We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x. The estimation of θ is based on an extended quasi score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in-y unbiased estimating functions. Of special interest is the case where the distribution of x depends only on a subvector α of θ, which may be considered a nuisance parameter. In general, α must be estimated simultaneously together with the rest of θ, but there are cases where α can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.We also study a number of special measurement error models in greater detail.
In a multivariate mean-variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is "extended" in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.
A polynomial structural measurement error model is considered. A goodness-of-fit test is constructed based on the quasi-likelihood estimator, which is asymptotically optimal in a large class of estimators. The power of the test is discussed. The test for the linear model with unknown nuisance parameters is studied in more detail. Similar test can be applied to much more general situation, where the estimator is constructed based on a score function.
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