In the existing literature [1, 2, 3, 4, 5, 6] problems of estimation have been discussed under the classical approach when the lifetime data following a lognormal distribution are observed under progressive type-II censoring, hybrid censoring, and based on record values. The Expectation Maximization (EM) algorithm and the idea of the missing information principle have been employed to obtain maximum likelihood estimates and associated interval estimates for normal or lognormal parameters. However, we observed a gap in the existing literature concerning the problems of estimation and prediction for the lognormal distribution under the Bayesian approach. In our work, we considered both informative and non-informative priors. Since Bayes estimators do not turn out to have a closed form, we considered the Lindley method [7] for this purpose. We further observed that importance sampling can be used to draw samples from the associated posterior densities. Therefore we generated samples from the associated posterior densities and obtained Bayes estimates and associated highest posterior density interval estimates using the idea of Chen and Shao [8]. We have also used OpenBUGS software for this purpose. Next, we obtained predictive estimates and associated predictive interval estimates to provide inferences about the censored and future samples respectively under one- and two-sample prediction. Finally, we analyzed various real data sets and conducted a simulation study to observe the performance of the proposed methods of estimation and prediction.

In literature [9, 10] reliability acceptance sampling plans have been constructed for the lognormal distribution when the data are observed under type-II and progressive type-II censoring schemes. We extended these plans to situations when the data are observed under progressive first-failure censoring [11], a generalization of type-II, progressive type-II, and first-failure censoring schemes. In fact, the plans in existing literature [10, 12, 13] were developed for a given proportion of removed units, and in our work, we contributed a discussion on selecting plans based on an optimal criterion. This work also considers cost constraints, and the proposed algorithms can be applied to the Weibull distribution as well.

Lin et al. [14] considered the lognormal distribution under progressive type-I interval censoring [15]. It is observed that the inverse Weibull and lognormal distributions exhibit a similar type of hazard rate. However, from a computational aspect, one advantage of considering the inverse Weibull distribution over lognormal is that the cumulative distribution function of the inverse Weibull distribution admits a closed form. The work of Palumbo and Pallotta [16] also motivates us to study the inverse Weibull distribution. We considered inverse Weibull distribution under progressive type-I interval censoring and obtained maximum likelihood and midpoint estimates using EM algorithm, with probability plot estimates serving as an initial guess for the unknown parameters. In our work, Bayes estimates are also obtained using Lindley and Tierney-Kadane [17] methods, and a discussion is presented on the selection of inspection times and optimal censoring. 

References:

[1] Balakrishnan N, Kannan N, Lin CT, Ng HKT. Point and interval estimation for gaussian distribution based on progressively type-ii censored samples. IEEE Transactions on Reliability, 52(1):90-95, 2003.

[2] Balakrishnan N, Mi J. Existence and uniqueness of the mles for normal distribution based on general progressively type-ii censored samples. Statistics & Probability Letters, 64(4):407-414, 2003.

[3] Ng HKT, Chan PS, Balakrishnan N. Estimation of parameters from progressively censored data using em algorithm. Computational Statistics & Data Analysis, 39(4):371-386, 2002.

[4] Dube S, Pradhan B, Kundu B. Parameter estimation of the hybrid censored log-normal distribution. Journal of Statistical Computation and Simulation, 81(3):275-287, 2011.

[5] Balakrishnan N, Chan PS. On the normal record values and associated inference. Statistics & Probability Letters, 39(1):73-80, 1998.

[6] Doostparast M, Deepak S, Zangoie A. Estimation with the lognormal distribution on the basis of records. Journal of Statistical Computation and Simulation, 83(12):2339-2351, 2013.

[7] Lindley DV. Approximate bayesian methods. Trabajos de estadistica y de investigacion operativa, 31(1):223-245, 1980.

[8] Chen MH, Shao QM. Monte carlo estimation of bayesian credible and hpd intervals. Journal of Computational and Graphical Statistics, 8(1):69-92, 1999.

[9] Schneider H. Failure-censored variables-sampling plans for lognormal and weibull distributions. Technometrics, 31(2):199-206, 1989.

[10] Balasooriya U, Balakrishnan N. Reliability sampling plans for lognormal distribution based on progressively censored samples. IEEE Transactions on Reliability, 49(2):199-203, 2000.

[11] Wu SJ, Kus C. On estimation based on progressive first-failure censored sampling. Computational Statistics & Data Analysis, 53(10):3659-3670, 2009.

[12] Balasooriya U, Saw SLC, Gadag V. Progressively censored reliability sampling plans for the Weibull distribution. Technometrics, 42(2):160-167, 2000.

[13] Wu SJ, Huang SR. Progressively first-failure censored reliability sampling plans with cost constraint. Computational Statistics & Data Analysis, 56(6):2018-2030, 2012.

[14] Lin CT, Wu SJS, Balakrishnan N. Planning life tests with progressive type-I interval censored data from the lognormal distribution. Journal of Statistical Planning and Inference, 139(1):54-61, 2009.

[15] Aggarwala R. Progressive interval censoring: some mathematical results with applications to inference. Communications in Statistics-Theory and Methods, 30(8-9):1921-1935, 2001.

[16] Palumbo B, Pallotta G. New approach to the identification of the inverse Weibull model. In 46-th Scientific meeting of the Italian Statistical Society, 2012.

[17] Tierney L, Kadane JB. Accurate approximations for posterior moments and marginal densities. Journal of the American Statistical Association, 81:82-85, 1986.

List of publications from Ph.D. thesis:

• S. Singh, Y. M. Tripathi, S. J. Wu : On estimating parameters of a progressively censored lognormal distribution, Journal of Statistical Computation and Simulation, Vol. 85 (6), 1071-1089, 2015. (View, Download)

• S. Singh, Y. M. Tripathi, S. J. Wu : Bayesian analysis for lognormal distribution under progressive type-II censoring. Hacettepe Journal of Mathematics and Statistics. Vol. 48(5), 1488-1504, 2019. (View, Download)

• S. Singh, Y. M. Tripathi : Bayesian estimation and prediction for a hybrid censored lognormal distribution, IEEE Transactions on Reliability, Vol. 65(2), 782-795, 2016. (View, Download)

• S. Singh, Y. M. Tripathi, S. J. Wu : Bayesian estimation and prediction based on lognormal record values. Journal of Applied Statistics, Vol. 44(5), 916-940, 2017. (View, Download)

• S. Singh, Y. M. Tripathi : Reliability sampling plans for a lognormal distribution under progressive first-failure censoring with cost constraint, Statistical Papers, Vol. 56(3), 773-817, 2015. (View, Download)

• S. Singh, Y. M. Tripathi : Estimating the parameters of an inverse Weibull distribution under progressive type-I interval censoring, Statistical Papers, Vol. 59(1), 21-56, 2018. (View, Download)

PhD Doctoral Committee: 

Dr. Yogesh Mani Tripathi (supervisor)

Dr. Ashish Kumar Upadhyay (chairman)

Dr. Prashant Kumar Srivastava (member)

Dr. Debshree Guha Adhya (member)

Dr. Shovan Bhaumik (member)

Dr. Nutan Kumar Tomar (Departmental PhD program coordinator)