Brazilian Journal of Probability and Statistics

Bayesian analysis of flexible measurement error models

Luz Marina Rondon and Heleno Bolfarine

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Abstract

This paper proposes the Bayesian inference for flexible measurement error models, in which their systematic components include explanatory variable vectors with and without measurement errors, as well as nonlinear effects that are approximated by using B-splines. The model investigated is the structural version, as the error-prone variables follow scale mixtures of normal distributions such as Student-$t$, slash, contaminated normal, Laplace and symmetric hyperbolic distributions. To draw samples of the posterior distribution of the model parameters, an MCMC algorithm is proposed. The performance of this algorithm is assessed through simulations. In addition, the function fmem() of the R package BayesGESM is presented, which provides an easy way to apply the methodology presented in this paper. The proposed methodology is applied to a real data set, which shows that ignoring measurement errors (i.e., analyze the data by using the traditional methodology) can lead to wrong conclusions.

Article information

Source
Braz. J. Probab. Stat., Volume 31, Number 3 (2017), 618-639.

Dates
Received: March 2015
Accepted: June 2016
First available in Project Euclid: 22 August 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1503388831

Digital Object Identifier
doi:10.1214/16-BJPS326

Mathematical Reviews number (MathSciNet)
MR3693983

Zentralblatt MATH identifier
1377.62152

Keywords
Bayesian analysis measurement error models semi-parametric models MCMC algorithm $B$-splines scale mixtures of normal distributions

Citation

Rondon, Luz Marina; Bolfarine, Heleno. Bayesian analysis of flexible measurement error models. Braz. J. Probab. Stat. 31 (2017), no. 3, 618--639. doi:10.1214/16-BJPS326. https://projecteuclid.org/euclid.bjps/1503388831


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References

  • Andrews, D. F. and Mallows, C. L. (1974). Scale mixtures of normal distributions. Journal of the Royal Statistical Society. Series B (Methodological) 36, 99–102.
  • Arellano-Valle, R. B., Bolfarine, H. and Labra, V. (1996). Ultrastructural elliptical models. Canadian Journal of Statistics 24, 207–216.
  • Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. Proceedings of the Royal Society. Series A, Mathematical, Physical and Engineering Sciences 353, 401–419.
  • Belsley, D. A., Kuh, E. and Welsch, R. E. (2005). Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. New York: Wiley.
  • Box, G. and Tiao, G. (1973). Bayesian Inference in Statistical Analysis. New York: John Wiley and Sons.
  • Cao, C. Z., Lin, J. G. and Zhu, X. X. (2012). On estimation of a heteroscedastic measurement error model under heavy-tailed distributions. Computational Statistics and Data Analysis 56, 438–448.
  • Carroll, R. J., Roeder, K. and Wasserman, L. (1999). Flexible parametric measurement error models. Biometrics 55, 44–54.
  • Carroll, R. J., Ruppert, D., Stefanski, L. A. and Crainiceanu, C. M. (2006). Measurement Error in Nonlinear Models: A Modern Perspective, 2nd ed. Boca Raton: Chapman and Hall.
  • Cheng, C. L. and Van Ness, J. W. (1999). Statistical Regression with Measurement Error. London: Arnold.
  • De Boor, C. (1978). A Practical Guide to Splines. Applied Mathematical Sciences. New York: Springer.
  • de Castro, M., Bolfarine, H. and Galea, M. (2013). Bayesian inference in measurement error models for replicated data. Environmetrics 24, 22–30.
  • Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with $B$-splines and penalties. Statistical Science 11, 89–121.
  • Fuller, M. A. (1987). Measurement Error Models. New York: Wiley.
  • Gamerman, D. and Lopes, H. F. (2006). Markov Chain Monte Carlo, 2nd ed. Boca Raton, FL: Chapman and Hall.
  • Gelfand, A., Dey, D. and Chang, H. (1992). Model determination using predictive distributions with implementation via sampling-based methods. Bayesian Statistics 4, 147–167.
  • Harrison, D. Jr. and Rubinfeld, D. L (1978). Hedonic housing prices and the demand for clean air. Journal of environmental economics and management 5, 81–102.
  • He, X., Fung, W. K. and Zhu, Z. (2005). Robust estimation in generalized partial linear models for clustered data. Journal of the American Statistical Association 100, 1176–1184.
  • Jörgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. Lecture Notes in Statistics 9. New York: Springer.
  • Kelly, B. C. (2007). Some aspects of measurement error in linear regression of astronomical data. The Astrophysical Journal 665, 1489–1506.
  • Kulathinal, S. B., Kuulasmaa, K. and Gasbarra, D. (2002). Estimation of an errors-in-variables regression model when the variances of the measurement errors vary betwwen the observations. Statistics in Medicine 21, 1089–1101.
  • Lemonte, A. J. and Patriota, A. G. (2011). Multivariate elliptical models with general parameterization. Statistical Methodology 8, 389–400.
  • Li, L., Palta, M. and Shao, J. (2004). A measurement error model with a Poisson distributed surrogate. Statistics in Medicine 23, 2527–2536.
  • Maronna, R. A., Martin, D. R. and Yohai, V. J. (2006). Robust Statistics: Theory and Methods. New York: Wiley.
  • Nadarajah, S. and Kotz, S. (2006). The exponentiated type distributions. Acta Applicandae Mathematicae 92, 97–111.
  • Patriota, A. G., Bolfarine, H. and de Castro, M. (2009). A heteroscedastic structural errors-in-variables model with equation error. Statistical Methodology 6, 408–423.
  • Peng, F. and Dey, D. K. (1995). Bayesian analysis of outlier problems using divergence measures. The Canadian Journal of Statistics 23, 199–213.
  • Rondon, L. M. and Bolfarine, H. (2014). BayesGESM: Bayesian Analysis of Generalized Elliptical Semiparametric Models. R package version 1.1. http://CRAN.R-project.org/package=BayesGESM.
  • Rogers, W. H. and Tukey, J. W. (1972). Understanding some long-tailed symmetrical distributions. Statistica Neerlandica 26, 211–226.
  • Spiegelhalter, D. J., Best, N. G., Carlin, B. P. and Var der Linde, A. (2002). Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society. Series B (Methodological) 64, 583–640.
  • Tanner, M. A. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation (with discussion). Journal of the American Statistical Association 82, 528–550.
  • Weiss, R. and Cook, R. (1992). A grafical case statistic for assessing posterior influence. Biometrika 79, 51–55.