This dissertation includes three components. Component 1 provides an estima- tion procedure for circadian parameters in cyanobacteria. Component 2 explores the relationship between baseline and amplitude by model selection under the framework of smoothing spline. Component 3 investigates properties of hypothesis testing. The following three paragraphs briefly summarize these three components, respectively. Varying coefficient models are frequently used in statistical modeling. We pro- pose a semiparametric varying coefficient periodic model which is suitable to study periodic patterns. This model has ample applications in the study of the cyanobac- teria circadian clock. To achieve the desired flexibility, the model we consider may not be globally identifiable. We propose to perform local approximations by kernel based methods and focus on estimating one solution that is biologically meaningful. Asymptotic properties are developed. Simulations show that the gain by our proce- dure over the commonly used method is substantial. The methodology is illustrated by an application to a cyanobacteria dataset. Smoothing spline can be implemented, but a direct application with the penalty selected by the generalized cross-validation often leads to non-convergence outcomes. We propose an adjusted cross-validation instead, which resolves the difficulties. Biol- ogists believe that the amplitude function of the periodic component is proportional to the baseline function. To verify this belief, we propose a full model without any assumptions regarding such a relationship, and two reduced models with the ratio of baseline and amplitude to be a constant and a quadratic function of time, respectively. We use model selection techniques, Akaike information criterion (AIC) and Schwarz Bayesian information criterion (BIC), to determine the optimal model. Simulations show that AIC and BIC select the correct model with high probabilities. Application to cyanobacteria data shows that the full model is the best model. To investigate the same problem in component 2 by a formal hypothesis testing procedure, we develop kernel based methods. In order to construct the test statistic, we derive the global degree of freedom for the residual sum of squares. Simulations show that the proposed tests perform well. We apply the proposed procedures to the data and conclude that the baseline and amplitude functions share no linear or quadratic relationship.

This thesis investigates higher order asymptotic properties of a semiparametric averaged derivative estimator. Classical parametric models assume that we know the distribution function of random variables of interest up to finite dimensional parameters, while nonparametric models do not assume this knowledge. Parametric estimators typically enjoy - consistency and asymptotic normality under certain conditions, while nonparametric estimators converge to the true functionals of interest slower than parametric ones. Semiparametric estimators, a compromise between the two, have been intensively studied since the 1970s. Some of them have been shown to have the same convergence rate as parametric estimators despite involving nonparametric functional estimates. Semiparametric methods often suit econometrics because economic theory typically does not provide the whole information on economic variables which parametric methods require, and a sample of very large size is rarely available in econometrics. This thesis treats a semiparametric averaged derivative estimator of single index models. Its first order asymptotic theory has been studied since late 1980s. It has been shown to be n-consistent and asymptotically normally distributed under certain regularity conditions despite involving a nonparametric density estimate. However its higher order properties could be affected by the property of nonparametric estimates. We obtain valid Edgeworth expansions for both normalized and studentized estimators, and moreover show the bootstrap distribution approximates the exact distribution of the estimator asymptotically as well as the Edgeworth expansion for the normalized statistics. We propose optimal bandwidth choices which minimize the normal approximation error using the expansion. We also examine the finite sample performance of the Edgeworth expansions by a Monte Carlo study.

Mixed models are powerful tools for the analysis of clustered data and many extensions of the classical linear mixed model with normally distributed response have been established. As with all parametric models, correctness of the assumed model is critical for the validity of the ensuing inference. Model robust regression techniques predict mean response as a convex combination of a parametric and a nonparametric model fit to the data. It is a semiparametric method by which incompletely or incorrectly specified parametric models can be improved through adding an appropriate amount of a nonparametric fit. We apply this idea of model robustness in the framework of the linear mixed model. The mixed model robust regression (MMRR) predictions we propose are convex combinations of predictions obtained from a standard normal-theory linear mixed model, which serves as the parametric model component, and a locally weighted maximum likelihood fit which serves as the nonparametric component. An application of this technique with real data is provided. / Ph. D.

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One of the essential features of financial time series data is volatility. It is often the case that, over time, structural changes occur in volatility, and an accurate estimation of the volatility of financial time series requires careful identification of the change-points. A common approach to modeling the volatility of time series data is based on the well-known Generalized Autoregressive Conditional Heteroscedastic (GARCH) model. Although the problem of change-point estimation of volatility dynamics derived from the GARCH model has been considered in the literature, these approaches rely on parametric assumptions of the conditional error distribution, which are frequently violated in financial time series. This misspecification of error distribution may lead to change-point detection inaccuracies, resulting in unreliable GARCH volatility estimates. In this dissertation, we introduce novel change-point detection algorithms based on a semiparametric GARCH model. The proposed semiparametric GARCH model retains the structural advantages of the GARCH process while incorporating the flexibility of nonparametric conditional error distribution. Consequently, the likelihood function and the corresponding volatility estimates obtained via this semiparametric approach are more accurate than the traditional Quasi-Maximum Likelihood Estimation (QMLE) method that relies on an assumed parametric error distribution. The main objective of the change-point estimation problem is to detect the exact number and locations of the change-points. This dissertation proposes an innovative semiparametric GARCH process in developing solutions for change-point estimation problems. Specifically, a penalized likelihood approach based on a semiparametric GARCH model and an efficient binary segmentation algorithm is developed to estimate the change points' locations. The results demonstrate that in terms of change-point identification and estimation accuracy for multiple GARCH process variations, the proposed semiparametric method outperforms the commonly used approaches to change-point analysis in financial data.

One form of model robust regression (MRR) predicts mean response as a convex combination of a parametric and a nonparametric prediction. MRR is a semiparametric method by which an incompletely or an incorrectly specified parametric model can be improved through adding an appropriate amount of a nonparametric fit. The combined predictor can have less bias than the parametric model estimate alone and less variance than the nonparametric estimate alone. Additionally, as shown in previous work for uncorrelated data with linear mean function, MRR can converge faster than the nonparametric predictor alone. We extend the MRR technique to the problem of predicting mean response for clustered non-normal data. We combine a nonparametric method based on local estimation with a global, parametric generalized estimating equations (GEE) estimate through a mixing parameter on both the mean scale and the linear predictor scale. As a special case, when data are uncorrelated, this amounts to mixing a local likelihood estimate with predictions from a global generalized linear model. Cross-validation bandwidth and optimal mixing parameter selectors are developed. The global fits and the optimal and data-driven local and mixed fits are studied under no/some/substantial model misspecification via simulation. The methods are then illustrated through application to data from a longitudinal study. / Ph. D.

Master of Science / Department of Statistics / Weixin Yao / In this report, we introduce the minimum Hellinger distance (MHD) estimation method and review its history. We examine the use of Hellinger distance to obtain a new efficient and robust estimator for a class of semiparametric mixture models where one component has known distribution while the other component and the mixing proportion are unknown. Such semiparametric mixture models have been used in biology and the sequential clustering algorithm. Our new estimate is based on the MHD, which has been shown to have good efficiency and robustness properties. We use simulation studies to illustrate the finite sample performance of the proposed estimate and compare it to some other existing approaches. Our empirical studies demonstrate that the proposed minimum Hellinger distance estimator (MHDE) works at least as well as some existing estimators for most of the examples considered and outperforms the existing estimators when the data are under contamination. A real data set application is also provided to illustrate the effectiveness of our proposed methodology.

Semiparametric regression has become very popular in the field of Statistics over the years. While on one hand more and more sophisticated models are being developed, on the other hand the resulting theory and estimation process has become more and more involved. The main problems that are addressed in this work are related to efficient inferential procedures in general semiparametric regression problems. We first discuss efficient estimation of population-level summaries in general semiparametric regression models. 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. Next, motivated from the problem of testing for genetic effects on complex traits in the presence of gene-environment interaction, we consider developing score test in general semiparametric regression problems that involves Tukey style 1 d.f form of interaction between parametrically and non-parametrically modeled covariates. We develop adjusted score statistics which are unbiased and asymptotically efficient and can be performed using standard bandwidth selection methods. In addition, to over come the difficulty of solving functional equations, we give easy interpretations of the target functions, which in turn allow us to develop estimation procedures that can be easily implemented using standard computational methods. Finally, we take up the important problem of estimation in a general semiparametric regression model when covariates are measured with an additive measurement error structure having normally distributed measurement errors. In contrast to methods that require solving integral equation of dimension the size of the covariate measured with error, we propose methodology based on Monte Carlo corrected scores to estimate the model components and investigate the asymptotic behavior of the estimates. For each of the problems, we present simulation studies to observe the performance of the proposed inferential procedures. In addition, we apply our proposed methodology to analyze nontrivial real life data sets and present the results.

Tropospheric ozone is one of the six criteria pollutants regulated by the United States Environmental Protection Agency under the Clean Air Act and has been linked with several adverse health effects, including mortality. Due to the strong dependence on weather conditions, ozone may be sensitive to climate change and there is great interest in studying the potential effect of climate change on ozone, and how this change may affect public health. In this paper we develop a Bayesian spatial model to predict ozone under different meteorological conditions, and use this model to study spatial and temporal trends and to forecast ozone concentrations under different climate scenarios. We develop a spatial quantile regression model that does not assume normality and allows the covariates to affect the entire conditional distribution, rather than just the mean. The conditional distribution is allowed to vary from site-to-site and is smoothed with a spatial prior. For extremely large datasets our model is computationally infeasible, and we develop an approximate method. We apply the approximate version of our model to summer ozone from 1997-2005 in the Eastern U.S., and use deterministic climate models to project ozone under future climate conditions. Our analysis suggests that holding all other factors fixed, an increase in daily average temperature will lead to the largest increase in ozone in the Industrial Midwest and Northeast. / Dissertation

Doctor of Philosophy / Department of Statistics / Weixin Yao / This dissertation consists of three parts that are related to semiparametric mixture models. In Part I, we construct the minimum profile Hellinger distance (MPHD) estimator for a class of semiparametric mixture models where one component has known distribution with possibly unknown parameters while the other component density and the mixing proportion are unknown. Such semiparametric mixture models have been often used in biology and the sequential clustering algorithm. In Part II, we propose a new class of semiparametric mixture of regression models, where the mixing proportions and variances are constants, but the component regression functions are smooth functions of a covariate. A one-step backfitting estimate and two EM-type algorithms have been proposed to achieve the optimal convergence rate for both the global parameters and nonparametric regression functions. We derive the asymptotic property of the proposed estimates and show that both proposed EM-type algorithms preserve the asymptotic ascent property. In Part III, we apply the idea of single-index model to the mixture of regression models and propose three new classes of models: the mixture of single-index models (MSIM), the mixture of regression models with varying single-index proportions (MRSIP), and the mixture of regression models with varying single-index proportions and variances (MRSIPV). Backfitting estimates and the corresponding algorithms have been proposed for the new models to achieve the optimal convergence rate for both the parameters and the nonparametric functions. We show that the nonparametric functions can be estimated as if the parameters were known and the parameters can be estimated with the same rate of convergence, n[subscript](-1/2), that is achieved in a parametric model.