EXTENDED ESTIMATING EQUATIONS AND EMPIRICAL LIKELIHOOD
Slud, Eric V
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Classic Estimating Equations (CEE) were first introduced by Godambe and have been widely used under both parametric and nonparametric settings. However, under some prominent semiparametric models, CEE cannot be used to identify certain low-dimensional parameters. We prove that under regularity conditions, for the Cox (1972) survival-time model, a CEE for the structural parameter does not exist; and under more restrictive conditions, a CEE for the structural parameter in the Accelerated Failure Time (AFT) model does not exist when lifetime is subject to random right censoring with unknown distribution. Motivated by this lack of coverage of CEE's for finite-dimensional parameters in semiparametric problems, we establish a method named Extended Estimating Equation (EEE). The EEE's relax the requirement in the CEE of which the estimating function must be a function of the independently identically distributed (i.i.d.) summands and instead allow the estimating function to incorporate ratio of the sums of functions depending on two of the i.i.d. arguments. To our knowledge, the broadest class of semiparametric models that can be investigated using EEE is the $\vphi$-transformation model class that we construct, where $\vphi$ is a given function of the covariate, structural parameter and random error with unknown hazard rate. With different choices of $\vphi$, the model can represent the general transformation model, nonlinear location-shift model, models incorporating cumulative integrated functions of times at risk and others. Inspired by Tsiatis's work in 1990, by defining martingale structure on the residual scale, we are able to prove the asymptotic linearity of the associated EEE, which leads to the asymptotic normality of the structural estimator. Another perspective from which to view EEE is to use it as a constraint in the Empirical Likelihood (EL) method. We first show that under the CEE setting, regardless of the continuity of the criterion function, there exists a neighborhood of the true structural parameter on which there always exists a probability vector that maximizes the EL. The same conclusion can be generalized to the EEE setting with continuous criterion function as well as the discontinuous criterion function with the martingale structure of the $\vphi$-transformation model or the Cox model. A point estimator for the structural parameter can be defined via maximizing the Profile Empirical Likelihood (pEL) associated with the EEE. We show that the pEL estimator is asymptotically normal, with asymptotic variance-covariance matrix identical to that of the Z-estimator obtained by directly solving for the root of EEE. Finally, we develop algorithms to compute and compare the Z-estimator and pEL estimator associated with the EEE and decide the minimal sample size for the two estimators to achieve asymptotic normality under three different parametric settings. Simulation shows a more symmetric covariate usually leads to a smaller threshold sample size, and the Z-estimator and pEL estimator are close in value and variance-covariance matrices. We also conclude that the pEL function tends to be much smoother, in settings where the EEE criterion function is non-smooth, than EEE itself, by comparing the plots of the projection of each function.