Abstract
In this paper, we discuss the extension of some diagnostic procedures to multivariate measurement error models with scale mixtures of skew-normal distributions (Lachos et al., Statistics 44:541–556, 2010c). This class provides a useful generalization of normal (and skew-normal) measurement error models since the random term distributions cover symmetric, asymmetric and heavy-tailed distributions, such as skew-t, skew-slash and skew-contaminated normal, among others. Inspired by the EM algorithm proposed by Lachos et al. (Statistics 44:541–556, 2010c), we develop a local influence analysis for measurement error models, following Zhu and Lee’s (J R Stat Soc B 63:111–126, 2001) approach. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and Cook’s well-known approach can be very difficult to apply to achieve local influence measures. Some useful perturbation schemes are also discussed. In addition, a score test for assessing the homogeneity of the skewness parameter vector is presented. Finally, the methodology is exemplified through a real data set, illustrating the usefulness of the proposed methodology.
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References
Andrews DF, Mallows CL (1974) Scale mixtures of normal distributions. J R Stat Soc B 36: 99–102
Azzalini A, Dalla-Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726
Branco M, Dey DK (2001) A general class of multivariate skew-elliptical distribution. J Multivar Anal 79: 93–113
Cancho VG, Dey DK, Lachos VH, Andrade MG (2010) Bayesian nonlinear regression models with scale mixtures of skew-normal distributions: estimation and case influence diagnostics. Comput Stat Data Anal 55: 588–602
Chipkevitch E, Nishimura R, Tu D, Galea-Rojas M (1996) Clinical measurement of testicular volume in adolescents: comparison of the reliability of 5 methods. J Urol 156: 2050–2053
Cook RD (1986) Assessment of local influence (with discussion). J R Stat Soc B 48: 133–169
Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman & Hall, London
Ho HJ, Lin TI (2010) Robust linear mixed models using the skew t distribution with application to schizophrenia data. Biom J 52: 449–469
Lachos VH, Abanto-Valle CA, Angolini T (2010a) On estimation and local influence analysis for measurement errors models under heavy-tailed distributions. Stat Pap. doi:10.1007/s00362-009-0270-4
Lachos VH, Ghosh P, Arellano-Valle RB (2010b) Likelihood based inference for skew-normal/independent linear mixed models. Stat Sin 20: 303–322
Lachos VH, Vilca-Labra FE, Bolfarine H, Ghosh P (2010c) Robust multivariate measurement error models based on scale mixtures of the skew-normal distribution. Statistics 44: 541–556
Lee SY, Xu L (2004) Influence analysis of nonlinear mixed-effects models. Comput Stat Data Anal 45: 321–341
Lucas A (1997) Robustness of the Student t based M-estimator. Commun Stat Theory Methods 26: 1165–1182
Magnus JR, Neudecker H (1988) Matrix differential calculus with applications in statistics and econometrics. Wiley, New York
Montenegro LC, Bolfarine H, Lachos VH (2009) Local influence analysis for skew-normal linear mixed models. Commun Stat Theory Methods 38: 484–496
Montenegro LC, Lachos VH, Bolfarine H (2010) Inference for a skew extension of the Grubbs model. Stat Pap 51: 701–715
Osorio F, Paula GA, Galea-Rojas M (2009) On estimation and influence diagnostics for the Grubbs’ model under heavy-tailed distributions. Comput Stat Data Anal 53: 1249–1263
Patriota AG, Bolfarine H (2010) Measurement error models with a general class of error distribution. Statistics 44: 119–127
Pinheiro JC, Liu CH, Wu YN (2001) Efficient algorithms for robust estimation in linear mixed-effects models using a multivariate t-distribution. J Comput Graph Stat 10: 249–276
Vilca-Labra FE, Aoki R, Zeller CB (2010) Hypotheses testing for structural calibration model. Stat Pap. doi:10.1007/s00362-009-0269-x
Xie FC, Wei BC, Lin JG (2009) Homogeneity diagnostics for skew-normal nonlinear regression models. Stat Probab Lett 79: 821–827
Zeller CB, Lachos VH, Vilca-Labra FE (2011) Local influence analysis for regression models with scale mixtures of skew-normal distributions. J Appl Stat 8: 343–368
Zhu HT, Lee S (2001) Local influence for incomplete-data models. J R Stat Soc B 63: 111–126
Zhu HT, Ibrahim JG, Lee SY, Zhang HP (2007) Perturbation selection and influence measures in local influence analysis. Ann Stat 35: 2565–2588
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Zeller, C.B., Carvalho, R.R. & Lachos, V.H. On diagnostics in multivariate measurement error models under asymmetric heavy-tailed distributions. Stat Papers 53, 665–683 (2012). https://doi.org/10.1007/s00362-011-0371-8
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DOI: https://doi.org/10.1007/s00362-011-0371-8