On estimation and local influence analysis for measurement errors models under heavy-tailed distributions

• Published in: Statistical papers (Berlin, Germany) Vol. 52; no. 3; pp. 567 - 590
• Main Authors: Lachos, V. H., Angolini, T., Abanto-Valle, C. A.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01-08-2011; Springer Nature B.V
• Subjects: Algorithms; Economic Theory/Quantitative Economics/Mathematical Methods; Economics; Error analysis; Finance; Gaussian; Insurance; Management; Mathematical models; Mathematics and Statistics; Measurement; Normal distribution; Operations Research/Decision Theory; Perturbation methods; Probability Theory and Stochastic Processes; Random errors; Random variables; Robustness; Statistical analysis; Statistics; Statistics for Business; Studies; Taguchi methods; Measurement error models; EM algorithm; Mahalanobis distance; Scale mixtures of normal distributions
• ISSN: 0932-5026; 1613-9798
• DOI: 10.1007/s00362-009-0270-4

Scale mixtures of normal distributions form a class of symmetric thick-tailed distributions that includes the normal one as a special case. In this paper we consider local influence analysis for measurement error models (MEM) when the random error and the unobserved value of the covariates jointly follow scale mixtures of normal distributions, providing an appealing robust alternative to the usual Gaussian process in measurement error models. In order to avoid difficulties in estimating the parameter of the mixing variable, we fixed it previously, as recommended by Lange et al. (J Am Stat Assoc 84:881–896, 1989) and Berkane et al. (Comput Stat Data Anal 18:255–267, 1994). The local influence method is used to assess the robustness aspects of the parameter estimates under some usual perturbation schemes. However, as the observed log-likelihood associated with this model involves some integrals, Cook’s well–known approach may be hard to apply to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (J R Stat Soc Ser B 63:121–126, 2001), which is based on the EM algorithm. Results obtained from a real data set are reported, illustrating the usefulness of the proposed methodology, its relative simplicity, adaptability and practical usage.