Semiparametric maximum likelihood for measurement error model regression

This paper presents an EM algorithm for semiparametric likelihood analysis of linear, generalized linear, and nonlinear regression models with measurement errors in explanatory variables. A structural model is used in which probability distributions are specified for (a) the response and (b) the measurement error. A distribution is also assumed for the true explanatory variable but is left unspecified and is estimated by nonparametric maximum likelihood. For various types of extra information about the measurement error distribution, the proposed algorithm makes use of available routines that would be appropriate for likelihood analysis of (a) and (b) if the true x were available. Simulations suggest that the semiparametric maximum likelihood estimator retains a high degree of efficiency relative to the structural maximum likelihood estimator based on correct distributional assumptions and can outperform maximum likelihood based on an incorrect distributional assumption. The approach is illustrated on three examples with a variety of structures and types of extra information about the measurement error distribution.     

A note on Gauss--Hermite quadrature

For Gauss—Hermite quadrature, we consider a systematicmethod for transforming the variable of integration so thatthe integrand is sampled in an appropriate region. The effectivenessof the quadrature then depends on the ratio of the integrandto some Gaussian density being a smooth function, well approximatedby a low-order polynomial. It is pointed out that, in this approach,order one Gauss-Hermite quadrature becomes the Laplace approximationmxHermitequadrature becomes the Laplace approximationmxHermite quadraturebecomes the Laplace approximationmxHermite quadrature becomesthe Laplace approximation. Thus the quadrature as implementedhere can be thought of as a higher-order Laplace approximation.