Abstract
The relationship between the linear errors-in-variables model and the corresponding ordinary linear model in statistical inference is studied. It is shown that normality of the distribution of covariate is a necessary and sufficient condition for the equivalence. Therefore, testing for lack-of-fit in linear errors-in-variables model can be converted into testing for it in the corresponding ordinary linear model under normality assumption. A test of score type is constructed and the limiting chi-squared distribution is derived under the null hypothesis. Furthermore, we discuss the power of the test and the choice of the weight function involved in the test statistic.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, T.W. Estimating linear statistical relationships. Ann. Statist., 12: 1–45 (1984)
Carroll, R.J., Ruppert, D. and Stefanski, L. A. Measurement error on nonlinear models. Monographs on statistics and applied probability, Chapman & Hall, 1995
Cook, R.D., Weisberg, S. Residual and influence in regression. Chapman and Hall, New York, 1982
Eubank, R.L., Hart, J.D. Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests. Biometrika, 80: 89–98 (1993)
Eubank, R.L., Spiegelman, C.H. Testing the goodness of fit of a linear model via nonparametric regression techniques. J. Amer. Statist. Assoc., 85: 387–392 (1990)
Fuller, W.A. Measurement error models. John Wiley, New York, 1987
Hall, P., Hart, J.D. Bootstrap test for difference between means in nonparametric regression, J. Amer. Statist. Assoc., 85: 1039–1049 (1990)
Härdle, W., Mammen, E. Comparing non-parametric versus parametric regression fits. Ann. Statist., 21: 1926–1947 (1993)
Härdle, W., Mammen, E., Müller, M. Testing parametric versus semiparametric modeling in generalized linear models. J. Amer. Statist. Assoc., 93: 1461–1474 (1998)
Hart, J.D. Nonparametric Smoothing and lack-of-fit tests. Springer Series in Statistics, New York, 1997
He, X., Ng, P. Quantile splines with several covariates. J. Statist. Plann. Inference, 75: 343–352 (1999)
He, X., Zhu, L.X. A lack-of-fit test for quantile regression. Journal of the American Statistical Association, 98: 1013–1022 (2003)
King, E. Testing the equality of two regression curves using linear smoothers. Statistics and Probability Letters, 12: 239–247 (1991)
Müller, H.G. Goodness-of-fit diagnostic for regression models. Scand. J. Statist., 19: 157–172 (1993)
Stute, W. Nonparametric model checks for regression. Ann. Statist., 25: 613–641 (1997)
Stute, W., Manteiga, G., Quindimil, M.P. Bootstrap approximations in model checks for regression. J. Amer. Statist. Asso., 93: 141–149 (1998)
Stute, W., Thies, S., Zhu, L.X. Model checks for regression: An innovation process approach. Ann. Statist., 26: 1916–1934 (1998)
Stute, W., Zhu, L.X. Model checks for generalized linear models. Scan. J. Statist., 29: 535–546 (2002)
Zhu, L.X. Model checking of dimension-reduction type for regression. Statist. Sinica, 13: 283–296 (2003)
Zhu, L.X., Ng, K.W. Checking the adequacy of a partial linear model. Statist. Sinica, 13: 763–781 (2003)
Author information
Authors and Affiliations
Additional information
Supported by the National Natural Science Foundation of China (No. 19771011) and Committee of Research and Conference Grant of the University of Hong Kong
Rights and permissions
About this article
Cite this article
Zhu, Lx., Cui, Hj. & Ng, K.W. Some Properties of A Lack-of-Fit Test for a Linear Errors in Variables Model. Acta Mathematicae Applicatae Sinica, English Series 20, 533–540 (2004). https://doi.org/10.1007/s10255-004-0190-y
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/s10255-004-0190-y