On Bayesian Analysis of Burr Type VII Distribution under Different Censoring Schemes

Abstract

This paper includes the Bayesian analysis of Burr type VII distribution. Three censoring schemes, namely, left censoring, singly type II censoring, and doubly type II censoring have been used for posterior estimation. The results of different censoring schemes have been compared with those under complete samples. The comparative study among the performance of different censoring schemes has also been made. Two noninformative (uniform and Jeffreys) priors have been assumed to derive the posterior distributions under each case. The performance of Bayes estimators has been compared in terms of posterior risks under a simulation study.

1. Introduction

Burr [1] introduced twelve forms of the Burr distribution. However, most of the authors have considered the estimation of Burr type XII distribution. The Burr type VII distribution has rarely received any attention. Wahed [2] presented Bayes estimators for the parameters of Burr type XII distribution under the symmetric squared error loss function and the asymmetric linear exponential loss function based on a simple prior distribution. As the estimator turns out to be ratios of integrals, different approximation techniques have been used to obtain approximate Bayes estimators. Real life example has been used to demonstrate the application of Burr type XII distribution. Dasgupta [3] discussed that under certain conditions, the distribution of Burr can be shown to follow an extreme value distribution. Hence, a result on extremal process based on stationary sequence has been proved. Some data sets have been analyzed, and applications of the results have been indicated. Makhdoom and Jafari [4] obtained Bayesian estimators for the shape parameter of the Burr Type XII distribution using grouped and ungrouped data and also consider relationship between them. Bayes point and interval estimators have been derived. Squared error and precautionary loss functions have been considered for the posterior analysis. Monte Carlo simulation has been used to compare the performance of different estimators. Panahi and Asadi [5] considered the statistical inferences based on a Type-II hybrid censored sample from a Burr type XII distribution. As the maximum likelihood estimators cannot be obtained in closed form, a simple fixed point type algorithm has been proposed to compute the maximum likelihood estimators. The approximate confidence intervals for the parameters based on the s-normal approximation to the asymptotic distribution of MLE have been constructed. Bayes estimates of the unknown parameters have also been obtained under the Linex loss function using two approximations. Monte Carlo simulations have been performed to observe the behavior of the proposed methods.

The probability density function of Burr type VII distribution is 𝑓(𝑥)=𝜃2^−𝜃sech²(𝑥){1+tanh(𝑥)}^𝜃−1,−∞<𝑥<∞,𝜃>0. (1) And the cumulative distribution function of the distribution is 𝐹(𝑥)=2^−𝜃{1+tanh(𝑥)}^𝜃. (2) The probability density function and cumulative distribution function of the Burr type VII distribution can be, respectively, presented as 𝑓(𝑥)=𝜃sech²(𝑥)𝑒^{−ln{1+tanh(𝑥)}}𝑒^{−𝜃ln{(1+tanh(𝑥))/2}−1},𝐹(𝑥)=𝑒^{−𝜃ln{(1+tanh(𝑥))/2}−1}. (3) Under inverse transformation method of simulation the random numbers can be generated from the distribution by using the following formula, where 𝑈 is uniformly distributed random variable: 𝑥=tanh⁻¹𝑢2^{−𝜃1/𝜃}−1, (4) where tanh⁻¹(𝑥)=(1/2)ln((1+𝑥)/(1−𝑥)).

2. Posterior Analysis under Complete Data

In this section, the Bayes estimators along with posterior risks have been derived under the assumption of uniform and Jeffreys priors using complete data. Squared error loss function (SELF) and precautionary loss function (PLF) have been used for estimation. In order to derive the Bayes estimators and corresponding risks, the first step is to obtain the likelihood function. The likelihood function for a sample of size “𝑛” observation is 𝐿𝜃∣𝑥∝𝜃^𝑛𝑒^−𝜃𝑇, (5) where 𝑇=∑^𝑛_𝑖=1ln{(1+tanh(𝑥_𝑖))/2}⁻¹.

The uniform prior is assumed to be 𝑝(𝜃)∝1;𝜃>0. (6) The posterior distribution under the assumption of uniform prior is 𝑝𝜃∣𝑥=[𝑇]^𝑛+1Γ()𝜃𝑛+1^𝑛𝑒^−𝜃𝑇,𝜃>0. (7) The Bayes estimators and corresponding posterior risks under squared error loss function (SELF) can be derived by using the following formulae: 𝜃_SELF=𝐸𝜃∣𝑥,𝜌𝜃_SELF=𝐸𝜃²∣𝑥−𝐸𝜃∣𝑥². (8) Bayes estimator and posterior risk under uniform prior using SELF are 𝜃_SELF=𝑛+1𝑇,𝜌𝜃_SELF=𝑛+1𝑇². (9) Similarly, The Bayes estimators and corresponding posterior risks under precautionary loss function (PLF) can be derived by using the following formulae: 𝜃_PLF=𝐸𝜃²∣𝑥^1/2,𝜌𝜃_PLF=2𝐸𝜃²∣𝑥^1/2−𝐸𝜃∣𝑥. (10) The derivations of Bayes estimators and associated posterior risks using precautionary loss function have not been presented in the paper. These can easily be derived from the concerned posterior distributions.

The Jeffreys prior has been derived to be 𝑝(𝜃)∝1𝜃,𝜃>0. (11) The posterior distribution under the assumption of Jeffreys prior is 𝑝𝜃∣𝑥=𝑇^𝑛Γ(𝑛)𝜃^𝑛−1𝑒^−𝜃𝑇,𝜃>0. (12) Bayes estimator and risk under Jeffreys prior using SELF are 𝜃_SELF=𝑛𝑇,𝜌𝜃_SELF=𝑛𝑇². (13)

3. Posterior Analysis under Left Censored Samples

Let 𝑋_𝑟+1,…,𝑋_𝑛 be last 𝑛−𝑟 order statistics from a sample of size 𝑛 from Burr type VII distribution. Then the likelihood function for the 𝑋_𝑟+1,…,𝑋_𝑛 observations is 𝐿𝜃∣𝑥∝𝐹𝑥_𝑟+1^𝑟𝑛_𝑖=𝑟+1𝑓𝑥_𝑖,𝐿𝜃∣𝑥∝𝜃^𝑛−𝑟𝑒^{−𝜃𝜉1𝑟}, (14) where 𝜉_1𝑟𝑥=𝑟ln1+tanh_𝑟+12⁻¹+^𝑛_𝑖=𝑟+1𝑥ln1+tanh_𝑖2⁻¹. (15) The posterior distribution under uniform prior is 𝑝𝜃∣𝑥=𝜉_1𝑟^{𝑛−𝑟+1}Γ()𝜃𝑛−𝑟+1^𝑛−𝑟𝑒^{−𝜃𝜉1𝑟}. (16) The Bayes estimator and posterior risk under uniform prior using SELF are 𝜃_SELF=𝑛−𝑟+1𝜉_1𝑟,𝜌𝜃_SELF=𝑛−𝑟+1𝜉_1𝑟². (17) The posterior distribution under Jeffreys prior is 𝑝𝜃∣𝑥=𝜉_1𝑟^𝑛−𝑟Γ()𝜃𝑛−𝑟^{𝑛−𝑟−1}𝑒^{−𝜃𝜉1𝑟}. (18) The Bayes estimator and posterior risk under Jeffreys prior using SELF are 𝜃_SELF=𝑛−𝑟𝜉_1𝑟,𝜌𝜃_{𝑆𝐸𝐿𝐹}=𝑛−𝑟𝜉_1𝑟². (19)

4. Posterior Analysis under Singly Type II Censored Samples

Suppose “𝑛” items are put on a life-testing experiment, and only first “𝑚” failure times have been observed, that is, 𝑥₁<𝑥₂⋯<𝑥_𝑚 and remaining “𝑛−𝑚” items are still working. Under the assumptions that the lifetimes of the items are independently and identically distributed Burr type VII random variable, the likelihood function of the observed data, is 𝐿𝜃∣𝑥∝^𝑚_𝑖=1𝑓𝑥_𝑖1−𝐹𝑥_𝑚^𝑛−𝑚,𝐿𝜃∣𝑥∝^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑚_𝑚𝑒^{−𝜃𝜓1𝑗}, (20)where𝜓_1𝑗=^𝑚_𝑖=1𝑥ln1+tanh_𝑖2⁻¹𝑥+𝑗ln1+tanh_𝑚2⁻¹. (21) The posterior distribution under uniform prior is 𝑝𝜃∣𝑥=1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑚_𝑚𝑒^{−𝜃𝜓1𝑗}, (22) where 𝐶₁=∑^𝑛−𝑚_𝑗=0(−1)^𝑗_𝑗^𝑛−𝑚(Γ(𝑚+1)/(𝜓_1𝑗)^𝑚+1).

The Bayes estimator and posterior risk under uniform prior using SELF are 𝜃_SELF=1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+2)𝜓_1𝑗^𝑚+2,𝜌𝜃_SELF=1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+3)𝜓_1𝑗^𝑚+3−1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+2)𝜓_1𝑗^𝑚+22. (23) The posterior distribution under Jeffreys prior is 𝑝𝜃∣𝑥=1𝐶₂^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑚_𝑚−1𝑒^{−𝜃𝜓1𝑗}, (24) where 𝐶₂=∑^𝑛−𝑚_𝑗=0(−1)^𝑗_𝑗^𝑛−𝑚(Γ(𝑚)/(𝜓_1𝑗)^𝑚).

The Bayes estimator and posterior risk under Jeffreys prior using SELF are 𝜃_SELF=1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+1)𝜓_1𝑗^𝑚+1,𝜌𝜃_SELF=1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+2)𝜓_1𝑗^𝑚+2−1𝐶₁^𝑛−𝑚_𝑗=0(−1)^𝑗𝑗𝑛−𝑚Γ(𝑚+1)𝜓_1𝑗^𝑚+12. (25)

5. Posterior Analysis under Doubly Type II Censored Samples

Consider a random sample of size “𝑛” from an Burr type 𝑋 distribution, and let 𝑥_𝑟,…,𝑥_𝑠 be the ordered observations remaining when the “𝑟−1” smallest observations and the “𝑛−𝑠” largest observations have been censored. The likelihood function for 𝜃 given the type II doubly censored sample 𝑥=(𝑥_𝑟,…,𝑥_𝑠) is 𝐿𝜃∣𝑥∝𝐹𝑥_𝑟∣𝜃^𝑟−11−𝐹𝑥_𝑠∣𝜃^{𝑠𝑛−𝑠}_𝑖=𝑟𝑓𝑥_𝑖,𝐿∣𝜃𝜃∣𝑥∝^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑠_𝑘𝑒^{−𝜃𝜓2𝑗},𝜓_2𝑗=()𝑥𝑟−1ln1+tanh_𝑟2⁻¹+^𝑠_𝑖=𝑟𝑥ln1+tanh_𝑖2⁻¹𝑥+𝑗ln1+tanh_𝑠2⁻¹, (26) and 𝑘=𝑠−𝑟+1.

The posterior distribution under uniform prior is 𝑝𝜃∣𝑥=1𝐶₃^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑠_𝑘𝑒^{−𝜃𝜓2𝑗}, (27) where 𝐶₃=∑^𝑛−𝑠_𝑗=0(−1)^𝑗_𝑗^𝑛−𝑠(Γ(𝑘+1)/(𝜓_2𝑗)^𝑘+1).

The Bayes estimator and posterior risk under uniform prior using SELF are 𝜃_SELF=1𝐶₃^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+2)𝜓_2𝑗^𝑘+2,𝜌𝜃_SELF=1𝐶₃^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+3)𝜓_2𝑗^𝑘+3−1𝐶₃^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+2)𝜓_2𝑗^𝑘+22. (28) The posterior distribution under Jeffreys prior is 𝑝𝜃∣𝑥=1𝐶₄^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝜃𝑛−𝑠_𝑘−1𝑒^{−𝜃𝜓2𝑗}, (29) where 𝐶₄=∑^𝑛−𝑠_𝑗=0(−1)^𝑗_𝑗^𝑛−𝑠(Γ(𝑘)/(𝜓_2𝑗)^𝑘).

The Bayes estimator and posterior risk under Jeffreys prior using SELF are 𝜃_SELF=1𝐶₄^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+1)𝜓_2𝑗^𝑘+1,𝜌𝜃_SELF=1𝐶₄^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+2)𝜓_2𝑗^𝑘+2−1𝐶₄^𝑛−𝑠_𝑗=0(−1)^𝑗𝑗𝑛−𝑠Γ(𝑘+1)𝜓_2𝑗^𝑘+12. (30)

6. Simulation Study

Simulation study is a useful technique to assess and compare the performance of different estimators numerically. A simulation study often reflects the patterns of the real life data. Here, the inverse transformation technique of simulation has been used for 𝑛=50,70,100, and 150 under the parametric space 𝜃∈(3,6). As a single sample cannot fully describe the behaviors and properties of the estimators, the samples have been replicated 1000 times, and an average of the results has been presented. The magnitude of posterior risks, associated with each estimator, has been underlined in the tables.

The results of simulation study, presented in Tables 1–8, are obtained under complete and 20% censored samples. It is immediate from the above study that the magnitude of risk decreases with the increase in the sample size irrespective of nature (complete and censored) of the samples. The estimated value of the parameter converges to the true value very rapidly under complete data; however, in case of censored data the convergence is not that good. While, increase in the true parametric value and the censoring rate imposes a negative impact on the convergence and performance of the estimates. It can also be observed that the performance, in terms of convergence and posterior risk, of precautionary loss function is better than squared error loss function. It is because that the posterior distributions are asymmetric. It can also be assessed that the magnitudes of risks associated with estimates under Jeffreys prior are lesser as compared to those under uniform prior. In comparison of censoring schemes it is found that the left censored samples provide the best results. The estimates under doubly censored samples stand at second position performance wise.

Table 1 

Bayes estimators and risks under uniform prior using SELF.

Table 2 

Bayes estimators and risks under uniform prior using PLF.

Table 3 

Bayes estimators and risks under Jeffreys prior using SELF.

Table 4 

Bayes estimators and risks under Jeffreys prior using PLF.

Table 5 

Bayes estimators and risks under uniform prior using SELF.

Table 6 

Bayes estimators and risks under uniform prior using PLF.

Table 7 

Bayes estimators and risks under Jeffreys prior using SELF.

Table 8 

Bayes estimators and risks under Jeffreys prior using PLF.

7. Conclusions and Recommendations

From the findings of simulation study it can be concluded that in order to estimate the parameter of Burr type VII distribution under a Bayesian framework using censored data, the use of Jeffreys prior, precautionary loss function, and left censored samples can be preferred. The study can further be extended by using some other loss functions; informative priors and mixture of two are more components of Burr type VII distribution. The work on the mixture of two components of Burr type VII distribution is continued.