Dissertations / Theses on the topic 'Local linear estimator'

In a nonparametric setting, the functional form of the relationship between the response variable and the associated predictor variables is unspecified; however it is assumed to be a smooth function. The main aim of nonparametric regression is to highlight an important structure in data without any assumptions about the shape of an underlying regression function. In regression, the random and fixed design models should be distinguished. Among the variety of nonparametric regression estimators currently in use, kernel type estimators are most popular. Kernel type estimators provide a flexible class of nonparametric procedures by estimating unknown function as a weighted average using a kernel function. The bandwidth which determines the influence of the kernel has to be adapted to any kernel type estimator. Our focus is on Nadaraya-Watson estimator and Local Linear estimator which belong to a class of kernel type regression estimators called local polynomial kerne l estimators. A closely related problem is the determination of an appropriate sample size that would be required to achieve a desired confidence level of accuracy for the nonparametric regression estimators. Since sequential procedures allow an experimenter to make decisions based on the smallest number of observations without compromising accuracy, application of sequential procedures to a nonparametric regression model at a given point or series of points is considered. The motivation for using such procedures is: in many applications the quality of estimating an underlying regression function in a controlled experiment is paramount; thus, it is reasonable to invoke a sequential procedure of estimation that chooses a sample size based on recorded observations that guarantees a preassigned accuracy. We have employed sequential techniques to develop a procedure for constructing a fixed-width confidence interval for the predicted value at a specific point of the independent variable. These fixed-width confidence intervals are developed using asymptotic properties of both Nadaraya-Watson and local linear kernel estimators of nonparametric kernel regression with data-driven bandwidths and studied for both fixed and random design contexts. The sample sizes for a preset confidence coefficient are optimized using sequential procedures, namely two-stage procedure, modified two-stage procedure and purely sequential procedure. The proposed methodology is first tested by employing a large-scale simulation study. The performance of each kernel estimation method is assessed by comparing their coverage accuracy with corresponding preset confidence coefficients, proximity of computed sample sizes match up to optimal sample sizes and contrasting the estimated values obtained from the two nonparametric methods with act ual values at given series of design points of interest. We also employed the symmetric bootstrap method which is considered as an alternative method of estimating properties of unknown distributions. Resampling is done from a suitably estimated residual distribution and utilizes the percentiles of the approximate distribution to construct confidence intervals for the curve at a set of given design points. A methodology is developed for determining whether it is advantageous to use the symmetric bootstrap method to reduce the extent of oversampling that is normally known to plague Stein's two-stage sequential procedure. The procedure developed is validated using an extensive simulation study and we also explore the asymptotic properties of the relevant estimators. Finally, application of our proposed sequential nonparametric kernel regression methods are made to some problems in software reliability and finance.

Uma das principais preocupações da macroeconomia é a compreensão da dinâmica da inflação no curto prazo. Entender como a inflação se relaciona com a atividade econômica é decisivo para traçar estratégias de desinflação, assim como, de determinação da trajetória de política monetária. Uma questão que surge é qual a forma exata da relação inflação-produto. Ou seja, podemos caracterizar essa relação como não linear? Se sim, qual a forma dessa não linearidade? Para responder a essas perguntas, estimou-se a relação inflação-produto de forma não-paramétrica através de um local linear kernel estimator. O resultado da estimação gerou uma forma funcional a qual foi aproximada pela estimação, via GMM, de uma curva de Phillips Novo-Keynesiana Híbrida. Essa abordagem foi aplicada para o Brasil a partir de 2000. As estimações sugeriram que a dinâmica da inflação brasileira é melhor descrita quando adiciona-se um termo cúbico relativo ao hiato do produto, ou seja, a inflação brasileira mostrou-se state-dependent.
One of the most important macroeconomics’ concerns is the comprehension about sort-run inflation dynamic. To understand how inflation relates to economic activity is crucial to decision-making in disinflation strategies, as well as in monetary policy paths. A question that arises is what does real form of relation inflation-output trade-off? Could one characterize it as a non-linear relation? If does, what is the shape of this non-linear relation? To answer those questions, we estimate the inflation-output relation non-parametrically using a local linear kernel estimator. The functional form achieved was approximated by a New-Keynesian Hybrid Phillips Curve, which one was estimated by GMM. This approach was applied to Brazil since 2000. We have found evidence that Brazilian inflation dynamic is better described adding a cubic term related to output gap, in other words, the Brazilian inflation is state-dependent.

This thesis consists of four papers that are related to commonly used propensity score-based estimators for average causal effects. The first paper starts with the observation that researchers often have access to data containing lots of covariates that are correlated. We therefore study the effect of correlation on the asymptotic variance of an inverse probability weighting and a matching estimator. Under the assumptions of normally distributed covariates, constant causal effect, and potential outcomes and a logit that are linear in the parameters we show that the correlation influences the asymptotic efficiency of the estimators differently, both with regard to direction and magnitude. Further, the strength of the confounding towards the outcome and the treatment plays an important role. The second paper extends the first paper in that the estimators are studied under the more realistic setting of using the estimated propensity score. We also relax several assumptions made in the first paper, and include the doubly robust estimator. Again, the results show that the correlation may increase or decrease the variances of the estimators, but we also observe that several aspects influence how correlation affects the variance of the estimators, such as the choice of estimator, the strength of the confounding towards the outcome and the treatment, and whether constant or non-constant causal effect is present. The third paper concerns estimation of the asymptotic variance of a propensity score matching estimator. Simulations show that large gains can be made for the mean squared error by properly selecting smoothing parameters of the variance estimator and that a residual-based local linear estimator may be a more efficient estimator for the asymptotic variance. The specification of the variance estimator is shown to be crucial when evaluating the effect of right heart catheterisation, i.e. we show either a negative effect on survival or no significant effect depending on the choice of smoothing parameters.   In the fourth paper, we provide an analytic expression for the covariance matrix of logistic regression with normally distributed regressors. This paper is related to the other papers in that logistic regression is commonly used to estimate the propensity score.

Local polynomial regression has received a great deal of attention in the past. It is a highly adaptable regression method when the true response model is not known. However, estimates obtained in this way are not guaranteed to be monotone. In some situations the response is known to depend monotonically upon some variables. Various methods have been suggested for constraining nonparametric local polynomial regression to be monotone. The earliest of these is known as the Pool Adjacent Violators algorithm (PAVA) and was first suggested by Brunk (1958). Kappenman (1987) suggested that a non-parametric estimate could be made monotone by simply increasing the bandwidth used until the estimate was monotone. Dette et al. (2006) have suggested a monotonicity constraint which they call the DNP method. Their method involves calculating a density estimate of the unconstrained regression estimate, and using this to calculate an estimate of the inverse of the regression function. Fan, Heckman and Wand (1995) generalized local polynomial regression to quasi-likelihood based settings. Obviously such estimates are not guaranteed to be monotone, whilst in many practical situations monotonicity of response is required. In this thesis I discuss how the above mentioned monotonicity constraint methods can be adapted to the quasi-likelihood setting. I am particularly interested in the estimation of monotone psychometric functions and, more generally, biological transducer functions, for which the response is often known to follow a distribution which belongs to the exponential family. I consider some of the key theoretical properties of the monotonised local linear estimators in the quasi-likelihood setting. I establish asymptotic expressions for the bias and variance for my adaptation of the DNP method (called the LDNP method) and show that this estimate is asymptotically normally distributed and first{-}order equivalent to competing methods. I demonstrate that this adaptation overcomes some of the problems with using the DNP method in likelihood based settings. I also investigate the choice of second bandwidth for use in the density estimation step. I compare the LDNP method, the PAVA method and the bandwidth method by means of a simulation study. I investigate a variety of response models, including binary, Poisson and exponential. In each study I calculate monotone estimates of the response curve using each method and compare their bias, variance, MSE and MISE. I also apply these methods to analysis of data from various hearing and vision studies. I show some of the deficiencies of using local polynomial estimates, as opposed to local likelihood estimates.

Adaptive filters are frequently employed in many applications, such as, system identification, adaptive echo cancellation (AEC), active noise control (ANC), adaptive beamforming, speech signal processing and other related problems, in which the statistic of the underlying signals is either unknown a priori, or slowly-varying. Given the observed signals under study, we shall consider, in this dissertation, the time-varying linear model with Gaussian or contaminated Gaussian (CG) noises. In particular, we focus on recursive local estimation and its applications in linear systems. We base our development on the concept of local likelihood function (LLF) and local posterior probability for parameter estimation, which lead to efficient adaptive filtering algorithms. We also study the convergence performance of these algorithms and their applications by theoretical analyses. As for applications, another important one is to utilize adaptive filters to obtain recursive hypothesis testing and model order selection methods. It is known that the maximum likelihood estimate (MLE) may lead to large variance or ill-conditioning problems when the number of observations is limited. An effective approach to address these problems is to employ various form of regularization in order to reduce the variance at the expense of slightly increased bias. In general, this can be viewed as adopting the Bayesian estimation, where the regularization can be viewed as providing a certain prior density of the parameters to be estimated. By adopting different prior densities in the LLF, we derive the variable regularized QR decomposition-based recursive least squares (VR-QRRLS) and recursive least M-estimate (VR-QRRLM) algorithms. An improved state-regularized variable forgetting factor QRRLS (SR-VFF-QRRLS) algorithm is also proposed. By approximating the covariance matrix in the RLS, new variable regularized and variable step-size transform domain normalized least mean square (VR-TDNLMS and VSS-TDNLMS) algorithms are proposed. Convergence behaviors of these algorithms are studied to characterize their performance and provide useful guidelines for selecting appropriate parameters in practical applications. Based on the local Bayesian estimation framework for linear model parameters developed previously, the resulting estimate can be utilized for recursive nonstationarity detection. This can be cast under the problem of hypothesis testing, as the hypotheses can be viewed as two competitive models between stationary and nonstationary to be selected. In this dissertation, we develop new regularized and recursive generalized likelihood ratio test (GLRT), Rao’s and Wald tests, which can be implemented recursively in a QRRLS-type adaptive filtering algorithm with low computational complexity. Another issue to be addressed in nonstationarity detection is the selection of various models or model orders. In particular, we derive a recursive method for model order selection from the Bayesian Information Criterion (BIC) based on recursive local estimation. In general, the algorithms proposed in this dissertation have addressed some of the important problems in estimation and detection under the local and recursive Bayesian estimation framework. They are intrinsically connected together and can potentially be utilized for various applications. In this dissertation, their applications to adaptive beamforming, ANC system and speech signal processing, e.g. adaptive frequency estimation and nonstationarity detection, have been studied. For adaptive beamforming, the difficulties in determining the regularization or loading factor have been explored by automatically selecting the regularization parameter. For ANC systems, to combat uncertainties in the secondary path estimation, regularization techniques can be employed. Consequently, a new filtered-x VR-QRRLM (Fx-VR-QRRLM) algorithm is proposed and the theoretical analysis helps to address challenging problems in the design of ANC systems. On the other hand, for ANC systems with online secondary-path modeling, the coupling effect of the ANC controller and the secondary path estimator is thoroughly studied by analyzing the Fx-LMS algorithm. For speech signal processing, new approaches for recursive nonstationarity detection with automatic model order selection are proposed, which provides online time-varying autoregressive (TVAR) parameter estimation and the corresponding stationary intervals with low complexity.
published_or_final_version
Electrical and Electronic Engineering
Doctoral
Doctor of Philosophy

L'objectif de cette thèse est d'étudier le problème de la détection de données dans le système de communication sans fil, à la fois pour le cas de l'information d'état de canal parfaite et imparfaite au niveau du récepteur. Comme on le sait, la complexité de MLSE est exponentielle en la mémoire de canal et la cardinalité de l'alphabet symbole est rapidement ingérable, ce qui force à recourir à des approches sousoptimales. Par conséquent, en premier lieu, nous proposons une nouvelle égalisation itérative lorsque le canal est inconnu à l'émetteur et parfaitement connu au niveau du récepteur. Ce récepteur est basé sur une approche de continuation, et exploite l'idée d'approcher une fonction originale de coût d'optimisation par une suite de fonctions plus dociles et donc de réduire la complexité de calcul au récepteur.En second lieu, en vue de la détection de données sous un canal dynamique linéaire, lorsque le canal est inconnu au niveau du récepteur, le récepteur doit être en mesure d'effectuer conjointement l'égalisation et l'estimation de canal. De cette manière, on formule une représentation de modèle état-espace combiné du système de communication. Par cette représentation, nous pouvons utiliser le filltre de Kalman comme le meilleur estimateur des paramètres du canal. Le but de cette section est de motiver de façon rigoureuse la mise en place du filltre de Kalman dans l'estimation des sequences de Markov par des canaux dynamiques Gaussien. Par la présente, nous interprétons et explicitons les approximations sous-jacentes dans les approaches heuristiques.Enfin, si nous considérons une approche plus générale pour le canal dynamique non linéaire, nous ne pouvons pas utiliser le filtre de Kalman comme le meilleur estimateur. Ici, nous utilisons des modèles commutation d’espace-état (SSSM) comme modèles espace-état non linéaires. Ce modèle combine le modèle de Markov caché (HMM) et le modèle espace-état linéaire (LSSM). Pour l'estimation de canal et la detection de données, l'approche espérance et maximisation (EM) est utilisée comme approche naturelle. De cette façon, le filtre de Kalman étendu (EKF) et les filtres à particules sont évités
The aim of this thesis is to study the problem of data detection in wireless communication system, for both case of perfect and imperfect channel state information at the receiver. As well known, the complexity of MLSE being exponential in the channel memory and in the symbol alphabet cardinality is quickly unmanageable and forces to resort to sub-optimal approaches. Therefore, first we propose a new iterative equalizer when the channel is unknown at the transmitter and perfectly known at the receiver. This receiver is based on continuation approach, and exploits the idea of approaching an original optimization cost function by a sequence of more tractable functions and thus reduce the receiver's computational complexity. Second, in order to data detection under linear dynamic channel, when the channel is unknown at the receiver, the receiver must be able to perform joint equalization and channel estimation. In this way, we formulate a combined state-space model representation of the communication system. By this representation, we can use the Kalman filter as the best estimator for the channel parameters. The aim in this section is to motivate rigorously the introduction of the Kalman filter in the estimation of Markov sequences through Gaussian dynamical channels. By this we interpret and make clearer the underlying approximations in the heuristic approaches. Finally, if we consider more general approach for non linear dynamic channel, we can not use the Kalman filter as the best estimator. Here, we use switching state-space model (SSSM) as non linear state-space model. This model combines the hidden Markov model (HMM) and linear state-space model (LSSM). In order to channel estimation and data detection, the expectation and maximization (EM) procedure is used as the natural approach. In this way extended Kalman filter (EKF) and particle filters are avoided

The sequential estimation of the states (filtering) and the corresponding simultaneous estimation of the states and fixed parameters of a dynamic state-space model, being linear or not, is an important probleminmany fields of research, such as in the area of finance. The main objective of this research is to estimate sequ entially and efficiently –from a Bayesian perspective via the particle filtering methodology– the states and/or the fixed parameters of a nonstandard dynamic state-spacemodel: one that is possibly nonlinear, non-stationary or non-Gaussian. The present thesis consists of seven chapters and is structured into two parts. Chapter 1 introduces basic concepts, themotivation, the purpose, and the outline of the thesis. Chapters 2-4, the first part of the thesis, focus on the estimation of the states. Chapter 2 provides a comprehensive review of themost classic algorithms (non-simulation based: KF, EKF, and UKF; and simulation based: SIS, SIR, ASIR, EPF, and UPF1) used for filtering solely the states of a dynamic statespacemodel. All these filters scattered in the literature are not only described in detail, but also placed in a unified notation for the sake of consistency, readability and comparability. Chapters 3 and 4 confirm the efficiency of the well-established particle filtering methodology, via extensive Monte Carlo (MC) studies, when estimating only the latent states for a dynamic state-space model, being linear or not. Also, complementary MC studies are conducted to analyze some relevant issues within the adopted approach, such as the degeneracy problem, the resampling strategy, or the possible impact on estimation of the number of particles used and the time series length. Chapter 3 specifically illustrates the performance of the particle filtering methodology in a linear and Gaussian context, using the exact Kalman filter as a benchmark. The performance of the four studied particle filter variants (SIR, SIRopt, ASIR, KPF, the latter being a special case of the EPF algorithm) is assessed using two apparently simple, but important time series processes: the so-called Local Level Model (LLM) and the AR(1) plus noise model, which are non-stationary and stationary, respectively. An exhaustive study on the effect of the signal-to-noise ratio (SNR) over the quality of the estimation is additionally performed. ComplementaryMC studies are conducted to assess the degree of degeneracy and the possible effect of increasing the number of particles and the time series length. Chapter 4 assesses and illustrates the performance of the particle filtering methodology in a nonlinear context. Specifically, a synthetic nonlinear, non Gaussian and non-stationary state space model taken from literature is used to illustrate the performance of the four competing particle filters under study (SIR, ASIR, EPF, UPF) in contraposition to two well-known non-simulation based filters (EKF, UKF). In this chapter, the residual and stratified resampling schemes are compared and the effect of increasing the number of particles is addressed. In the second part (Chapters 5 and 6), extensive MC studies are carried out, but the main goal is the simultaneous estimation of states and fixed model parameters for chosen non-standard dynamic models. This area of research is still very active and it is within this area where this thesis contributes themost. Chapter 5 provides a partial survey of particle filter variants used to conduct the simultaneous estimation of states and fixed parameters. Such filters are an extension of those previously adopted for estimating solely the states. Additionally, a MC study is carried out to estimate the state (level) and the two fixed variance parameters of the non-stationary local level model; we use four particle filter variants (LW, SIRJ, SIRoptJ, KPFJ), six typical settings of the SNR and two settings for the discount factor needed in the jittering step. In this chapter, the SIRJ particle filter variant is proposed as an alternative to the well-established filter of Liu West (LW PF). The combined use of a Kalman-based proposal distribution and a jittering step is proposed and explored, which gives rise to the particle filter variant called: the Kalman Particle Filter plus Jittering (KPFJ). Chapter 6 focuses on estimating the states and three fixed parameters of the non-standard basic stochastic volatility model known as stochastic autoregressive volatility model of order one: SARV(1). After an introduction and detailed description of the stylized features of financial time series, the estimation ability of two competing particle filter variants (SIRJ vs LW(Liu andWest)) is shown empirically using simulated data. The chapter ends with an application to real data sets from the financial area: the Spanish IBEX 35 returns index and the Europe Brent Spot prices (in dollars). The contribution in chapters 5 and 6 is to propose new variants of particle filters, such as the KPFJ, the SIRJ, and the SIRoptJ (a special case of the SIRJ that uses an optimal proposal distribution) that have developed along this work. The thesis also suggests that the so-called EPFJ (Extended Particle Filter with Jittering) and the UPFJ (Unscented Particle Filter with Jittering) algorithms could be reasonable choices when dealingwith highly nonlinearmodels. In this part, also relevant issueswithin the particle filteringmethodology are discussed, such as the potential impact on estimation of the discount factor parameter, the time series length, and the number of particles used. Throughout this work, pseudo-codes are written for all filters studied and are implemented in RLanguage. The reported findings are obtained as the result of extensive MC studies, considering a variety of case-scenarios described in the thesis. The intrinsic characteristics of the model at hand guided -according to suitability– the choice of filters in each specific situation. The comparison of filters is based on the RMSE, the elapsed CPU-time and the degree of degeneracy. Finally, Chapter 7 includes the discussion, contributions, and future lines of research. Some complementary theoretical and practical aspects are presented in the appendix.
L’estimació seqüencial dels estats (filtratge) i la corresponent estimació simultània dels estats i els paràmetres fixos d’unmodel dinàmic formulat en forma d’espai d’estat –sigui lineal o no– constitueix un problema de rellevada importància enmolts camps, com ser a l’àrea de finances. L’objectiu principal d’aquesta tesi és el d’estimar seqüencialment i de manera eficient –des d’un punt de vista bayesià i usant lametodologia de filtratge de partícules– els estats i/o els paràmetres fixos d’unmodel d’espai d’estat dinàmic no estàndard: possiblement no lineal, no gaussià o no estacionari. El present treball consisteix de 7 capítols i s’organitza en dues parts. El Capítol 1 hi introdueix conceptes bàsics, lamotivació, el propòsit i l’estructura de la tesi. La primera part d’aquesta tesi (capítols 2 a 4) se centra únicament en l’estimació dels estats. El Capítol 2 presenta una revisió exhaustiva dels algorismes més clàssics no basats en simulacions (KF, EKF, UKF2) i els basats en simulacions (SIS, SIR, ASIR, EPF, UPF). Per a aquests filtres, tots esmentats en la literatura, amés de descriure’ls detalladament, s’ha unificat la notació amb l’objectiu que aquesta sigui consistent i comparable entre els diferents algorismes implementats al llarg d’aquest treball. Els capítols 3 i 4 se centren en la realització d’estudis Monte Carlo (MC) extensos que confirmen l’eficiència de la metodologia de filtratge de partícules per estimar els estats latents d’un procés dinàmic formulat en forma d’espai d’estat, sigui lineal o no. Alguns estudis MC complementaris es duen a terme per avaluar diferents aspectes de la metodologia de filtratge de partícules, com ser el problema de la degeneració, l’elecció de l’estratègia de remostreig, el nombre de partícules usades o la grandària de la sèrie temporal. Específicament, el Capítol 3 il·lustra el comportament de la metodologia de filtratge de partícules en un context lineal i gaussià en comparació de l’òptim i exacte filtre de Kalman. La capacitat de filtratge de les quatre variants de filtre de partícules estudiades (SIR, SIRopt, ASIR, KPF; l’últim sent un cas especial de l’algorisme EPF) es va avaluar sobre la base de dos processos de sèries temporals aparentment simples però importants: els anomenats Local Level Model (LLM) i el AR (1) plus noise, que són no estacionari i estacionari, respectivament. Aquest capítol estudia en profunditat temes rellevants dins de l’enfocament adoptat, coml’impacte en l’estimació de la relació entre el senyal i el soroll (SNR: signal-to-noise-ratio, en aquesta tesi), de la longitud de la sèrie temporal i del nombre de partícules. El Capítol 4 avalua i il·lustra el comportament de la metodologia de filtratge de partícules en un context no lineal. En concret, s’utilitza un model d’espai d’estat no lineal, no gaussià i no estacionari pres de la literatura per il·lustrar el comportament de quatre filtres de partícules (SIR, ASIR, EPF, UPF) en contraposició a dos filtres no basats en simulació ben coneguts (EKF, UKF). Aquí es comparen els esquemes de remostreig residual i estratificat i s’avalua l’efecte d’augmentar el nombre de partícules. A la segona part (capítols 5 i 6), es duen a terme també estudis MC extensos, però ara l’objectiu principal és l’estimació simultània dels estats i paràmetres fixos de certsmodels seleccionats. Aquesta àrea de recerca segueix sentmolt activa i és on aquesta tesi hi contribueixmés. El Capítol 5 proveeix una revisió parcial dels mètodes per dur a terme l’estimació simultània dels estats i paràmetres fixos a través de la metodologia de filtratge de partícules. Aquests filtres són una extensió d’aquells adoptats anteriorment només per estimar els estats. Aquí es realitza un estudi MC per estimar l’estat (nivell) i els dos paràmetres de variància del model LLM no estacionari; s’utilitzen quatre variants (LW, SIRJ, SIRoptJ, KPFJ) de filtre de partícules, sis escenaris típics del SNR i dos escenaris per a l’anomenat factor de descompte necessari en el pas de diversificació. En aquest capítol, es proposa la variant de filtre de partícules SIRJ (Sample Importance Resampling with Jittering) com a alternativa al filtre de referència de Liu iWest (LWPF). També es proposa i explora l’ús combinat d’una distribució d’importància basada en el filtre de Kalman i un pas de diversificació (jittering) que dóna lloc a la variant del filtre de partícules anomenada Kalman Particle Filteringwith Jittering (KPFJ). El Capítol 6 se centra en l’estimació dels estats i dels paràmetres fixos delmodel bàsic no estàndard de volatilitat estocàstica denominat Stochastic autoregressive model of order one: SARV (1). Després d’una introducció i descripció detallada de les característiques pròpies de sèries temporals financeres, es demostra mitjançant estudis MC la capacitat d’estimació de dues variants de filtre de partícules (SIRJ vs. LW(Liu iWest)) utilitzant dades simulades. El capítol acaba amb una aplicació a dos conjunts de dades reals dins de l’àrea financera: l’índex de rendiments espanyol IBEX 35 i els preus al comptat (en dòlars) del Brent europeu. La contribució en els capítols 5 i 6 consisteix en proposar noves variants de filtres de partícules, compoden ser el KPFJ, el SIRJ i el SIRoptJ (un cas especial de l’algorisme SIRJ utilitzant una distribució d’importància òptima) que s’han desenvolupat al llarg d’aquest treball. També se suggereix que els anomenats filtres de partícules EPFJ (Extended Particle Filter with Jittering) i UPFJ (Unscented Particle Filter with Jittering) podrien ser opcions raonables quan es tracta de models altament no lineals; el KPFJ sent un cas especial de l’algorisme EPFJ. En aquesta part, també es tracten aspectes rellevants dins de la metodologia de filtratge de partícules, com ser l’impacte potencial en l’estimació de la longitud de la sèrie temporal, el paràmetre de factor de descompte i el nombre de partícules. Al llarg d’aquest treball s’han escrit (i implementat en el llenguatge R) els pseudo-codis per a tots els filtres estudiats. Els resultats presentats s’obtenenmitjançant simulacionsMonte Carlo (MC) extenses, tenint en compte variats escenaris descrits en la tesi. Les característiques intrínseques del model baix estudi van guiar l’elecció dels filtres a comparar en cada situació específica. Amés, la comparació dels filtres es basa en el RMSE (RootMean Square Error), el temps de CPU i el grau de degeneració. Finalment, el Capítol 7 presenta la discussió, les contribucions i les línies futures de recerca. Alguns aspectes teòrics i pràctics complementaris es presenten en els apèndixs.
La estimación secuencial de los estados (filtrado) y la correspondiente estimación simultánea de los estados y los parámetros fijos de un modelo dinámico formulado en forma de espacio de estado –sea lineal o no– constituye un problema de relevada importancia enmuchos campos, como ser en el área de finanzas. El objetivo principal de esta tesis es el de estimar secuencialmente y de manera eficiente –desde un punto de vista bayesiano y usando la metodología de filtrado de partículas– los estados y/o los parámetros fijos de un modelo de espacio de estado dinámico no estándar: posiblemente no lineal, no gaussiano o no estacionario. El presente trabajo consta de 7 capítulos y se organiza en dos partes. El Capítulo 1 introduce conceptos básicos, la motivación, el propósito y la estructura de la tesis. La primera parte de esta tesis (capítulos 2 a 4) se centra únicamente en la estimación de los estados. El Capítulo 2 presenta una revisión exhaustiva de los algoritmos más clásicos no basados en simulaciones (KF, EKF,UKF3) y los basados en simulaciones (SIS, SIR, ASIR, EPF, UPF). Para todos estos filtros, mencionados en la literatura, además de describirlos en detalle, se ha unificado la notación con el objetivo de que ésta sea consistente y comparable entre los diferentes algoritmos implementados a lo largo de este trabajo. Los capítulos 3 y 4 se centran en la realización de estudios Monte Carlo (MC) extensos que confirman la eficiencia de la metodología de filtrado de partículas para estimar los estados latentes de un proceso dinámico formulado en forma de espacio de estado, sea lineal o no. Algunos estudios MC complementarios se llevan a cabo para evaluar varios aspectos de la metodología de filtrado de partículas, como ser el problema de la degeneración, la elección de la estrategia de remuestreo, el número de partículas usadas o el tamaño de la serie temporal. Específicamente, el Capítulo 3 ilustra el comportamiento de lametodología de filtrado de partículas en un contexto lineal y gaussiano en comparación con el óptimo y exacto filtro de Kalman. La capacidad de filtrado de las cuatro variantes de filtro de partículas estudiadas (SIR, SIRopt, ASIR, KPF; el último siendo un caso especial del algoritmo EPF) se evaluó en base a dos procesos de series temporales aparentemente simples pero importantes: los denominados Local Level Model (LLM) y el AR (1) plus noise, que son no estacionario y estacionario, respectivamente. Este capítulo estudia en profundidad temas relevantes dentro del enfoque adoptado, como el impacto en la estimación de la relación entre la señal y el ruido (SNR: signal-to-noise-ratio, en esta tesis), de la longitud de la serie temporal y del número de partículas. El Capítulo 4 evalúa e ilustra el comportamiento de la metodología de filtrado de partículas en un contexto no lineal. En concreto, se utiliza un modelo de espacio de estado no lineal, no gaussiano y no estacionario tomado de la literatura para ilustrar el comportamiento de cuatro filtros de partículas (SIR, ASIR, EPF, UPF) en contraposición a dos filtros no basados en simulación bien conocidos (EKF, UKF). Aquí se comparan los esquemas de remuestreo residual y estratificado y se evalúa el efecto de aumentar el número de partículas. En la segunda parte (capítulos 5 y 6), se llevan a cabo también estudios MC extensos, pero ahora el objetivo principal es la estimación simultánea de los estados y parámetros fijos de ciertos modelos seleccionados. Esta área de investigación sigue siendo muy activa y es donde esta tesis contribuye más. El Capítulo 5 provee una revisión parcial de losmétodos para llevar a cabo la estimación simultánea de los estados y parámetros fijos a través de lametodología de filtrado de partículas. Dichos filtros son una extensión de aquellos adoptados anteriormente sólo para estimar los estados. Aquí se realiza un estudio MC para estimar el estado (nivel) y los dos parámetros de varianza del modelo LLM no estacionario; se utilizan cuatro variantes (LW, SIRJ, SIRoptJ, KPFJ) de filtro de partículas, seis escenarios típicos del SNR y dos escenarios para el llamado factor de descuento necesario en el paso de diversificación. En este capítulo, se propone la variante de filtro de partículas SIRJ (Sample Importance resampling with Jittering) como alternativa al filtro de referencia de Liu y West (LW PF). También se propone y explora el uso combinado de una distribución de importancia basada en el filtro de Kalman y un paso de diversificación (jittering) que da lugar a la variante del filtro de partículas denominada Kalman Particle Filteringwith Jittering (KPFJ). El Capítulo 6 se centra en la estimación de los estados y de los parámetros fijos del modelo básico no estándar de volatilidad estocástica denominado Stochastic autoregressivemodel of order one: SARV (1). Después de una introducción y descripción detallada de las características propias de series temporales financieras, se demuestra mediante estudios MC la capacidad de estimación de dos variantes de filtro de partículas (SIRJ vs. LW (Liu y West)) utilizando datos simulados. El capítulo termina con una aplicación a dos conjuntos de datos reales dentro del área financiera: el índice de rendimientos español IBEX 35 y los precios al contado (en dólares) del Brent europeo. La contribución en los capítulos 5 y 6 consiste en proponer nuevas variantes de filtros de partículas, como pueden ser el KPFJ, el SIRJ y el SIRoptJ (Caso especial del algoritmo SIRJ utilizando una distribución de importancia óptima) que se han desarrollado a lo largo de este trabajo. También se sugiere que los llamados filtros de partículas EPFJ (Extended Particle Filter with Jittering) y UPFJ (Unscented Particle Filter with Jittering) podrían ser opciones razonables cuando se trata de modelos altamente no lineales; el KPFJ siendo un caso especial del algoritmo EPFJ. En esta parte, también se tratan aspectos relevantes dentro de lametodología de filtrado de partículas, como ser el impacto potencial en la estimación de la longitud de la serie temporal, el parámetro de factor de descuento y el número de partículas. A lo largo de este trabajo se han escrito (e implementado en el lenguaje R) los pseudo-códigos para todos los filtros estudiados. Los resultados presentados se obtienen mediante simulaciones Monte Carlo (MC) extensas, teniendo en cuenta variados escenarios descritos en la tesis. Las características intrínsecas del modelo bajo estudio guiaron la elección de los filtros a comparar en cada situación específica. Además, la comparación de los filtros se basa en el RMSE (Root Mean Square Error), el tiempo de CPU y el grado de degeneración. Finalmente, el Capítulo 7 presenta la discusión, las contribuciones y las líneas futuras de investigación. Algunos aspectos teóricos y prácticos complementarios se presentan en los apéndices.

Nous nous intéressons dans cette thèse à l'estimation, dans un cadre non paramétrique, d'unis fonction de régression présentant des discontinuités et, plus précisément aux problèmes de dé tection de ruptures, d'estimation des paramètres de rupture (nombre, localisations, ainplitudesj et de segmentation de la fonction de régression (reconstitution de la fonction). La méthode utilisée est basée sur les propriétés du processsus de saut estimé, y(t), défini en tout i comme lt. Différence entre un estimateur à droite et un estimateur à gauche, ces estimateurs étant obtenus régression linéaire locale. - Dans un premier temps, nous considérons la situation d'une seule discontinuité et étudions les propriétés de l'estimateur de l'amplitude de la discontinuité lorsque la localisation est connue. Nous donnons l'expression de l'erreur quadratique moyenne asymptotique et montrons la convergence et la normalité asymptotique de l'estimateur. Lorsque la localisation r n'est pas connue, nous construisons un estimateur de r à l'aide du processus de déviation locale associé à 7(t) et montrons que cet estimateur converge avec une vitesse en n,-r ou arbitrairement proche de r-r selon le noyau utilisé. Nous proposons ensuite trois tests d'existence d'une rupture : un test strictement local, un test local et un test global, tous trois définis en terme d'une statistique construite à (aide du processus de saut estimé. Concernant le problème d'estimation du nombre de ruptures nous élaborons une procédure permettant à la fois d'estimer le nombre y de ruptures et les localisations rr. . . . Ry. Nous montrons la convergence presque sûre de ces estimateurs et donnons aussi des résultats sur les vitesses de convergence. Enfin nous proposons une méthode de reconstitution d'une fonction de régression présentant des discontinuités basée sur la segmentation des observations. Nous montrons qu'en utilisant la procédure d'estimation du nombre de ruptures et des localisations développée auparavant, nous obtenons un estimateur de la fonction de régression qui a la même vitesse de convergence qu'en (absence de ruptures. Des expérimentations numériques sont fournies pour chacun des problèmes étudiés de manière à mettre en évidence les propriétés des procédures étudiées et leur sensibilité aux divers paramètres

Dans cette thèse, nous nous intéressons à développer des méthodes robustes et efficaces dans l’estimation non paramétrique de la fonction de régression. Le modèle considéré ici est le modèle censuré aléatoirement à droite qui est le plus utilisé dans différents domaines pratiques. Dans un premier temps, nous proposons un nouvel estimateur de la fonction de régression en utilisant la méthode linéaire locale. Nous étudions sa convergence uniforme presque sûre avec vitesse. Enfin, nous comparons ses performances avec celles de l’estimateur de la régression à noyau classique à l’aide de simulations. Dans un second temps, nous considérons l’estimateur de la fonction de régression par erreur relative (RER en anglais), basé sur la minimisation de l’erreur quadratique relative moyenne. Ainsi, nous établissons la convergence uniforme presque sûre (sur un compact) avec vitesse de l’estimateur défini pour des observations indépendantes et identiquement distribuées. En outre, nous prouvons sa normalité asymptotique en explicitant le terme de variance. Enfin, nous conduisons une étude de simulations pour confirmer nos résultats théoriques et nous appliquons notre estimateur sur des données réelles. Par la suite, nous étudions la convergence uniforme presque sûre (sur un compact) avec vitesse de l’estimateur RER pour des observations soumises à une structure de dépendance du type α-mélange. Une étude de simulation montre le bon comportement de l’estimateur étudié. Des prévisions sur données générées sont réalisées pour illustrer la robustesse de notre estimateur. Enfin, nous établissons la normalité asymptotique de l’estimateur RER pour des observations α-mélangeantes où nous construisons des intervalles de confiance afin de réaliser une étude de simulations qui valide nos résultats. Pour conclure, le fil conducteur de cette modeste contribution, hormis l’analyse des données censurées est la proposition de deux méthodes de prévision alternative à la régression classique. La première approche corrige les effets de bord crée par les estimateurs à noyaux classiques et réduit le biais. Tandis que la seconde est plus robuste et moins affectée par la présence de valeurs aberrantes dans l’échantillon
In this thesis, we are interested in developing robust and efficient methods in the nonparametric estimation of the regression function. The model considered here is the right-hand randomly censored model which is the most used in different practical fields. First, we propose a new estimator of the regression function by the local linear method. We study its almost uniform convergence with rate. We improve the order of the bias term. Finally, we compare its performance with that of the classical kernel regression estimator using simulations. In the second step, we consider the regression function estimator, based on theminimization of the mean relative square error (called : relative regression estimator). We establish the uniform almost sure consistency with rate of the estimator defined for independent and identically distributed observations. We prove its asymptotic normality and give the explicit expression of the variance term. We conduct a simulation study to confirm our theoretical results. Finally, we have applied our estimator on real data. Then, we study the almost sure uniform convergence (on a compact set) with rate of the relative regression estimator for observations that are subject to a dependency structure of α-mixing type. A simulation study shows the good behaviour of the studied estimator. Predictions on generated data are carried out to illustrate the robustness of our estimator. Finally, we establish the asymptotic normality of the relative regression function estimator for α-mixing data. We construct the confidence intervals and perform a simulation study to validate our theoretical results. In addition to the analysis of the censored data, the common thread of this modest contribution is the proposal of two alternative prediction methods to classical regression. The first approach corrects the border effects created by classical kernel estimators and reduces the bias term. While the second is more robust and less affected by the presence of outliers in the sample

Ce document synthétise mes travaux de thèse de doctorat en Automatique Productiqueà Grenoble INP (Institut National Polytechnique), thèse préparée au sein dudépartement automatique du laboratoire GIPSA-lab (Grenoble Image Parole Signal etAutomatique). Ce travail s’inscrit dans le cadre du contrôle local et de la supervisiondes systèmes de trafic routier. Les principales contributions portent sur la modélisation,la supervision et la commande locale des systèmes de trafic routier.La contribution apportée à la modélisation du trafic est l’ajout d’un modèle d’incertitudesur le modèle CTM (Cell Transmission Model [Daganzo, 1994]). Ce nouveaumodèle permet de prendre en compte les incertitudes sur différents paramètres dumodèle pour in-fine proposer de nouvelles stratégies de commandes commutées robustes.Outre cette approche de modélisation, nous proposons un niveau de supervisionpermettant d’une part d’estimer en temps réel le mode de fonctionnement et d’autrepart de détecter, localiser et estimer certaines fautes sur le système. L’estimation dynamiquede mode de fonctionnement nous permet de connaître l’état de congestion (ou denon-congestion) de l’aménagement routier considéré. Nous sommes en mesure de détecterdes fautes telles que des chutes de vitesse ou des chutes de capacité survenant sur la route.Enfin, nous proposons deux lois de commandes locales basées sur la théorie dessystèmes à commutations. Ainsi, le schéma de contrôle s’adaptera dynamiquementaux changements de propriétés du système. Ces lois de commande ont pour objet des’insérer dans un schéma de régulation hiérarchique
This document synthesizes my Phd thesis work in Automatic Control in Grenoble-INP. This thesis has been prepared in the automatic control department of thelaboratory GIPSA-lab. This work is situated in the area of traffic systems control andsupervision. Our contributions are about modeling, supervision and local traffic control.The CTM traffic model has been extended with a model of uncertainties. Thisnews model allows us to take into account the uncertain parameters of the model, topropose new robust switched control law.In addition to this modeling approach, we propose some developments on supervisionof trafic systems. On one hand, we can estimate the operating mode of thesystem in real time and on the other hand to estimate some faults on the system. Thedynamical estimation of the operating mode allows us to know the state of congestion(or non congestion) of the road. We are able to estimate faults such as speed fall andcapacities drop that may appear.Finally, we propose two control laws based on switching systems control. The developedcontrollers adapt their geometry to the properties of the system. The purposeof these controllers is to be inserted in a hierarchic control scheme

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FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais
O objetivo deste trabalho é apresentar algumas contribuições para a melhoria do processo de estimação por máxima verossimilhança via algoritmo EM em modelos misturas assimétricas com regressão, além de realizar neles a análise de influência local e global. Essas contribuições, em geral de natureza computacional, visam à resolução de problemas comuns na modelagem estatística de maneira mais eficiente. Dentre elas está a substituição de métodos utilizados nas versões dos algoritmos GEM por outras que reduzem o problema aproximadamente a um algoritmo EM clássico nos principais exemplos das distribuições misturas de escala assimétricas de normais. Após a execução do processo de estimação, discutiremos ainda as principais técnicas existentes para o diagnóstico de pontos influentes com as adaptações necessárias aos modelos em foco. Desejamos com tal abordagem acrescentar ao tratamento dessa classe de modelos estatísticos a análise de regressão nas distribuições mais recentes na literatura. Também esperamos abrir caminho para o uso de técnicas similares em outras classes de modelos.
The objective of this work is to present some contributions to improvement the process of maximum likelihood estimation via the EM algorithm in skew mixtures models with regression, as well as to execute in them the global and local influence analysis. These contributions, usually with computational nature, aim to solving common problems in statistical modeling more efficiently. Among them is the replacement of used methods in the versions of the GEM algorithm by other techniques that reduce the problem approximately to a classic EM algorithm in the main examples of skew scale mixtures of normals distributions. After performing the estimation process, we will also discuss the main existing techniques for the diagnosis of influential points with the necessaries adaptations to the models in focus. We wish with this approach to add for the treatment of this statistical model class the regression analysis in the most recent distributions in the literature. We too hope to paving the way for use of similar techniques in other models classes.

Dans la premiere partie, on presente un test du nombre de modes d'une loi de probabilite reelle, on teste la bimodalite (ou plus) contre l'unimodalite. Dans la seconde partie, un estimateur du minimum d'entropie sous une contrainte non lineaire est obtenu. La convergence et la loi limite de cet estimateur sont etablies. Dans la derniere partie, on obtient un equivalent asymptotique de la probabilite que la somme de n variables aleatoires independantes appartienne a un compact, sous certaines conditions. Ce resultat est utilise pour obtenir des probabilites de deviations pour la moyenne empirique et les m-estimateurs

Neste trabalho, apresentamos os modelos simétricos parcialmente lineares AR(1), que generalizam os modelos parcialmente lineares para a presença de erros autocorrelacionados seguindo uma estrutura de autocorrelação AR(1) e erros seguindo uma distribuição simétrica ao invés da distribuição normal. Dentre as distribuições simétricas, podemos considerar distribuições com caudas mais pesadas do que a normal, controlando a curtose e ponderando as observações aberrantes no processo de estimação. A estimação dos parâmetros do modelo é realizada por meio do critério de verossimilhança penalizada, que utiliza as funções escore e a matriz de informação de Fisher, sendo todas essas quantidades derivadas neste trabalho. O número efetivo de graus de liberdade e resultados assintóticos também são apresentados, assim como procedimentos de diagnóstico, destacando-se a obtenção da curvatura normal de influência local sob diferentes esquemas de perturbação e análise de resíduos. Uma aplicação com dados reais é apresentada como ilustração.
In this master dissertation, we present the symmetric partially linear models with AR(1) errors that generalize the normal partially linear models to contain autocorrelated errors AR(1) following a symmetric distribution instead of the normal distribution. Among the symmetric distributions, we can consider heavier tails than the normal ones, controlling the kurtosis and down-weighting outlying observations in the estimation process. The parameter estimation is made through the penalized likelihood by using score functions and the expected Fisher information. We derive these functions in this work. The effective degrees of freedom and asymptotic results are also presented as well as the residual analysis, highlighting the normal curvature of local influence under different perturbation schemes. An application with real data is given for illustration.

The aim of this work is to present an inference and diagnostic study of an extension of the lifetime distribution family proposed by Birnbaum and Saunders (1969a,b). This extension is obtained by considering a skew-elliptical distribution instead of the normal distribution. Specifically, in this work we develop a Birnbaum-Saunders (BS) distribution type based on scale mixtures of skew-normal distributions (SMSN). The resulting family of lifetime distributions represents a robust extension of the usual BS distribution. Based on this family, we reproduce the usual properties of the BS distribution, and present an estimation method based on the EM algorithm. In addition, we present regression models associated with the BS distributions (based on scale mixtures of skew-normal), which are developed as an extension of the sinh-normal distribution (Rieck and Nedelman, 1991). For this model we consider an estimation and diagnostic study for uncensored data.
O objetivo deste trabalho é apresentar um estudo de inferência e diagnóstico em uma extensão da família de distribuições de tempos de vida proposta por Birnbaum e Saunders (1969a,b). Esta extensão é obtida ao considerar uma distribuição skew-elíptica em lugar da distribuição normal. Especificamente, neste trabalho desenvolveremos um tipo de distribuição Birnbaum-Saunders (BS) baseda nas distribuições mistura de escala skew-normal (MESN). Esta família resultante de distribuições de tempos de vida representa uma extensão robusta da distribuição BS usual. Baseado nesta família, vamos reproduzir as propriedades usuais da distribuição BS, e apresentar um método de estimação baseado no algoritmo EM. Além disso, vamos apresentar modelos de regressão associado à distribuições BS (baseada na distribuição mistura de escala skew-normal), que é desenvolvida como uma extensão da distribuição senh-normal (Rieck e Nedelman, 1991), para estes vamos considerar um estudo de estimação e diagnóstisco para dados sem censura.

This thesis is focused on local polynomial smoothers of the conditional vari- ance function in a heteroscedastic nonparametric regression model. Both mean and variance functions are assumed to be smooth, but neither is assumed to be in a parametric family. The basic idea is to apply a local linear regression to squa- red residuals. This method, as we have shown, has high minimax efficiency and it is fully adaptive to the unknown conditional mean function. However, the local linear estimator may give negative values in finite samples which makes variance estimation impossible. Hence Xu and Phillips proposed a new variance estimator that is asymptotically equivalent to the local linear estimator for interior points but is guaranteed to be non-negative. We also established asymptotic results of both estimators for boundary points and proved better asymptotic behavior of the local linear estimator. That motivated us to propose a modification of the local li- near estimator that guarantees non-negativity. Finally, simulations are conducted to evaluate the finite sample performances of the mentioned estimators.

碩士
國立東華大學
應用數學系
87
In the case of the random design nonparametric regression, the local linear estimator（LLE）is the most popular kernel regression function estimator. However, there is a drawback to the LLE. That is, in some cases , the associated inverse matrix to the LLE may not exist. For exam- ple, there is only one design point falling in the compact window around the point at which the regression function value is estimated. To correct for the drawback to the LLE, we apply the idea of principle component analysis to the LLE, and propose the local linear principle component re-gression function estimator (LLPCRFE). Simulation studies demonstrate that the LLPCRFE has better performance than the ordinary LLE.

博士
國立東華大學
應用數學系
92
There are two topics in this paper. One topic is integrated cross-validation, and the other topic is a study of local linear estimators. For the random design nonparametric regression, cross-validation is a popular bandwidth selector. It is constructed by using the criterion of "weighted" integrated square error. In practice, however, the weighting scheme by the design density in the criterion causes that its associated cross-validation function puts more emphasis in regions with more data, gives little attention to regions with few data, but has no consideration for regions without data. In such a case, the value of the cross-validated bandwidth depends on the distribution of the design points, but is independent of the location of the interval on which the regression function value is estimated. Hence, if there are sparse regions in the realization of the design, then the resulting cross-validated bandwidth is usually not large enough in magnitude such that its corresponding kernel regression function estimate has rough appearance in these sparse regions. To avoid this drawback to cross-validation, we suggest using the criterion of "unweighted" integrated square error to construct the bandwidth selector. Under the criterion, a bandwidth selector called integrated cross-validation is proposed, and the resulting bandwidth is shown to be asymptotically optimal. Empirical studies demonstrate that the kernel regression function estimate obtained by using our proposed bandwidth is better than that employing the ordinary cross-validated bandwidth, in both senses of having smoother appearance and yielding smaller sample unweighted integrated square error. In nonparametric regression, local linear estimators have a number of advantages; however, Seifert and Gasser (1996) claimed that the local linear estimators with compactly supported kernels have unbounded finite sample conditional variance when the design regions are sparse or the realization has locally a small variance and/or a skew empirical distribution. In regression analysis, if the multicollinearity exists among explanatory variables , it can have a large influence on the estimation of regression coefficients. We find that the variance of local linear estimators may become larger and larger when multicollinearity and sparseness exist. The reason why the drawback may occur is that the multicollinearity and sparseness can lead to the inverse of the sample covariance matrix of the considered explanatory variables that may not exist or be unstable in numberical computation, which will make the variance of estimators become extremely large. In order to solve such a problem, the local linear ridge regression estimators are suggested to help. Mathematical proofs and simulation studies both demonstrate that the local linear ridge regression estimators can overcome the drawback caused by multicollinearity and sparseness.

博士
國立東華大學
應用數學系
90
In the field of random design nonparametric regression, we examine two kernel estimators involving, respectively, piecewise linear interpolation of kernel regression function estimates and local ridge regression. Efforts dedicated to understanding their properties bring forth the following main messages. The kernel estimate of a regression function inherits its smoothness properties from the kernel function chosen by the investigator. Nevertheless, practical regression function estimates are often presented in interpolated form, using the exact kernel estimates only at some equally spaced grids of points. The asymptotic integrated mean square error (AIMSE) properties of such polygon type estimate, namely kernel regression function polygons (KRFP), are investigated. Call the "optimal kernel" the minimizer of the AIMSE. Epanechnikov kernel is not the optimal kernel unless for the case that the distance between every two consecutive grids is of smaller order in magnitude than the bandwidth used by the kernel regression function estimator. If the distance and bandwidth are of the same order in magnitude, we obtain the optimal kernel from the class of degree-two polynomials through numerical calculations. In this case, the best AIMSE performances deteriorate as the distance is increased to reduce the computational effort. When the distance is of larger order in magnitude than the bandwidth, then uniform kernel serves as the optimal kernel for KRFP. Local linear estimator (LLE) has many attractive asymptotic features. In finite sample situations, however, its conditional variance may become arbitrarily large. To cope with this difficulty, which can translate into the spurious rough appearance of the regression function estimate when design becomes sparse or clustered, Seifert and Gasser (1996)suggest "ridging" the LLE and propose the local linear ridge regression estimator (LLRRE). In this dissertation, local and numerical properties of the LLRRE are studied. It is shown that its finite sample mean square errors, both conditional and unconditional, are bounded above by finite constants. If the ridge regression parameters are not selected properly, then the resulting LLRRE suffers some drawbacks. For example, it is asymptotically biased and has boundary effects, and fails to inherit the nice asymptotic bias quality of the LLE. Letting the ridge parameters depend on sample size and converge to 0 as the sample size increases, we are able to ensure LLRRE the nice asymptotic features of the LLE under some mild conditions. Simulation studies demonstrate that the LLRRE using cross-validated bandwidth and ridge parameters could have smaller sample mean integrated square error than the LLE using cross-validated bandwidth, in reasonable sample sizes.