In this paper we discuss inferential aspects and the local influence analysis of the multivariate null intercept measurement error model where the unobserved value of the covariate (latent variable) follows a skew-normal distribution. In order to develop the hypotheses testing of interest and the local influence diagnostics, closed-form expressions of the marginal likelihood, the score function and the observed information matrix are presented. Additionally, an EM-type algorithm for evaluating the unrestricted and restricted maximum likelihood estimates of the parameters under equality constraints on the regression coefficients is examined. Also, we derive the appropriate matrices to assess the local influence on the parameters estimate under different perturbation schemes. The results and methods are applied to a dental clinical trial presented in Hadgu and Koch [Journal of Biopharmaceutical Statistic 9 (1999) 161–178].
Aoki, R., Bolfarine, H. and Singer, J. M. (2001). Null intercept measurement error regression models., Test 10, 441–454.
Aoki, R., Bolfarine, H., Achcar, J. A. and Pinto, D. L. (2003). Bayesian analysis of a multivariate null intercept error-in-variables regression model., Journal of Biopharmaceutical Statistics 13, 767–775.
Aoki, R., Pinto, J. D. L. and Achcar, J. A. (2006). Mixture of normal distributions in multivariate null intercept measurement error model., Journal of Biopharmaceutical Statistics 16, 785–802.
Aoki, R., Singer, J. M. and Bolfarine, H. (2007). Local influence for measurement error regression models for the analysis of pretest/posttest data., Journal of Applied Statistics 15, 317–330.
Arellano-Valle, R. B., Bolfarine, H. and Lachos, V. H. (2005). Skew-normal linear mixed models., Journal of Data Science 3, 415–438.
Arellano-Valle, R. B. and Genton, M. G. (2005). On fundamental skew distributions., Journal of Multivariate Analysis 96, 93–116.
Arellano-Valle, R. B., Ozan, S., Bolfarine, H. and Lachos, V. H. (2005). Skew normal measurement error models., Journal of Multivariate Analysis 98, 265–281.
Azzalini, A. (1985). A class of distributions which includes the normal ones., Scandinavian Journal of Statistics 12, 171–178.
Mathematical Reviews (MathSciNet): MR808153
Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution., Journal of the Royal Statistical Society 61, 579–602.
Azzalini, A. and Capitanio, A. (2003). Distributions generated and perturbation of symmetry with emphasis on the multivariate skew, t-distribution. Journal of the Royal Statistical Society, Series B 61, 367–389.
Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution., Biometrika 83, 715–726.
Bradley, R. A. and Gart, J. J. (1962). The asymptotic properties of ml estimators when sampling from associated populations., Biometrika 49, 205–214.
Mathematical Reviews (MathSciNet): MR137259
Branco, M. D. and Dey, D. K. (2001). A general class of multivariate skew-elliptical distributions., Journal of Multivariate Analysis 79, 99–113.
Chan, L. K. and Mak, T. K. (1979). On the maximun likelihood estimation of a linear structural relationship when the intercept is known., Journal of Multivariate Analyis 9, 304–313.
Mathematical Reviews (MathSciNet): MR538410
Chen, J. T., Gupta, A. K. and Troskie, C. G. (2003). The distribution of stock returns when the market is up., Communication in Statistics—Theory and Methods 32, 1541–1558.
Cheng, C. L. and Van-Ness, J. W. (1999)., Statistical Regression with Measurement Error. London: Arnold.
Cook, R. D. (1977). Detection of influential observations in linear regression., Technometrics 19, 15–18.
Mathematical Reviews (MathSciNet): MR436478
Cook, R. D. (1986). Assessment of local influence (with discussion)., Journal of the Royal Statistical Society, Series B 48, 133–169.
Mathematical Reviews (MathSciNet): MR867994
Cox, D. R. (2006)., Principles of Statistical Inference. New York: Cambridge Univ. Press.
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the em algorithm., Journal of the Royal Statistical Society, Series B 39, 1–22.
Mathematical Reviews (MathSciNet): MR501537
Escobar, E. and Meeker, W. (1992). Assessing influence in regression analysis with censured data., Biometrics 48, 507–528.
Fahrmeir, L. and Klinger, J. (1994). Estimating and testing generalized linear models under inequality restrictions., Statistical Papers 35, 211–229.
Genton, M. G. (2004)., Skew-Elliptical Distributions and Their Applications. A Journey Beyond Normality. New York: Chapman & Hall/CRC.
Hadgu, A. and Koch, G. (1999). Application of generalized estimating equations to a dental randomized clinical trial., Journal of Biopharmaceutical Statistic 9, 161–178.
Lachos, V. H., Ghosh, P. and Arellano-Valle, R. B. (2010). Likelihood based inference for skew–normal/independent linear mixed models., Statistica Sinica 20, 303–322.
Lachos, V. H., Montenegro, L. C. and Bolfarine, H. (2008). Inference and assessment of local influence in skew-normal null intercept measurement error model., Journal of Statistical Computation and Simulation 78, 395–419.
Liseo, B. and Loperfido, N. (2006). A note on reference priors for the scalar skew-normal distribution., Journal of Statistical Planning and Inference 136, 373–389.
Loperfido, N. (2010). Canonical transformation of skew-normal variates., Test 10, 146–165.
Meeker, W. Q. and Escobar, L. A. (1995). Teaching about approximate confidence regions based on maximum likelihood estimation., The American Statistician 49, 48–53.
Patefield, W. M. (1985). Information from the maximized likelihood function., Biometrics 72, 664–668.
Mathematical Reviews (MathSciNet): MR817581
Pawitan, Y. (2001)., In All Likelihood: Statistical Modellling and Inference Using Likelihood. Oxford: Clarendon Press.
Pewsey, A. (2000). Problems of inference for Azzalini’s skew-normal distribution., Journal of Applied Statistics 27, 859–870.
Sartori, N. (2006). Bias prevention of maximum likelihood estimates for scalar skew normal and skew-t distributions., Journal of Statistical Planning and Inference 136, 4259–4275.
Sen, P. K. and Singer, J. M. (1993)., Large Sample Methods in Statistics: An Introduction with Applications. New York: Chapman & Hall.
Turesky, S., Gilmore, N. D. and Glickman, J. (1970). Reduced plaque formation by the chloromethyl analogue of victamine c., Journal of Periodontology 41, 41–43.