Local and omnibus goodness‐of‐fit tests in classical measurement error models

We consider functional measurement error models, i.e. models where covariates are measured with error and yet no distributional assumptions are made about the mismeasured variable. We propose and study a score-type local test and an orthogonal series-based, omnibus goodness-of-fit test in this context, where no likelihood function is available or calculated-i.e. all the tests are proposed in the semiparametric model framework. We demonstrate that our tests have optimality properties and computational advantages that are similar to those of the classical score tests in the parametric model framework. The test procedures are applicable to several semiparametric extensions of measurement error models, including when the measurement error distribution is estimated non-parametrically as well as for generalized partially linear models. The performance of the local score-type and omnibus goodness-of-fit tests is demonstrated through simulation studies and analysis of a nutrition data set.

[1]  A. Tsiatis,et al.  Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions , 2005 .

[2]  A. Tsiatis,et al.  ON CLOSED FORM SEMIPARAMETRIC ESTIMATORS FOR MEASUREMENT ERROR MODELS , 2006 .

[3]  R. Carroll,et al.  Locally Efficient Estimators for Semiparametric Models With Measurement Error , 2006 .

[4]  A goodness-of-fit test for a polynomial errors-in-variables model , 2004 .

[5]  D. Ruppert,et al.  Measurement Error in Nonlinear Models , 1995 .

[6]  Anastasios A. Tsiatis,et al.  Locally efficient semiparametric estimators for functional measurement error models , 2004 .

[7]  Pranab Kumar Sen,et al.  On M tests in linear models , 1982 .

[8]  P. Hall,et al.  Testing the suitability of polynomial models in errors-in-variables problems , 2007, 0803.3007.

[9]  Raymond J. Carroll,et al.  Conditional scores and optimal scores for generalized linear measurement-error models , 1987 .

[10]  Jianqing Fan,et al.  Nonparametric regression with errors in variables , 1993 .

[11]  A F Subar,et al.  Design and serendipity in establishing a large cohort with wide dietary intake distributions : the National Institutes of Health-American Association of Retired Persons Diet and Health Study. , 2001, American journal of epidemiology.

[12]  Frequentist-Bayes Lack-of-Fit Tests Based on Laplace Approximations , 2009 .

[13]  Gerda Claeskens,et al.  Goodness-of-Fit Tests in Mixed Models , 2009 .

[14]  C. Heyde.  Quasi-likelihood and its application : a general approach to optimal parameter estimation , 1998 .

[15]  A. Tsiatis.  Semiparametric Theory and Missing Data , 2006 .

[16]  Raymond J Carroll,et al.  Structure of dietary measurement error: results of the OPEN biomarker study. , 2003, American journal of epidemiology.

[17]  Jeffrey D. Hart,et al.  Nonparametric Smoothing and Lack-Of-Fit Tests , 1997 .

[18]  Jinfang Wang,et al.  Numerical Methods for Nonlinear Estimating Equations , 2003 .

[19]  Ryan Louis Janicki,et al.  Statistical Inference Based On Estimating Functions in Exact and Misspecified Models , 2009 .

[20]  D. Hall.  Measurement Error in Nonlinear Models: A Modern Perspective , 2008 .

[21]  P. Hall,et al.  Semiparametric estimators of functional measurement error models with unknown error , 2007 .

[22]  Raymond J. Carroll,et al.  Measurement error in nonlinear models: a modern perspective , 2006 .

[23]  Marc G. Genton,et al.  Explicit estimating equations for semiparametric generalized linear latent variable models , 2010 .