Weighted empirical likelihood in some two-sample semiparametric models with various types of censored data

In this article, the weighted empirical likelihood is applied to a general setting of two-sample semiparametric models, which includes biased sampling models and case-control logistic regression models as special cases. For various types of censored data, such as right censored data, doubly censored data, interval censored data and partly interval-censored data, the weighted empirical likelihood-based semiparametric maximum likelihood estimator $(\tilde{\theta}_n,\tilde{F}_n)$ for the underlying parameter $\theta_0$ and distribution $F_0$ is derived, and the strong consistency of $(\tilde{\theta}_n,\tilde{F}_n)$ and the asymptotic normality of $\tilde{\theta}_n$ are established. Under biased sampling models, the weighted empirical log-likelihood ratio is shown to have an asymptotic scaled chi-squared distribution for censored data aforementioned. For right censored data, doubly censored data and partly interval-censored data, it is shown that $\sqrt{n}(\tilde{F}_n-F_0)$ weakly converges to a centered Gaussian process, which leads to a consistent goodness-of-fit test for the case-control logistic regression models.Comment: Published in at http://dx.doi.org/10.1214/009053607000000695 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Identification and Inference of Nonlinear Models Using Two Samples with Arbitrary Measurement Errors

This paper considers identification and inference of a general latent nonlinear model using two samples, where a covariate contains arbitrary measurement errors in both samples, and neither sample contains an accurate measurement of the corresponding true variable. The primary sample consists of some dependent variables, some error-free covariates and an error-ridden covariate, where the measurement error has unknown distribution and could be arbitrarily correlated with the latent true values. The auxiliary sample consists of another noisy measurement of the mismeasured covariate and some error-free covariates. We first show that a general latent nonlinear model is nonparametrically identified using the two samples when both could have nonclassical errors, with no requirement of instrumental variables nor independence between the two samples. When the two samples are independent and the latent nonlinear model is parameterized, we propose sieve quasi maximum likelihood estimation (MLE) for the parameter of interest, and establish its root-n consistency and asymptotic normality under possible misspecification, and its semiparametric efficiency under correct specification. We also provide a sieve likelihood ratio model selection test to compare two possibly misspecified parametric latent models. A small Monte Carlo simulation and an empirical example are presented.Data combination, Nonlinear errors-in-variables model, Nonclassical measurement error, Nonparametric identification, Misspecified parametric latent model, Sieve likelihood estimation and inference

Robust inference for threshold regression models

This paper considers robust inference in threshold regression models when the practitioners do not know whether at the threshold point the true specification has a kink or a jump, nesting previous works that assume either continuity or discontinuity at the threshold. We find that the parameter values under the kink restriction are irregular points of the Hessian matrix, destroying the asymptotic normality and inducing the cube-root convergence rate for the threshold estimate. However, we are able to obtain the same asymptotic distribution as Hansen (2000) for the quasi-likelihood ratio statistic for the unknown threshold. We propose to construct confidence intervals for the threshold by bootstrap test inversion. Finite sample performances of the proposed procedures are examined through Monte Carlo simulations and an economic empirical application is given

Robust misspecification tests for the Heckman’s two-step estimator

We construct and evaluate LM and Neyman’s C(α) tests based on bivariate Edgeworth expansions for the consistency of the Heckman’s two-step estimator in selection models, that is, for the marginal normality and linearity of the conditional expectation of the error terms. The proposed tests are robust to local misspecification in nuisance distributional parameters. Monte Carlo results show that instead of testing bivariate normality, testing marginal normality and linearity of the conditional expectations separately have a better size performance. Moreover, the robust variants of the tests have better size and similar power to non-robust tests, which determines that these tests can be successfully applied to detect specific departures from the null model of bivariate normality. We apply the tests procedures to women’s labor supply data

High dimensional generalized empirical likelihood for moment restrictions with dependent data

This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over-identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting

Model Assessment Tools for a Model False World

A standard goal of model evaluation and selection is to find a model that approximates the truth well while at the same time is as parsimonious as possible. In this paper we emphasize the point of view that the models under consideration are almost always false, if viewed realistically, and so we should analyze model adequacy from that point of view. We investigate this issue in large samples by looking at a model credibility index, which is designed to serve as a one-number summary measure of model adequacy. We define the index to be the maximum sample size at which samples from the model and those from the true data generating mechanism are nearly indistinguishable. We use standard notions from hypothesis testing to make this definition precise. We use data subsampling to estimate the index. We show that the definition leads us to some new ways of viewing models as flawed but useful. The concept is an extension of the work of Davies [Statist. Neerlandica 49 (1995) 185--245].Comment: Published in at http://dx.doi.org/10.1214/09-STS302 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org