Dissertations / Theses on the topic 'Semiparametric theory'

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.

Bayesian nonparametric modelling has been widely applied to statistics and econometrics due to the various simulation methods that have been developed and in particular of Markov Chain Monte Carlo (MCMC) techniques. This thesis develops novel Bayesian nonparametric ordinal-response models and proposes efficient MCMC algorithms to estimate them. In chapter 21, we set up a model for inference on panel ordered data and apply it to sovereign credit ratings. In chapter 3, a model for ordinal-valued time series data is considered and is used to examine contagion across stock markets. Using real and simulated data, we show that the proposed models provide a great deal of flexibility in modelling and overcome the standard weakness of Bayesian methods due to the usual parametric assumptions.

Thesis (Ph. D.)--University of Missouri-Columbia, 2007.
The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on November 27, 2007) Vita. Includes bibliographical references.

Nous appliquons la théorie asymptotique des expériences statistiques à des problèmes liés aux valeurs extrêmes. Quatre modèles semi-paramétriques sont envisagés. Tout d'abord le modèle d'échantillonnage de fonction de répartition de type Pareto. L'index de Pareto est le paramètre d'intérêt tandis que la fonction à variation lente, qui intervient dans la décomposition de la fonction de survie, joue le rôle de nuisance. Nous considérons ensuite des observations i.i.d. de fonction de répartition de type Weibull. Le troisième modèle étudié est un modèle de régression. On considère des couples d'observations $(Y_i,X_i)$ indépendants, les v.a. $X_i$ sont i.i.d. de loi connue et on suppose que la fonction de répartition de la loi de $Y$ conditionnellement à $X$ est de type Pareto, avec une fonction à variation lente et un index $gamma$ qui dépendent de $X$. On fait l'hypothèse que la fonction $gamma$ a une forme quelconque mais connue, qui dépend d'un paramètre $\
Doctorat en sciences, Orientation statistique
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Semiparametric models play important roles in the field of biological statistics. In this dissertation, two types of semiparametic models are to be studied. One is the partially linear model, where the parametric part is a linear function. We are to investigate the two common estimation methods for the partially linear models when the data is correlated — longitudinal or clustered. The other is a semiparametric model where a latent covariate is incorporated in a mixed effects model. We will propose a semiparametric approach for estimation of this model and apply it to the study on colon carcinogenesis. First, we study the profilekernel and backfitting methods in partially linear models for clustered/longitudinal data. For independent data, despite the potential rootn inconsistency of the backfitting estimator noted by Rice (1986), the two estimators have the same asymptotic variance matrix as shown by Opsomer and Ruppert (1999). In this work, theoretical comparisons of the two estimators for multivariate responses are investigated. We show that, for correlated data, backfitting often produces a larger asymptotic variance than the profilekernel method; that is, in addition to its bias problem, the backfitting estimator does not have the same asymptotic efficiency as the profilekernel estimator when data is correlated. Consequently, the common practice of using the backfitting method to compute profilekernel estimates is no longer advised. We illustrate this in detail by following Zeger and Diggle (1994), Lin and Carroll (2001) with a working independence covariance structure for nonparametric estimation and a correlated covariance structure for parametric estimation. Numerical performance of the two estimators is investigated through a simulation study. Their application to an ophthalmology dataset is also described. Next, we study a mixed effects model where the main response and covariate variables are linked through the positions where they are measured. But for technical reasons, they are not measured at the same positions. We propose a semiparametric approach for this misaligned measurements problem and derive the asymptotic properties of the semiparametric estimators under reasonable conditions. An application of the semiparametric method to a colon carcinogenesis study is provided. We find that, as compared with the corn oil supplemented diet, fish oil supplemented diet tends to inhibit the increment of bcl2 (oncogene) gene expression in rats when the amount of DNA damage increases, and thus promotes apoptosis.

This dissertation considers semiparametric spectral estimates of temporal dependence in time series. Semiparametric frequency domain methods rely on a local parametric specification of the spectral density in a neighbourhood of the frequency of interest. Therefore, such methods can be applied to the analysis of singularities in the spectral density at frequency zero to identify long memory. They can also serve as the basis for the estimation of regular parts of the spectrum. One thereby avoids inconsistency that might arise from misspecification of dynamics at frequencies other than the frequency under focus. In case of long financial time series, the loss of efficiency with respect to fully parametric methods (or full band estimates) may be offset by the greater robustness properties. However, if semiparametric frequency domain methods are to be valid tools for inference on financial time series, they need to allow for conditional heteroscedasticity which is now recognized as a dominant feature of asset returns. This thesis provides a general specification which allows the time series under investigation to exhibit this type of behaviour. Two statistics are considered. The weighted periodogram statistic provides asymptotically normal point estimates of the spectral density at zero frequency for weakly dependent processes. The local Whittle (or local frequency domain maximum likelihood) estimate provides asymptotically normal estimates of long memory in possibly strongly dependent processes. The asymptotic results hold irrespective of the behaviour of the spectral density at non zero frequencies. The asymptotic variances are identical to those that obtain under conditional homogeneity in the distribution of the innovations to the observed process. In semiparametric frequency domain estimation, the choice of bandwidth is crucial. Indeed, it determines the asymptotic efficiency of the procedure. Optimal choices of bandwidth are derived, balancing asymptotic bias and asymptotic variance. Feasible versions of these optimal band-widths are proposed, and their performance is assessed in an extensive Monte Carlo study where the innovations to the observed process are simulated under numerous parametric submodels of the general specification, covering a wide range of persistence properties both in the levels and in the squares of the observed process. The techniques described above are applied to the analysis of temporal dependence and persistence in intra-day foreign exchange rate returns and their volatilities. While no strong indication of returns predictability is found in the former, a clear pattern arises in the latter, indicating that intra-day exchange rate returns are well described as martingale differences with weakly stationary and fractionally cointegrated long memory volatilities.

Thesis (Ph. D.) -- University of Maryland, College Park, 2007.
Thesis research directed by: Dept. of Mathematics. Title from t.p. of PDF. Includes bibliographical references. Published by UMI Dissertation Services, Ann Arbor, Mich. Also available in paper.

L'objectif général de cette thèse est d'introduire de nouveaux concepts ou d'étendre certaines procédures statistiques déjà existantes touchant à la notion de profondeur statistique.

Celle-ci, originellement introduite afin de généraliser la notion de médiane et de fournir naturellement un ordre (depuis un centre, vers l'extérieur) dans un contexte multivarié, a, depuis son développement, démontré ses nombreuses qualités, tant en termes de robustesse, que d'utilité dans de nombreuses procédures inférentielles.

Les résultats proposés dans ce travail se développent le long de trois axes.

Pour commencer, la thèse s'intéresse à la classification supervisée. La profondeur a, en effet, déjà été utilisée avec succès dans ce contexte. Cependant, jusqu'ici, les outils développés restaient limités aux distributions elliptiques, constituant ainsi une sévère restriction des méthodes utilisant les fonctions de profondeur, qui, pour la plupart, sont par essence nonparamétrique. La première partie de cette thèse propose donc une nouvelle méthode de classification, fondée sur la profondeur, dont on montrera qu'elle est essentiellement universellement convergente. En particulier, la règle de discrimination proposée se fonde sur les idées utilisées dans la classification par plus proches voisins, en introduisant cependant des voisinages fondés sur la profondeur, mieux à même de cerner le comportement des populations sous-jacentes.

Ces voisinages d'un point quelconque, et surtout l'information sur le comportement local de la distribution en ce point qu'ils apportent, ont été réutilisés dans la seconde partie de ce travail. Plusieurs auteurs ont en effet reconnu certaines limitations aux fonctions de profondeur, de par leur caractère global et la difficulté d'étudier par leur biais des distributions multimodales ou à support convexe. Une nouvelle définition de profondeur locale est donc développée et étudiée. Son utilité dans différents problèmes d'inférence est également explorée.

Enfin, la thèse s'intéresse au paramètre de forme pour les distributions elliptiques. Ce paramètre d'importance est utilisé dans de nombreuses procédures statistiques (analyse en composantes principales, analyse en corrélations canoniques, entre autres) et aucune fonction de profondeur pour celui-ci n'existait à ce jour. La profondeur de forme est donc définie et ses propriétés sont étudiées. En particulier, on montrera que le cadre général de la profondeur paramétrique n'est pas suffisant en raison de la présence du paramètre de nuisance (d'influence non nulle) qu'est l'échelle. Une application inférentielle est présentée dans le cadre des tests d'hypothèses.
Doctorat en Sciences
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A practical problem often encountered with observed count data is the presence of excess zeros. Zero-inflation in count data can easily be handled by zero-inflated models, which is a two-component mixture of a point mass at zero and a discrete distribution for the count data. In the presence of predictors, zero-inflated Poisson (ZIP) regression models are, perhaps, the most commonly used. However, the fully parametric ZIP regression model could sometimes be restrictive, especially with respect to the mixing proportions. Taking inspiration from some of the recent literature on semiparametric mixtures of regressions models for flexible mixture modeling, we propose a semiparametric ZIP regression model. We present an "EM-like" algorithm for estimation and a summary of asymptotic properties of the estimators. The proposed semiparametric models are then applied to a data set involving clandestine methamphetamine laboratories and Alzheimer's disease.

La causalité est un concept fondamental en science et en philosophie. Dans un contexte où la collecte massive de données de grande complexité s’impose dans tous les domaines, les statistiques jouent un rôle crucial dans l'inférence des causes et des effets. Cette thèse explore des méthodes avancées d'inférence causale. Elle met l'accent sur l'apprentissage de politiques d’action (“politiques” dans la suite), les variables instrumentales (IV), et les approches de différences en différences (DiD).Les méthodes IV et DiD sont utilisées par les chercheurs en épidémiologie, médecine, biostatistique, économétrie et sciences sociales quantitatives. Elles reposent sur des hypothèses restrictives, telles que, d’une part, l'exigence que l'IV n’ait aucun effet direct sur le résultat autre qu’à travers le traitement et, d’autre part, l'hypothèse de tendances parallèles en DiD, qui peut être violée en présence de confusion non mesurée.Dans ce contexte, cette thèse propose une approche innovante de DiD instrumentalisée pour l'apprentissage de politiques. Cette combinaison permet de relâcher certaines des hypothèses clés des méthodes IV et DiD conventionnelles. Des résultats d'identification novateurs pour les politiques optimales en présence de confusion non mesurée sont établis, et une gamme d'estimateurs (de Wald ; par pondération inverse des probabilités ; semi-paramétriques efficaces et multiplement robustes) sont introduits. Des garanties théoriques multiplement robustes sont fournies, incluant le taux cubique de convergence pour les politiques paramétriques et une inférence statistique valide avec des algorithmes de machine learning (ML) flexibles pour l'estimation des paramètres de nuisance. Ces méthodes sont en outre étendues à la configuration de données de panel.La majorité des méthodes d'inférence causale dans la littérature dépendent fortement de trois hypothèses causales standard pour identifier les effets causaux et les politiques optimales. Bien que des progrès aient été réalisés pour relâcher les hypothèses de consistance et de non-confusion, les avancées pour traiter les violations de l'hypothèse de positivité sont restées limitées.Dans ce contexte, cette thèse présente un cadre novateur d'apprentissage des politiques qui ne repose pas sur l'hypothèse de positivité, se concentrant plutôt sur des politiques dynamiques et stochastiques pratiques pour des applications réelles. Des politiques de score de propension incrémentale, ajustant les scores de propension par des paramètres individualisés, sont proposées. Leur analyse ne met en jeu que les hypothèses de consistance et de non-confusion. Ce cadre améliore le concept d'effets d'intervention incrémentale, l'adaptant aux contextes de politique de traitement individualisée, et utilise la théorie semi-paramétrique pour développer des fonctions d'influence efficaces et des estimateurs ML dédiés. Des méthodes pour optimiser les politiques en maximisant la fonction de valeur sous des contraintes spécifiques sont également introduites.De plus, le régime de traitement individualisé optimal (ITR) appris d'une population source peut ne pas se généraliser bien à une population cible en raison des décalages de covariables. Un cadre d'apprentissage par transfert est proposé pour l'estimation de l'ITR dans des populations hétérogènes avec des données de survie censurées à droite, que l’on rencontre fréquemment dans les études cliniques. Un estimateur doublement robuste pour la fonction de valeur ciblée est proposé, qui accommode une large classe de fonctionnelles de distributions de survie. Pour une classe pré-spécifiée d'ITRs, un taux cubique de convergence pour le paramètre estimé indexant l'ITR optimal est établi. L'utilisation de procédures de cross-fitting (ajustement croisé) assure la consistance et la normalité asymptotique de l'estimateur de valeur optimal proposé, y compris lorsque l’on a recours à des méthodes ML flexibles pour estimer des paramètres de nuisance
Causality is a fundamental concept in science and philosophy, and with the increasing complexity of data collection and structure, statistics plays a pivotal role in inferring causes and effects. This thesis delves into advanced causal inference methods, with a focus on policy learning, instrumental variables (IV), and difference-in-differences (DiD) approaches.The IV and DiD methods are critical tools widely used by researchers in fields like epidemiology, medicine, biostatistics, econometrics, and quantitative social sciences. However, these methods often face challenges due to restrictive assumptions, such as the IV's requirement to have no direct effect on the outcome other than through the treatment, and the parallel trends assumption in DiD, which may be violated in the presence of unmeasured confounding.In that context, this thesis introduces an innovative instrumented DiD approach to policy learning, which combines these two natural experiments to relax some of the key assumptions of conventional IV and DiD methods. To the best of our knowledge, the thesis presents the first comprehensive study of policy learning under the DiD setting. The direct policy search approach is proposed to learn optimal policies, based on the conditional average treatment effect estimators using instrumented DiD. Novel identification results for optimal policies under unmeasured confounding are established. Moreover, a range of estimators, including a Wald estimator, inverse probability weighting (IPW) estimators, and semiparametric efficient and multiply robust estimators, are introduced. Theoretical guarantees for these multiply robust policy learning approaches are provided, including the cubic rate of convergence for parametric policies and valid statistical inference with flexible machine learning algorithms for nuisance parameter estimation. These methods are further extended to the panel data setup.The majority of causal inference methods in the literature heavily depend on three standard causal assumptions to identify causal effects and optimal policies. While there has been progress in relaxing the consistency and unconfoundedness assumptions, addressing the violations of the positivity assumption has seen limited advancements.In that context, this thesis presents a novel policy learning framework that does not rely on the positivity assumption, instead focusing on dynamic and stochastic policies that are practical for real-world applications. Incremental propensity score policies, which adjust propensity scores by individualized parameters, are proposed, requiring only the consistency and unconfoundedness assumptions. This approach enhances the concept of incremental intervention effects, adapting it to individualized treatment policy contexts, and employs semiparametric theory to develop efficient influence functions and debiased machine learning estimators. Methods to optimize policy by maximizing the value function under specific constraints are also introduced.Additionally, the optimal individualized treatment regime (ITR) learned from a source population may not generalize well to a target population due to covariate shifts. A transfer learning framework is proposed for ITR estimation in heterogeneous populations with right-censored survival data, which is common in clinical studies and motivated by medical applications. This framework characterizes the efficient influence function (EIF) and proposes a doubly robust estimator for the targeted value function, accommodating a broad class of survival distribution functionals. For a pre-specified class of ITRs, a cubic rate of convergence for the estimated parameter indexing the optimal ITR is established. The use of cross-fitting procedures ensures the consistency and asymptotic normality of the proposed optimal value estimator, even with flexible machine learning methods for nuisance parameter estimation

In medical, behavioral, and social-psychological sciences, latent variable models are useful in handling variables that cannot be directly measured by a single observed variable, but instead are assessed through a number of observed variables. Traditional latent variable models are usually based on parametric assumptions on both relations between outcome and explanatory latent variables, and error distributions. In this thesis, semiparametric models with Bayesian P-splines are developed to relax these rigid assumptions.
In the fourth part of the thesis, the methodology developed in the third part is further extended to a varying coefficient model with latent variables. Varying coefficient model is a class of flexible semiparametric models in which the effects of covariates are modeled dynamically by unspecified smooth functions. A transformation varying coefficient model can handle arbitrarily distributed dynamic data. A simulation study shows that our proposed method performs well in the analysis of this complex model.
In the last part of the thesis, we propose a finite mixture of varying coefficient models to analyze dynamic data with heterogeneity. A simulation study demonstrates that our proposed method can explore possible existence of different groups in a dynamic data, where in each group the dynamic influences of covariates on the response variables have different patterns. The proposed method is applied to a longitudinal study concerning the effectiveness of heroin treatment. Distinct patterns of heroin use and treatment effect in different patient groups are identified.
In the second part of the thesis, a latent variable model is proposed to relax the first assumption, in which unknown additive functions of latent variables in the structural equation are modeled by Bayesian P-splines. The estimation of nonparametric functions is based on powerful Markov chain Monte Carlo (MCMC) algorithm with block update scheme. A simulation study shows that the proposed method can handle much wider situation than traditional models. The proposed semiparametric latent variable model is applied to a study on osteoporosis prevention and control. Some interesting functional relations, which may be overlooked by traditional parametric latent variable models, are revealed.
In the third part of the thesis, a transformation model is developed to relax the second assumption, which usually assumes the normality of observed variables and random errors. In our proposed model, the nonnormal response variables are transformed to normal by unknown functions modeled with Bayesian P-splines. This semiparametric transformation model is shown to be applicable to a wide range of statistical analysis. The model is applied to a study on the intervention treatment of polydrug use in which the traditional model assumption is violated because many observed variables exhibit serious departure from normality.
Lu, Zhaohua.
Adviser: Xin-Yuan Song.
Source: Dissertation Abstracts International, Volume: 72-04, Section: B, page: .
Thesis (Ph.D.)--Chinese University of Hong Kong, 2010.
Includes bibliographical references (leaves 119-130).
Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.
Abstract also in Chinese.

This dissertation develops efficient semiparametric estimation of parameters and expectations in dynamic nonlinear systems and analyzes the role of environmental factors in productivity growth accounting. The first essay considers the estimation of a general class of dynamic nonlinear systems. The semiparametric efficiency bound and efficient score are established for the problems. Using an M-estimator based on the efficient score, the feasible form of the semiparametric efficient estimators is worked out for several explicit assumptions regarding the degree of dependence between the predetermined variables and the disturbances of the model. Using this result, the second essay develops semiparametric estimation of the expectation of known functions of observable variables and unknown parameters in the class of dynamic nonlinear models. The semiparametric efficiency bound for this problem is established and an estimator that achieves the bound is worked out for two explicit assumptions. For the assumption of independence, the residual-based predictors proposed by Brown and Mariano (1989) are shown to be semiparametric efficient. Under unconditional mean zero assumption, I proposed an improved heteroskedastic autocorrelation consistent estimator. The third essay explores the directional distance function method to analyze productivity growth. The method explicitly evaluates the role of undesirable outputs of the economy, such as carbon dioxide and other green-house gases, have on the frontier production process which we specify as a piecewise linear and convex boundary function. We decompose productivity growth into efficiency change (catching up) and technology change (innovation). We test the statistical significance of the estimates using recently developed bootstrap method. We also explore implications for growth of total factor productivity in the OECD and Asia economies.

This dissertation proposes a suite of novel Bayesian semiparametric estimators for a proportional hazard function associated with the gaptimes, or inter-arrival times, of a counting process in survival analysis. The Cox model is applied and extended in order to identify the subsequent effect of an event on future events in a system with renewal. The estimators may also be applied, without changes, to model the effect of a point treatment on subsequent events, as well as the effect of an event on subsequent events in neighboring subjects. These Bayesian semiparametric estimators are used to analyze the survival and reliability of the New York City electric grid. In particular, the phenomenon of "infant mortality," whereby electrical supply units are prone to immediate recurrence of failure, is flexibly quantified as a period of increased risk. In this setting, the Cox model removes the significant confounding effect of seasonality. Without this correction, infant mortality would be misestimated due to the exogenously increased failure rate during summer months and times of high demand. The structural assumptions of the Bayesian estimators allow the use and interpretation of sparse event data without the rigid constraints of standard parametric models used in reliability studies.

New semiparametric panel estimation methods have been developed, which make minimal distributional or functional form assumptions on the model. These estimators are illustrated in an efficiency analysis of the banking industry. This analysis finds that functional form and distributional assumptions are important in efficiency estimation. Moreover, models of time varying efficiency are relevant and indicate productivity movements in the banking industry.

Multiple-outcome longitudinal data are abundant in clinical investigations. For example, infections with different pathogenic organisms are often tested concurrently, and assessments are usually taken repeatedly over time. It is therefore natural to consider a multivariate modeling approach to accommodate the underlying interrelationship among the multiple longitudinally measured outcomes. This dissertation proposes a multivariate semiparametric modeling framework for such data. Relevant estimation and inference procedures as well as model selection tools are discussed within this modeling framework. The first part of this research focuses on the analytical issues concerning binary data. The second part extends the binary model to a more general situation for data from the exponential family of distributions. The proposed model accounts for the correlations across the outcomes as well as the temporal dependency among the repeated measures of each outcome within an individual. An important feature of the proposed model is the addition of a bivariate smooth function for the depiction of concurrent nonlinear and possibly interacting influences of two independent variables on each outcome. For model implementation, a general approach for parameter estimation is developed by using the maximum penalized likelihood method. For statistical inference, a likelihood-based resampling procedure is proposed to compare the bivariate nonlinear effect surfaces across the outcomes. The final part of the dissertation presents a variable selection tool to facilitate model development in practical data analysis. Using the adaptive least absolute shrinkage and selection operator (LASSO) penalty, the variable selection tool simultaneously identifies important fixed effects and random effects, determines the correlation structure of the outcomes, and selects the interaction effects in the bivariate smooth functions. Model selection and estimation are performed through a two-stage procedure based on an expectation-maximization (EM) algorithm. Simulation studies are conducted to evaluate the performance of the proposed methods. The utility of the methods is demonstrated through several clinical applications.

Semiparametric nonlinear mixed-effects (NLME) models are flexible for modelling complex longitudinal data. Covariates are usually introduced in the models to partially explain inter-individual variations. Some covariates, however, may be measured with substantial errors. Moreover, the responses may be missing and the missingness may be nonignorable. In this thesis, we develop approximate maximum likelihood inference in the following three problems: (1) semiparametric NLME models with measurement errors and missing data in time-varying covariates; (2) semiparametric NLME models with covariate measurement errors and outcome-based informative missing responses; (3) semiparametric NLME models with covariate measurement errors and random-effect-based informative missing responses. Measurement errors, dropouts, and missing data are addressed simultaneously in a unified way. For each problem, we propose two joint model methods to simultaneously obtain approximate maximum likelihood estimates (MLEs) of all model parameters. Some asymptotic properties of the estimates are discussed. The proposed methods are illustrated in a HIV data example. Simulation results show that all proposed methods perform better than the commonly used two-step method and the naive method.
Science, Faculty of
Statistics, Department of
Graduate