
0.337
50
0.992
0.761
0.718
0.901
100
5
75
0.745
0.895
0.651
0.763
98
0.604
0.612
0.581
0.522It is noted that the posterior mean decreases whenn is increasing. Similarly the posterior variance of the parameters has the same behaviors when the sample size becomes large.



---

*Source: 250393-2012-11-18.xml*