Thesis (Ph. D.)--University of Washington, 2006. / Vita. Includes bibliographical references (p. 107-111).

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 178-188).

This thesis develops some new di.erence-based methods for nonparametric regression models. The .rst part of this thesis focuses on the variance estimation for nonparametric models with various settings. In Chapter 2, a uni.ed framework of variance estimator is proposed for a model with smooth mean function. This framework combines the higher order di.erence sequence with least squares method and greatly extends the literature, including most of existing methods as special cases. We derive the asymp­totic mean squared errors and make both theoretical and numerical comparison for various estimators within the system. Based on the dramatic interaction of ordinary di.erence sequences and least squares method, we eventually .nd a uniformly sat­isfactory estimator for all the settings, solving the challenging problem of sequence selection. In Chapter 3, three methods are developed for the variance estimation in the repeated measurement setting. Both their asymptotic properties and .nite sample performance are explored. The sequencing method is shown to be the most adaptive while the sample variance method and the partitioning method are shown to outperform in certain cases. In Chapter 4, we propose a pairwise regression method for estimating the residual variance. Speci.cally, we regress the squared di.erence between observations on the squared distance between design points, and then es­timate the residual variance as the intercept. Unlike most existing di.erence-based estimators that require a smooth regression function, our method applies to regres­sion models with jump discontinuities. And it also applies to the situations where the design points are unequally spaced. The smoothness assumption of the nonparametric regression function is quite critical for the curve .tting and the residual variance estimation. The second part (Chapter 5) concentrates on the discontinuities detection for the mean function. In particular, we revisit the di.erence-based method in M¨uller and Stadtm¨uller (1999) and propose to improve it. To achieve the goal, we .rst reveal that their method is less e.cient due to the inappropriate choice of the response variable in their linear regression model. We then propose a new regression model for estimating the resid­ual variance and the total amount of discontinuities simultaneously. In both theory and simulations, we show that the proposed variance estimator has a smaller MSE compared to their estimator, whereas the e.ciency of the estimators for the total amount of discontinuities remain unchanged. Finally, we construct a new test proce­dure for detection using the newly proposed estimations; and via simulation studies, we demonstrate that our new test procedure outperforms the existing one in most settings. At the beginning of Chapter 6, a series of new di.erence sequences is de.ned to complete the span between the optimal sequence and the ordinary sequence. The vari­ance estimators using proposed sequences are shown to be quite robust and achieve smallest mean square errors for most of general settings. Then, the di.erence-based methods for variance function estimation are generally discussed. Keywords: Asymptotic normality, Di.erence-based estimator, Di.erence sequence, Jump point, Least square, Nonparametric regression, Pairwise regression, Repeated measurement, Residual variance

The semi-parametric model enjoys a relatively flexible structure and keeps some of the simplicity in the statistical analysis. Hence, there are abundance discussions on semi-parametric models in the literature. The concept of partial consistency was firstly brought up in Neyman and Scott (1948). It was said the in cases where infinite parameters are involved, consistent estimators are always attainable for those "structural" parameters. The "structural' parameters are finite and govern infinite samples. Since the nonparametric model can be regarded as a parametric model with infinite parameters, then the semi-parametric model can be easily transformed into a infinite-parametric model with some "structural" parameters. Therefore, based on this idea, we develop several new methods for the estimating and model checking problems in semi-parametric models. The implementation of applying partial consistency is through the method "local average". We consider the nonparametric part as piecewise constant so that infinite parameters are created. The "structural" parameters shall be the parametric part, the model residual variance and so on. Due to the partial consistency phenomena, classical statistic tools can then be applied to obtain consistent estimators for those "structural" parameters. Furthermore, we can take advantage of the rest of parameters to estimate the nonparametric part. In this thesis, we take the varying coefficient model as the example. The estimation of the functional coefficient is discussed and relative model checking methods are presented. The proposed new methods, no matter for the estimation or the test, have remarkably lessened the computation complexity. At the same time, the estimators and the tests get satisfactory asymptotic statistical properties. The simulations we conducted for the new methods also support the asymptotic results, giving a relatively efficient and accurate performance. What's more, the local average method is easy to understand and can be flexibly applied to other type of models. Further developments could be done on this potential method. In Chapter 2, we introduce a local average method to estimate the functional coefficients in the varying coefficient model. As a typical semi-parametric model, the varying coefficient model is widely applied in many areas. The varying coefficient model could be seen as a more flexible version of classical linear model, while it explains well when the regression coefficients do not stay constant. In addition, we extend this local average method to the semi-varying coefficient model, which consists of a linear part and a varying coefficient part. The procedures of the estimations are developed, and their statistical properties are investigated. Plenty of simulations and a real data application are conducted to study the performance of the proposed method. Chapter 3 is about the local average method in variance estimation. Variance estimation is a fundamental problem in statistical modeling and plays an important role in the inferences in model selection and estimation. In this chapter, we have discussed the problem in several nonparametric and semi-parametric models. The proposed method has the advantages of avoiding the estimation of the nonparametric function and reducing the computational cost, and can be easily extended to more complex settings. Asymptotic normality is established for the proposed local average estimators. Numerical simulations and a real data analysis are presented to illustrate the finite sample performance of the proposed method. Naturally, we move to the model checking problem in Chapter 4, still taking varying coefficient models as an example. One important and frequently asked question is whether an estimated coefficient is significant or really "varying". In the literature, the relative hypothesis tests usually require fitting the whole model, including the nuisance coefficients. Consequently, the estimation procedure could be very compute-intensive and time-consuming. Thus, we bring up several tests which can avoid unnecessary functions estimation. The proposed tests are very easy to implement and their asymptotic distributions under null hypothesis have been deduced. Simulations are also studied to show the properties of the tests.

Distribution-free (nonparametric) control charts can be useful to the quality practitioner when the underlying distribution is not known. The term nonparametric is not intended to imply that there are no parameters involved, in fact, quite the contrary. While the term distribution-free seems to be a better description of what we expect from these charts, that is, they remain valid for a large class of distributions, nonparametric is perhaps the term more often used. In the statistics literature there is now a rather vast collection of nonparametric tests and confidence intervals and these methods have been shown to perform well compared to their normal theory counterparts. Remarkably, even when the underlying distribution is normal, the efficiency of some nonparametric tests relative to the corresponding (optimal) normal theory methods can be as high as 0.955 (see e.g. Gibbons and Chakraborti (2010) page 218). For some other heavy-tailed and skewed distributions, the efficiency can be 1.0 or even higher. It may be argued that nonparametric methods will be ‘less efficient’ than their parametric counterparts when one has a complete knowledge of the process distribution for which that parametric method was specifically designed. However, the reality is that such information is seldom, if ever, available in practice. Thus it seems natural to develop and use nonparametric methods in statistical process control (SPC) and the quality practitioners will be well advised to have these techniques in their toolkits. In this thesis we only propose univariate nonparametric control charts designed to track the location of a continuous process since very few charts are available for monitoring the scale and simultaneously monitoring the location and scale of a process. Chapter 1 gives a brief introduction to SPC and provides background information regarding the research conducted in this thesis. This will aid in familiarizing the reader with concepts and terminology that are helpful to the following chapters. Details are given regarding the three main classes of control charts, namely the Shewhart chart, the cumulative sum (CUSUM) chart and the exponentially weighted moving average (EWMA) chart. We begin Chapter 2 with a literature overview of Shewhart-type Phase I control charts followed by the design and implementation of these charts. A nonparametric Shewhart-type Phase I control chart for monitoring the location of a continuous variable is proposed. The chart is based on the pooled median of the available Phase I samples and the charting statistics are the counts (number of observations) in each sample that are less than the pooled median. The derivations recognize that in Phase I the signalling events are dependent and that more than one comparison is © University of Pretoria v made against the same estimated limits simultaneously; this leads to working with the joint distribution of a set of dependant random variables. An exact expression for the false alarm probability is given in terms of the multivariate hypergeometric distribution and this is used to provide tables for the control limits. Some approximations are discussed in terms of the univariate hypergeometric and the normal distributions. In Chapter 3 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are known or specified. This is referred to as the ‘standard(s) known’ case and is denoted Case K. Two nonparametric Phase II control charts are considered in this chapter, with the first one being a nonparametric exponentially weighted moving average (NPEWMA)-type control chart based on the sign (SN) statistic. A Markov chain approach (see e.g. Fu and Lou (2003)) is used to determine the run-length distribution of the chart and some associated performance characteristics (such as the average, standard deviation, median and other percentiles). In order to aid practical implementation, tables are provided for the chart’s design parameters. An extensive simulation study shows that on the basis of minimal required assumptions, robustness of the in-control run-length distribution and out-of-control performance, the proposed NPEWMA-SN chart can be a strong contender in many applications where traditional parametric charts are currently used. Secondly, we consider the NPEWMA chart that was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see e.g. Gibbons and Chakraborti (2010) page 195). This is called the nonparametric exponentially weighted moving average signed-rank (NPEWMA-SR) chart. In their article important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this thesis we address these issues with a more in-depth study of the NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the chart’s design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. In Chapter 4 Phase II control charts are introduced and considered for the case when the underlying parameters of the process distribution are unknown and need to be estimated. This is referred to as the ‘standard(s) unknown’ case and is denoted Case U. Two nonparametric Phase II control charts are proposed in this chapter. They are a Phase II NPEWMA-type control chart and a nonparametric cumulative sum (NPCUSUM)-type control chart, based on the exceedance statistics, © University of Pretoria vi respectively, for detecting a shift in the location parameter of a continuous distribution. The exceedance statistics can be more efficient than rank-based methods when the underlying distribution is heavy-tailed and / or right-skewed, which may be the case in some applications, particularly with certain lifetime data. Moreover, exceedance statistics can save testing time and resources as they can be applied as soon as a certain order statistic of the reference sample is available. We also investigate the choice of the order statistics (percentile), from the reference (Phase I) sample that defines the exceedance statistic. It is observed that other choices, such as the third quartile, can play an important role in improving the performance of these exceedance charts. It is seen that these exceedance charts perform as well as and, in many cases, better than its competitors and thus can be a useful alternative chart in practice. Chapter 5 wraps up this thesis with a summary of the research carried out and offers concluding remarks concerning unanswered questions and / or future research opportunities. © University / Thesis (PhD)--University of Pretoria, 2013. / gm2013 / Statistics / restricted

In this study I explore the feminist critique of Reinhold Niebuhr’s theology of sin, both to understand what the Niebuhrian and feminist understandings of sin talk about, and to see whether or not, or to what extent they are tenable in theory and in practice. Niebuhr’s feminist critics argue that Niebuhr’s claim of pride as the primary human sin fits only with men’s experience; women’s sin, they contend, is not self-inflation but self-loss. While I acknowledge the value of Niebuhr’s feminist critics’ interpretation of sin, this study provides a Niebuhrian response to the feminist critique. My main contention is that by overemphasizing women’s sin of passivity, some feminist theologians go too far to deny women’s capability of committing sin actively against others and the divine in both socio-moral and religio-theological aspect. The total rejection of the applicability of pride to women’s situation, I contend, undermines the profoundness of the feminist critique. I firstly give detailed expositions of Niebuhr’s theology of sin and the feminist critique of Niebuhr’s theology of sin respectively. The main discrepancies between the Niebuhrian and feminist understandings of sin will be laid out. Then I respond to some feminist criticisms by pointing out that the feminist misreading of Niebuhr on the topics of pride, the self, love, justice and the family is prevalent. I also question the two presuppositions of the feminist critique—the idea of women’s innocence and the spirit of secularity. These two presuppositions, I argue, contain in them some insoluble dilemmas that cause trouble for understanding women’s secular and religious experience. Lastly, I try to pull the insights of Niebuhr and his feminist critics together to form a more integrated view of women’s sin

This thesis presented a useful tool in regression. Nonparametric linear regression techniques were described in the general context of regression. A comparison of two of these techniques, kernel regression and iterative regression, showed various aspects of nonparametric linear regressors.

In this dissertation, we utilize the discrete Fourier analysis on axially symmetric data generation and nonparametric estimation. We first represent the axially symmetric process as Fourier series on circles with the Fourier random coefficients expressed as circularlysymmetric complex random vectors. We develop an algorithm to generate the axially symmetric data that follow the given covariance function. Our simulation study demonstrates that our approach performs comparable with the classical approach using the given axially symmetric covariance function directly, while at the same time significantly reducing computational costs. For the second contribution of this dissertation, we apply the discrete Fourier transform to provide the nonparametric estimation on the covariance function of the above circularly-symmetric complex random vectors under gridded data structure. Our results show that these estimates has closely related to the simultaneous diagonalization of circulant matrices. The simulation study shows that our proposed estimates match well with their theoretical counterparts. Finally through the Fourier transform of the original gridded data, the covariance estimator of an axially symmetric process based on the method of moments can be represented as a quadratic form of transformed data that is associated with a rotation matrix.

Functional data analysis is a eld in statistics that analyzes data which are dependent on time or space and from which inference can be conducted. Functional data analysis methods can estimate residuals from functional regression models that in turn require robust univariate density estimators for error density estimation. The accurate estimation of the error density from the residuals allows evaluation of the performance of functional regression estimation. Kernel density estimation using maximum likelihood cross-validation and Bayesian bandwidth selection techniques with a Gaussian kernel are reproduced and compared to least-squares cross-validation and plug-in bandwidth selection methods with an Epanechnikov kernel. For simulated data, Bayesian bandwidth selection methods for kernel density estimation are shown to give the minimum mean expected square error for estimating the error density, but are computationally ine cient and may not be adequately robust for real data. The (bounded) Epanechnikov kernel function is shown to give similar results as the Gaussian kernel function for error density estimation after functional regression. When the functional regression model is applied to a chemometric data set, the local least-squares cross-validation method, used to select the bandwidth for the functional regression estimator, is shown to give a signi cantly smaller mean square predicted error than that obtained with Bayesian methods. / Thesis / Master of Science (MSc)

Nonparametric tests are used to test hypotheses when the data at hand violate one or more of the assumptions for parametric tests procedures. The test is an ordered alternative (nondecreasing) when there is prior information about the data. It assumes that the underlying distributions are of the same type and therefore differ in location. For example, in dose-response studies, animals are assigned to k groups corresponding to k doses of an experimental drug. The effect of the drug on the animals is likely to increase or decrease with increasing doses. In this case, the ordered alternative is appropriate for the study. In this paper, we propose eight new nonparametric tests useful for testing against nondecreasing order alternatives for a mixed design involving randomized complete block and balanced incomplete block design. These tests involve various modifications of the Jonckheere-Terpstra test (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s test (1995). Three, four and five treatments were considered with different location parameters under different scenarios. For three and four treatments, 6,12, and 18 blocks were used for the simulation, while 10, 20, and 30 blocks were used for five treatments. Different tests performed best under different block combinations, but overall the standardized last for Alvo outperformed the other test when the number of treatments and number of missing observations per block increases. A simulation study was conducted comparing the powers of the various modification of Jonckheere-Terpstra (Jonckheere(1952), Terpstra(1954)) and Alvo and Cabilio’s (1995) tests under different scenarios. Recommendations are made.