In this paper, we introduce the random deterioration rate model with measurement error in order to incorporate the variability among different components. The motivation behind the random variable model is to capture the randomness in the individual differences across the population. This model incorporates only sample uncertainty of the degradation, and no temporal variability is included. The measurement error models appear to overcome this problem. The random rate analysis is based on repeated measurements of failure sizes generated by a degradation process over time in a components population. Some characteristics of the random deterioration rate model based on the inverse Gaussian distribution and subject to measurement error, are examined. We carry out simulation studies to (i) assess the performance of the maximum likelihood estimates obtained through the Gaussian quadrature along with Quasi-Newton optimization method; and (ii) examine the effects of model misspecification on the model selection criteria’s performance, as well as on the lifetime prediction’s accuracy and precision. The potentiality of the proposed model is illustrated through two real data sets.
References
Anderson, T. W. and Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association 49, 765–769. 0059.13302 10.1080/01621459.1954.10501232Anderson, T. W. and Darling, D. A. (1954). A test of goodness of fit. Journal of the American Statistical Association 49, 765–769. 0059.13302 10.1080/01621459.1954.10501232
Broyden, C. G. (1970). The convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications 6, 76–90. 0223.65023 10.1093/imamat/6.1.76Broyden, C. G. (1970). The convergence of a class of double-rank minimization algorithms. Journal of the Institute of Mathematics and Its Applications 6, 76–90. 0223.65023 10.1093/imamat/6.1.76
Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. New York, NY. 1005.62007Burnham, K. P. and Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd ed. New York, NY. 1005.62007
Carrasco, J. M. F., Ferrari, S. L. P. and Arellano-Valle, R. B. (2014). Errors-in-variables beta regression models. Journal of Applied Statistics 41, 1530–1547. 07267785 10.1080/02664763.2014.881784Carrasco, J. M. F., Ferrari, S. L. P. and Arellano-Valle, R. B. (2014). Errors-in-variables beta regression models. Journal of Applied Statistics 41, 1530–1547. 07267785 10.1080/02664763.2014.881784
Crowther, M. J., Andersson, T. M.-L., Lambert, P. C., Abrams, K. R. and Humphreys, K. (2016). Joint modelling of longitudinal and survival data: Incorporating delayed entry and an assessment of model misspecification. Statistics in Medicine 35, 1193–1209.Crowther, M. J., Andersson, T. M.-L., Lambert, P. C., Abrams, K. R. and Humphreys, K. (2016). Joint modelling of longitudinal and survival data: Incorporating delayed entry and an assessment of model misspecification. Statistics in Medicine 35, 1193–1209.
Fenyvesi, L., Lu, H. and Jack, T. R. (2004). Prediction of corrosion defect growth on operating pipelines. In 2004 International Pipeline Conference, 225–230. American Society of Mechanical Engineers.Fenyvesi, L., Lu, H. and Jack, T. R. (2004). Prediction of corrosion defect growth on operating pipelines. In 2004 International Pipeline Conference, 225–230. American Society of Mechanical Engineers.
Freitas, M. A., Luíza, M., de Toledo, G., Colosimo, E. A. and Pires, M. C. (2009). Using degradation data to assess reliability: A case study on train wheel degradation. Quality and Reliability Engineering International 25, 607–629.Freitas, M. A., Luíza, M., de Toledo, G., Colosimo, E. A. and Pires, M. C. (2009). Using degradation data to assess reliability: A case study on train wheel degradation. Quality and Reliability Engineering International 25, 607–629.
Freitas, M. A., Colosimo, E. A., dos Santos, T. R. and Pires, M. C. (2010). Reliability assessment using degradation models: Bayesian and classical approaches. Pesquisa Operacional 30, 194–219.Freitas, M. A., Colosimo, E. A., dos Santos, T. R. and Pires, M. C. (2010). Reliability assessment using degradation models: Bayesian and classical approaches. Pesquisa Operacional 30, 194–219.
Huyse, L. and Roodselaar, V. (2010). Effects of inline inspection sizing uncertainties on the accuracy of the largest features and corrosion rate statistics. In 2010 8th International Pipeline Conference (IPC2010). Calgary, Alberta, Canada.Huyse, L. and Roodselaar, V. (2010). Effects of inline inspection sizing uncertainties on the accuracy of the largest features and corrosion rate statistics. In 2010 8th International Pipeline Conference (IPC2010). Calgary, Alberta, Canada.
Kass, R. E. and Steffey, D. (1989). Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). Journal of the American Statistical Association 84, 717–726.Kass, R. E. and Steffey, D. (1989). Approximate Bayesian inference in conditionally independent hierarchical models (parametric empirical Bayes models). Journal of the American Statistical Association 84, 717–726.
Lesaffre, E. and Spiessens, B. (2001). On the effect of the number of quadrature points in a logistic random effects model: An example. Journal of the Royal Statistical Society Series C Applied Statistics 50, 325–335. 1112.62307 10.1111/1467-9876.00237Lesaffre, E. and Spiessens, B. (2001). On the effect of the number of quadrature points in a logistic random effects model: An example. Journal of the Royal Statistical Society Series C Applied Statistics 50, 325–335. 1112.62307 10.1111/1467-9876.00237
Lu, C. J. and Meeker, W. Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics 35, 161–174. 0775.62271 10.1080/00401706.1993.10485038Lu, C. J. and Meeker, W. Q. (1993). Using degradation measures to estimate a time-to-failure distribution. Technometrics 35, 161–174. 0775.62271 10.1080/00401706.1993.10485038
Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. New York, NY: John Wiley & Sons. Chapter 13. 0949.62086Meeker, W. Q. and Escobar, L. A. (1998). Statistical Methods for Reliability Data. New York, NY: John Wiley & Sons. Chapter 13. 0949.62086
Morita, L. H. M., Tomazella, V. L. D., Ramos, P. L., Ferreira, P. H. and Louzada, F. (2020). Supplement to “The random deterioration rate model with measurement error based on the inverse Gaussian distribution.” https://doi.org/10.1214/20-BJPS468SUPPMorita, L. H. M., Tomazella, V. L. D., Ramos, P. L., Ferreira, P. H. and Louzada, F. (2020). Supplement to “The random deterioration rate model with measurement error based on the inverse Gaussian distribution.” https://doi.org/10.1214/20-BJPS468SUPP
Nelson, K. P., Lipsitz, S. R., Fitzmaurice, G. M., Ibrahim, J., Parzen, M. and Strawderman, R. (2006). Use of the probability integral transformation to fit nonlinear mixed-effects models with nonnormal random effects. Journal of Computational and Graphical Statistics 15, 39–57.Nelson, K. P., Lipsitz, S. R., Fitzmaurice, G. M., Ibrahim, J., Parzen, M. and Strawderman, R. (2006). Use of the probability integral transformation to fit nonlinear mixed-effects models with nonnormal random effects. Journal of Computational and Graphical Statistics 15, 39–57.
Pandey, M. D. and Lu, D. (2013). Estimation of parameters of degradation growth rate distribution from noisy measurement data. Structural Safety 43, 60–69.Pandey, M. D. and Lu, D. (2013). Estimation of parameters of degradation growth rate distribution from noisy measurement data. Structural Safety 43, 60–69.
Peng, W., Li, Y.-F., Yang, Y.-J., Huang, H.-Z. and Zuo, M. J. (2014). Inverse Gaussian process models for degradation analysis: A Bayesian perspective. Reliability Engineering & Systems Safety 130, 175–189.Peng, W., Li, Y.-F., Yang, Y.-J., Huang, H.-Z. and Zuo, M. J. (2014). Inverse Gaussian process models for degradation analysis: A Bayesian perspective. Reliability Engineering & Systems Safety 130, 175–189.
Piessens, R., de Doncker-Kapenga, E., Überhuber, C. W. and Kahaner, D. K. (2012). Quadpack: A Subroutine Package for Automatic Integration, Vol. 1. Berlin: Springer. 0508.65005Piessens, R., de Doncker-Kapenga, E., Überhuber, C. W. and Kahaner, D. K. (2012). Quadpack: A Subroutine Package for Automatic Integration, Vol. 1. Berlin: Springer. 0508.65005
Pinheiro, J. C. and Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics 4, 12–35.Pinheiro, J. C. and Bates, D. M. (1995). Approximations to the log-likelihood function in the nonlinear mixed-effects model. Journal of Computational and Graphical Statistics 4, 12–35.
Rondon, L. M. and Bolfarine, H. (2017). Bayesian analysis of flexible measurement error models. Brazilian Journal of Probability and Statistics 31, 618–639. 1377.62152 10.1214/16-BJPS326 euclid.bjps/1503388831Rondon, L. M. and Bolfarine, H. (2017). Bayesian analysis of flexible measurement error models. Brazilian Journal of Probability and Statistics 31, 618–639. 1377.62152 10.1214/16-BJPS326 euclid.bjps/1503388831
Sakamoto, Y., Ishiguro, M. and Kitagawa, G. (1986). Akaike Information Criterion Statistics, Vol. 1, 1st ed. Netherlands: Springer. 0608.62006Sakamoto, Y., Ishiguro, M. and Kitagawa, G. (1986). Akaike Information Criterion Statistics, Vol. 1, 1st ed. Netherlands: Springer. 0608.62006
Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics 6, 461–464. 0379.62005 10.1214/aos/1176344136 euclid.aos/1176344136Schwarz, G. (1978). Estimating the dimension of a model. The Annals of Statistics 6, 461–464. 0379.62005 10.1214/aos/1176344136 euclid.aos/1176344136
Xiao, N.-C., Huang, H.-Z., Wang, Z., Liu, Y. and Zhang, X.-L. (2012). Unified uncertainty analysis by the mean value first order saddlepoint approximation. Structural and Multidisciplinary Optimization 46, 803–812. 1274.90127 10.1007/s00158-012-0794-4Xiao, N.-C., Huang, H.-Z., Wang, Z., Liu, Y. and Zhang, X.-L. (2012). Unified uncertainty analysis by the mean value first order saddlepoint approximation. Structural and Multidisciplinary Optimization 46, 803–812. 1274.90127 10.1007/s00158-012-0794-4
