Abstract
In this paper, we discuss inferential aspects for the Grubbs model when the unknown quantity x (latent response) follows a skew-normal distribution, extending early results given in Arellano-Valle et al. (J Multivar Anal 96:265–281, 2005b). Maximum likelihood parameter estimates are computed via the EM-algorithm. Wald and likelihood ratio type statistics are used for hypothesis testing and we explain the apparent failure of the Wald statistics in detecting skewness via the profile likelihood function. The results and methods developed in this paper are illustrated with a numerical example.
Similar content being viewed by others
References
Arellano-Valle RB, Genton MG (2005) Fundamental skew distributions. J Multivar Anal 96: 93–116
Arellano-Valle RB, Bolfarine H, Lachos VH (2005a) Skew-normal linear mixed models. J Data Sci 3: 415–438
Arellano-Valle RB, Ozan S, Bolfarine H, Lachos VH (2005b) Skew-normal measurement error models. J Multivar Anal 96: 265–281
Azzalini A (1985) A class of distributions which includes the normal ones. Scand J Stat 12: 171–178
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew normal distribution. J R Stat Soc Ser B 61: 579–602
Azzalini A, Dalla-Valle A (1996) The multivariate skew-normal distribution. Biometrika 83: 715–726
Barnett VD (1969) Simultaneous pairwise linear structural relationships. Biometrics 25: 129–142
Bedrick EJ (2001) An efficient scores test for comparing several measuring devices. J Qual Technol 33: 96–103
Bolfarine H, Galea-Rojas M (1995) Maximum likelihood estimation of simultaneous pairwise linear structural ralationships. Biom J 37: 673–689
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
Cheng CL, Ness V (1999) Statistical regression with measurement error, 1st edn. Oxford University Press, Oxford
Christensen R, Blackwood L (1993) Test for precision and acurracy of multiple measuring devices. Technometrics 35: 411–420
De Castro M, Lachos VH, Galea-Rojas M (2008) Heteroscedastic skew-normal measurement error models. Commun Stat Theory Methods (submitted)
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM-algorithm. J R Stat Soc Ser B 39: 1–22
DiCiccio TJ, Monti AC (2004) Inferential aspects of the skew exponential power distribution. J Am Stat Assoc 99: 439–450
Fuller WA (1987) Measurement error models. Wiley, New York
Grubbs FE (1948) On estimating precision of measuring instruments and product variability. J Am Stat Assoc 43: 243–264
Grubbs FE (1973) Errors of measurement, precision, accuracy and the statistical comparison of measuring instruments. Technometrics 15: 53–66
Grubbs FE (1983) Grubbs estimator. Encycloped Stat Sci 3: 542–549
Gupta AK, Chen JT (2004) A class of multivariate skew-normal models. Ann Inst Stat Math 56: 305–315
Henze N (1986) A probabilistic representation of the skew-normal distribution. Scand J Stat 13: 271–275
Johnson NL, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New York
Lachos VH, Bolfarine H, Arellano-Valle RB, Montenegro LC (2007a) Likelihood based inference for multivariate skew-normal regression models. Commun Stat Theory Methods 36: 1769–1786
Lachos VH, Vilca F, Galea M (2007b) Influence diagnostics for the Grubbs model. Stat Pap 48: 419–436
Lin TI, Lee JC (2007) Estimation and prediction in linear mixed models with skew-normal random effects for longitudinal data. Statistics in Medicine. Available online early view
Meeker WQ, Escobar LA (1995) Teaching about approximate confidence regions based on maximum likelihood estimation. Am Stat 49: 48–53
Nel DG (1980) On matrix differentiation in statistics. S Afr Stat J 14: 137–193
Pawitan Y (2000) A reminder of the fallibility of the Wald statistic: likelihood explanation. Am Stat 54: 54–56
Shyr I, Gleser L (1986) Inference about comparative precision in linear structural relationships. J Stat Plan Inference 14: 339–358
Theobald CM, Mallison JR (1978) Comparative calibration, linear structural relationship and congeneric measurements. Biometrics 34: 35–45
Zhang D, Davidian M (2001) Linear mixed models with flexible distributions of random effects for longitudinal data. Biometrics 57: 795–802
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Montenegro, L.C., Lachos, V.H. & Bolfarine, H. Inference for a skew extension of the Grubbs model. Stat Papers 51, 701–715 (2010). https://doi.org/10.1007/s00362-008-0157-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00362-008-0157-9