Nonlinear regression models based on the normal mean-variance mixture of Birnbaum-Saunders distribution. (English) Zbl 1368.62184
Summary: This paper presents a new extension of nonlinear regression models constructed by assuming the normal mean-variance mixture of Birnbaum-Saunders distribution for the unobserved error terms. A computationally analytical EM-type algorithm is developed for computing maximum likelihood estimates. The observed information matrix is derived for obtaining the asymptotic standard errors of parameter estimates. The practical utility of the methodology is illustrated through both simulated and real data sets.
References:
| [1] | Aitken, A. C., On Bernoullis numerical solution of algebraic equations, Proceedings of the Royal Society of Edinburgh, 46, 289-305 (1926) · JFM 52.0098.05 |
| [2] | Atkinson, A., Two graphical displays for outlying and influential observations in regression, Biometrika, 68, 13-20 (1981) · Zbl 0462.62049 |
| [3] | Barndorff-Nielsen, O.; Halgreen, C., Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions, Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 38, 309-311 (1977) · Zbl 0403.60026 |
| [4] | Birnbaum, Z. W.; Saunders, S. C., A new family of life distributions, Journal of Applied Probability, 6, 319-327 (1969) · Zbl 0209.49801 |
| [5] | Böhning, D.; Dietz, E.; Schaub, R.; Schlattmann, P.; Lindsay, B., The distribution of the likelihood ratio for mixtures of densities from the one-parameter exponential family, Annals of the Institute of Statistical Mathematics, 46, 373-388 (1994) · Zbl 0802.62017 |
| [6] | Cancho, V. G.; Dey, D. K.; Lachos, V. H.; Andrade, M. G., Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: Estimation and case influence diagnostics, Computational Statistics & Data Analysis, 55, 588-602 (2011) · Zbl 1247.62083 |
| [7] | Cancho, V. G.; Lachos, V. H.; Ortega, E. M.M., A nonlinear regression model with skew-normal errors, Statistical Papers, 51, 547-558 (2010) · Zbl 1247.62160 |
| [8] | Contreras-Reyes, J. E.; Arellano-Valle, R. B., Growth estimates of cardinalfish (Epigonus crassicaudus) based on scale mixtures of skew-normal distributions, Fisheries Research, 147, 137-144 (2013) |
| [9] | Dempster, A. P.; Laird, N. M.; Rubin, D. B., Maximum likelihood from incomplete data via the EM algorithm (with discussion), Journal of the Royal Statistical Society. Series B. Statistical Methodology, 39, 1-38 (1977) · Zbl 0364.62022 |
| [10] | Desmond, A. F., On the relationship between two fatigue-life models, IEEE Transactions on Reliability, 35, 167-169 (1986) · Zbl 0592.62089 |
| [11] | Fagundes, R. A.; De Souza, R. M.; Cysneiros, F. J.A., Robust regression with application to symbolic interval data, Engineering Applications of Artificial Intelligence, 26, 564-573 (2013) |
| [12] | Ferreira, C. S.; Bolfarine, H.; Lachos, V. H., Skew scale mixtures of normal distributions: properties and estimation, Statistical Methodology, 8, 154-171 (2011) · Zbl 1213.62023 |
| [13] | Ferreira, C. S.; Lachos, V. H., Nonlinear regression models under skew scale mixtures of normal distributions, Statistical Methodology, 33, 131-146 (2016) · Zbl 1487.62074 |
| [15] | Garay, A. M.; Lachos, V. H.; Abanto-Valle, C. A., Nonlinear regression models based on scale mixtures of skew-normal distributions, Journal of the Korean Statistical Society, 50, 115-124 (2011) · Zbl 1296.62141 |
| [16] | Garay, A. M.; Lachos, V. H.; Lin, T. I., Nonlinear censored regression models with heavy-tailed distributions, Statistics and its Interface, 9, 281-293 (2016) · Zbl 1405.62094 |
| [17] | Good, I. J., The population frequencies of species and the estimation of population parameters, Biometrika, 40, 237-260 (1953) · Zbl 0051.37103 |
| [18] | Ho, H. J.; Pyne, S.; Lin, T. I., Maximum likelihood inference for mixtures of skew Student-\(t\)-normal distributions through practical EM-type algorithms, Statistics and Computing, 22, 287-299 (2012) · Zbl 1322.62087 |
| [19] | Jamalizadeh, A.; Lin, T. I., A general class of scale-shape mixtures of skew-normal distributions: properties and estimation, Computational Statistics (2016) |
| [20] | Lange, K. L.; Sinsheimer, J. S., Normal/independent distributions and their applications in robust regression, Journal of Computational and Graphical Statistics, 2, 175-198 (1993) |
| [21] | Lin, T. I.; Ho, H. J.; Lee, C. R., Flexible mixture modelling using the multivariate skew-\(t\)-normal distribution, Statistics and Computing, 24, 531-546 (2014) · Zbl 1325.62113 |
| [22] | Lindsay, B. G., Mixture models: Theory, geometry and applications (1995), Institute of Mathematical Statistics: Institute of Mathematical Statistics Hayward, CA · Zbl 1163.62326 |
| [23] | Liu, C.; Rubin, D. B., The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence, Biometrika, 81, 633-648 (1994) · Zbl 0812.62028 |
| [24] | López Quintero, F. O.; Contreras-Reyes, J. E.; Wiff, R.; Arellano-Valle, R. B., Flexible Bayesian analysis of the von Bertalanffy growth function with the use of a log-skew-\(t\) distribution, Fishery Bulletin, 115, 13-26 (2017) |
| [25] | Louis, T. A., Finding the observed information when using the EM algorithm, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 44, 226-232 (1982) · Zbl 0488.62018 |
| [26] | McNeil, A.; Frey, R.; Embrechts, P., Quantitative risk management: Concepts, techniques and tools (2005), Princeton University Press · Zbl 1089.91037 |
| [27] | McNicholas, P. D.; Murphy, T. B.; McDaid, A. F.; Frost, D., Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models, Computational Statistics & Data Analysis, 54, 711-723 (2010) · Zbl 1464.62131 |
| [28] | Meilijson, I., A fast improvement to the EM algorithm to its own terms, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 51, 127-138 (1989) · Zbl 0674.65118 |
| [29] | Meng, X. L.; Rubin, D. B., Maximum likelihood estimation via the ECM algorithm: a general framework, Biometrika, 80, 267-278 (1993) · Zbl 0778.62022 |
| [30] | Pourmousa, R.; Jamalizadeh, A.; Rezapour, M., Multivariate normal mean-variance mixture distribution based on Birnbaum-Saunders distribution, Journal of Statistical Computation and Simulation, 85, 2736-2749 (2015) · Zbl 1457.62164 |
| [31] | Wang, W. L.; Lin, T. I., Bayesian analysis of multivariate \(t\) linear mixed models with missing responses at random, Journal of Statistical Computation and Simulation, 85, 3594-3612 (2015) · Zbl 1510.62305 |
| [32] | Wang, W. L.; Lin, T. I., Multivariate-\(t\) nonlinear mixed models with application to censored multi-outcome AIDS studies, Biostatistics (2017) |
| [33] | Wang, W. L.; Lin, T. I.; Lachos, V. H., Extending multivariate-\(t\) linear mixed models for multiple longitudinal data with censored responses and heavy tails, Statistical Methods in Medical Research (2015) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
