Yanyuan Ma

Highly Robust Estimation of the Autocovariance Function

In this paper, the problem of the robustness of the sample autocovariance function is addressed. We propose a new autocovariance estimator, based on a highly robust estimator of scale. Its robustness properties are studied by means of the influence function, and a new concept of temporal breakdown point. As the theoretical variance of the estimator does not have a closed form, we perform a simulation study. Situations with various size of outliers are tested. They confirm the robustness properties of the new estimator. An S-Plus function for the highly robust autocovariance estimator is made available on the Web at http://www-math.mit.edu/~yanyuan/Genton/Time/time.html. At the end, we analyze a time series of monthly interest rates of an Austrian bank.

A Semiparametric Approach to Dimension Reduction

We provide a novel and completely different approach to dimension-reduction problems from the existing literature. We cast the dimension-reduction problem in a semiparametric estimation framework and derive estimating equations. Viewing this problem from the new angle allows us to derive a rich class of estimators, and obtain the classical dimension reduction techniques as special cases in this class. The semiparametric approach also reveals that in the inverse regression context while keeping the estimation structure intact, the common assumption of linearity and/or constant variance on the covariates can be removed at the cost of performing additional nonparametric regression. The semiparametric estimators without these common assumptions are illustrated through simulation studies and a real data example. This article has online supplementary material.

Locally Efficient Semiparametric Estimators for Generalized Skew-Elliptical Distributions

We consider a class of generalized skew-normal distributions that is useful for selection modeling and robustness analysis and derive a class of semiparametric estimators for the location and scale parameters of the central part of the model. We show that these estimators are consistent and asymptotically normal. We present the semiparametric efficiency bound and derive the locally efficient estimator that achieves this bound if the model for the skewing function is correctly specified. The estimators that we propose are consistent and asymptotically normal even if the model for the skewing function is misspecified, and we compute the loss of efficiency in such cases. We conduct a simulation study and provide an illustrative example. Our method is applicable to generalized skew-elliptical distributions.

A functional generalized method of moments approach for longitudinal studies with missing responses and covariate measurement error

Covariate measurement error and missing responses are typical features in longitudinal data analysis. There has been extensive research on either covariate measurement error or missing responses, but relatively little work has been done to address both simultaneously. In this paper, we propose a simple method for the marginal analysis of longitudinal data with time-varying covariates, some of which are measured with error, while the response is subject to missingness. Our method has a number of appealing properties: assumptions on the model are minimal, with none needed about the distribution of the mismeasured covariate; implementation is straightforward and its applicability is broad. We provide both theoretical justification and numerical results.

Locally Efficient Estimators for Semiparametric Models With Measurement Error

We derive constructive locally efficient estimators in semiparametric measurement error models. The setting is one in which the likelihood function depends on variables measured with and without error, where the variables measured without error can be modeled nonparametrically. The algorithm is based on backfitting. We show that if one adopts a parametric model for the latent variable measured with error and if this model is correct, then the estimator is semiparametric efficient; if the latent variable model is misspecified, then our methods lead to a consistent and asymptotically normal estimator. Our method further produces an estimator of the nonparametric function that achieves the standard bias and variance property. We extend the methodology to allow estimation of parameters in the measurement error model by additional data in the form of replicates or instrumental variables. The methods are illustrated through a simulation study and a data example, where the putative latent variable distribution is a shifted lognormal, but concerns about the effects of misspecification of this assumption and the linear assumption of another covariate demand a more model-robust approach. A special case of wide interest is the partial linear measurement error model. If one assumes that the model error and the measurement error are both normally distributed, then our estimator has a closed form. When a normal model for the unobservable variable is also posited, our estimator becomes consistent and asymptotically normally distributed for the general partially linear measurement error model, even without any of the normality assumptions under which the estimator is originally derived. We show that the method in fact reduces to a same estimator as that of Liang et al., thus demonstrating a previously unknown optimality property of their method.

Efficient estimation of population-level summaries in general semiparametric regression models

This article considers a wide class of semiparametric regression models in which interest focuses on population-level quantities that combine both the parametric and the nonparametric parts of the model. Special cases in this approach include generalized partially linear models, generalized partially linear single-index models, structural measurement error models, and many others. For estimating the parametric part of the model efficiently, profile likelihood kernel estimation methods are well established in the literature. Here our focus is on estimating general population-level quantities that combine the parametric and nonparametric parts of the model (e.g., population mean, probabilities, etc.). We place this problem in a general context, provide a general kernel-based methodology, and derive the asymptotic distributions of estimates of these population-level quantities, showing that in many cases the estimates are semiparametric efficient. For estimating the population mean with no missing data, we show that the sample mean is semiparametric efficient for canonical exponential families, but not in general. We apply the methods to a problem in nutritional epidemiology, where estimating the distribution of usual intake is of primary interest and semiparametric methods are not available. Extensions to the case of missing response data are also discussed.

Saddlepoint Test in Measurement Error Models

We develop second-order hypothesis testing procedures in functional measurement error models for small or moderate sample sizes, where the classical first-order asymptotic analysis often fails to provide accurate results. In functional models no distributional assumptions are made on the unobservable covariates and this leads to semiparametric models. Our testing procedure is derived using saddlepoint techniques and is based on an empirical distribution estimation subject to the null hypothesis constraints, in combination with a set of estimating equations which avoid a distribution approximation. The validity of the method is proved in theorems for both simple and composite hypothesis tests, and is demonstrated through simulation and a farm size data analysis.

Cure Rate Model With Mismeasured Covariates Under Transformation

Cure rate models explicitly account for the survival fraction in failure time data. When the covariates are measured with errors, naively treating mismeasured covariates as error-free would cause estimation bias and thus lead to incorrect inference. Under the proportional hazards cure model, we propose a corrected score approach as well as its generalization, and implement a transformation on the mismeasured covariates toward error additivity and/or normality. The corrected score equations can be easily solved through the backfitting procedure, and the biases in the parameter estimates are successfully eliminated. We show that the proposed estimators for the regression coefficients are consistent and asymptotically normal. We conduct simulation studies to examine the finite-sample properties of the new method and apply it to a real data set for illustration.

Robustness properties of dispersion estimators

In this paper, we derive the influence function of dispersion estimators, based on a scale approach. The relation between the gross-error sensitivity of dispersion estimators and the one of the underlying scale estimator is described. We show that for the bivariate Gaussian distributions, the asymptotic variance of covariance estimators is minimal in the independent case, and is strictly increasing with the absolute value of the underlying covariance. The behavior of the asymptotic variance of correlation estimators seems to be the opposite, i.e. maximal for independent data, and strictly decreasing with the absolute value of the underlying correlation. In particular, dispersion estimators based on M-estimators of scale are studied closely. The one based on the median absolute deviation is the most B-robust in the class of symmetric estimators. Some other examples are analyzed, based on the maximum likelihood and the Welsch estimator of scale.

Functional and Structural Methods With Mixed Measurement Error and Misclassification in Covariates

Covariate measurement imprecision or errors arise frequently in many areas. It is well known that ignoring such errors can substantially degrade the quality of inference or even yield erroneous results. Although in practice both covariates subject to measurement error and covariates subject to misclassification can occur, research attention in the literature has mainly focused on addressing either one of these problems separately. To fill this gap, we develop estimation and inference methods that accommodate both characteristics simultaneously. Specifically, we consider measurement error and misclassification in generalized linear models under the scenario that an external validation study is available, and systematically develop a number of effective functional and structural methods. Our methods can be applied to different situations to meet various objectives.