Abstract
Some corrections are made for the article mentioned above. These errors occured due to the floating errors in the computation of the Bayes risk which involves terms of large magnitude and opposite sign as sample size increases, that resulted in loss of accuracy in fixed precision computational tool such as Fortran language. For accurate numerical calculation, we now use the symbolic math package, Maple, to implement the calculation which can supply numerical answers accurate to any required degree of precision. Based on the new tables, some comments made earlier in Sec. 4 have been changed suitably.
The following items are the corrections to be made for the article “Exact Bayesian Variable Sampling Plans for the Exponential Distribution Based on Type-I and Type-II Hybrid Censored Samples,” pp. 1101–1116 in issue 37(6) by Lin et al. These errors occured due to the floating errors in the computation of the Bayes risk in Eq. (3) on p. 1104. In fact, the Bayes risk in Eq. (3) involves terms of large magnitude and opposite sign as sample size increases, that resulted in loss of accuracy in fixed precision computational tool such as Fortran language. For accurate numerical calculation, we now use the symbolic math package, Maple, to implement the calculation which can supply numerical answers accurate to any required degree of precision.
The corrections have affected the minimum Bayes risks R(n 0, r 0, T 0, ξ0) under the proposed sampling plans, and the comparison with the results based on ordinary Type-I and Type-II censored samples and thus, some of the comments made on pp. 1107 and 1115 (Secs. 4.1 and 4.2) need to be changed suitably.
On p. 1107, the last eight lines in Sec. 4.1 should be changed as follows. From these values, we observe that the minimum Bayes risks R(n 0, r 0, T 0, ξ0) under the proposed sampling plans based on Type-I hybrid censored samples match exactly with those of the plans based on the ordinary Type-I censored samples (Lin et al., Citation2010). Moreover, we find that the efficiencies under both cases are close to 1 and in most cases being more than 90%. With regard to the coefficients a 2 and C r , the plan based on Type-I and Type-II hybrid censored samples are not robust in certain range of the values (with the lowest efficiency achieved being 62.3%). It can also be observed that the plan based on Type-I hybrid censoring is slightly more robust than the one based on Type-II hybrid censoring. This reveals that the proposed optimal sampling plans, especially those based on Type-I hybrid censoring, are quite robust with regard to changes in the parameters and coefficients in the model. | |||||
On pp. 1107 and 1115, Sec. 4.2 should be changed as follows. Lin et al. (Citation2010) corrected their comments in Lin et al. (Citation2008a) that the minimum Bayes risks of their proposed sampling plans under Type-I censoring are in general as efficient as those of the sampling plans of Lam (Citation1994) in addition to having the life-test to be shorter in some cases. Hence, we like to compare the proposed sampling plans based on Type-I and Type-II hybrid censored samples with those based on ordinary Type-I and Type-II censored samples in the work of Lam (Citation1994) and Lam (Citation1990) (which we abbreviate as Lam1 and Lam2), respectively. Their relative efficiencies are evaluated and listed under the heading Eff2 in our tables. | |||||
The procedure of calculation of these relative efficiencies is as follows. For the Type-I case, we will take the value of T 0 in the tables to be the fixed time for conventional Type-I censored sampling plan in Lam1, and then the relative efficiency of this plan to that of the optimal Type-I hybrid censoring (given in the table) is calculated. Similarly, we take n 0 and r 0 to be fixed for conventional Type-II censored sampling in Lam2, and then the relative efficiency of this plan to that of the optimal Type-II hybrid censoring (given in the table) is calculated.
To be specific, let us consider the values given in bold in Table . We now have T 0 = 0.6808 in the optimal sampling plan under Type-I hybrid censoring, and n 0 = 4 and r 0 = 4 in the optimal sampling plan under Type-II hybrid censoring. Taking T 0 = 0.6808 to be the fixed time for conventional Type-I censored sampling plan in Lam1, we obtain the relative efficiency of this plan to that of the optimal Type-I hybrid censoring plan to be 24.9451/24.9893 = 99.8%. Similarly, we take n 0 = 4 and r 0 = 4 for conventional Type-II censored sampling plan in Lam2, and then find the relative efficiency of this plan to that of the optimal Type-II hybrid censoring plan to be 24.8419/24.8420 = 100%.
Table 1 The minimum Bayes risks and optimal sampling plans for a 0 = 2.0, a 1 = 2.0, a 2 = 2.0, C s = 0.5, C r = 30, and some selected values of a and b
Table 2 The minimum Bayes risks and optimal sampling plans for a = 2.5, b = 0.8, a 1 = 2.0, a 2 = 2.0, C s = 0.5, C r = 30, and some selected values of a 0
Table 3 The minimum Bayes risks and optimal sampling plans for a = 2.5, b = 0.8, a 0 = 2.0, a 2 = 2.0, C s = 0.5, C r = 30, and some selected values of a 1
Table 4 The minimum Bayes risks and optimal sampling plans for a = 2.5, b = 0.8, a 0 = 2.0, a 1 = 2.0, C s = 0.5, C r = 30, and some selected values of a 2
Table 5 The minimum Bayes risks and optimal sampling plans for a = 2.5, b = 0.8, a 0 = 2.0, a 1 = 2.0, a 2 = 2.0, C r = 30, and some selected values of C s
Table 6 The minimum Bayes risks and optimal sampling plans for a = 2.5, b = 0.8, a 0 = 2.0, a 1 = 2.0, a 2 = 2.0, C s = 0.5, and some selected values of C r
We observe that the relative efficiencies are in the range of 97.2–99.9% (99.9–105.8)% in the Type-I (Type-II) case. We can therefore conclude that for exponential distribution, the proposed hybrid censoring plans are generally as efficient as those based on Lam1 in the Type-I case, and more efficient than those based on Lam2 in the Type-II case.
Notes
†The obtained value of ξ0 in this case was negative leading to a meaningless risk in the comparison.
†The obtained value of ξ0 in this case was negative leading to a meaningless risk in the comparison.
*Visiting Professor at King Saud University, Riyadh, Saudi Arabia, and National Central University, Taiwan.
References
- Lam , Y. ( 1990 ). An optimal single variable sampling plan with censoring . The Statistician 39 : 53 – 67 .
- Lam , Y. ( 1994 ). Bayesian variable sampling plans for the exponential distribution with Type-I censoring . The Annals of Statistics 22 : 696 – 711 .
- Lin , C. T. , Huang , Y. L. , Balakrishnan , N. ( 2008a ). Exact Bayesian variable sampling plans for exponential distribution under Type-I censoring . In: Huber , C. , Limnios , N. , Mesbah , M. , Nikulin , M. , eds. Mathematical Methods for Survival Analysis, Reliability and Quality of Life . London : Hermes , pp. 151 – 162 .
- Lin , C. T. , Huang , Y. L. , Balakrishnan , N. ( 2008b ). Exact Bayesian variable sampling plans for the exponential distribution based on Type-I and Type-II hybrid censored samples . Communications in Statistics – Simulation and Computation 37 : 1101 – 1116 .
- Lin , C. T. , Huang , Y. L. , Balakrishnan , N. ( 2010 ). Corrections on “Exact Bayesian Variable Sampling Plans For Exponential Distribution under Type-I Censoring”. Technical Report, Department of Mathematics, Tamkang University, Tamsui, Taiwan. http://test.math.tku.edu.tw/math.tku/chinese/images/doc/Chien/CETypeI.pdf
- *Visiting Professor at King Saud University, Riyadh, Saudi Arabia, and National Central University, Taiwan.