Medical cost analysis is an important part of treatment evaluation. Since resources are limited in society, it is important new treatments are developed with proper costconsiderations. The mean has been mostly accepted as a measure of the medical cost analysis. However, it is well known that cost data is highly skewed and the mean could be highly influenced by outliers. Therefore, in many situations the mean cost alone cannot offer complete information about medical costs. The quantiles (e.g., the first quartile, median and third quartile) of medical costs could better represent the typical costs paid by a group of individuals, and could provide additional information beyond mean cost. For a specified patient population, cost estimates are generally determined from the beginning of treatments until death or end of the study period. A number of statistical methods have been proposed to estimate medical cost. Since medical cost data are skewed to the right, normal approximation based confidence intervals can have much lower coverage probability than the desired nominal level when the cost data are moderately or severely skewed. Additionally, we note that the variance estimators of the cost estimates are analytically complicated. In order to address some of the above issues, in the first part of the dissertation we propose two empirical likelihood-based confidence intervals for the mean medical costs: One is an empirical likelihood interval (ELI) based on influence function, the other is a jackknife empirical likelihood (JEL) based interval. We prove that under very general conditions, −2log (empirical likelihood ratio) has an asymptotic standard chi squared distribution with one degree of freedom for mean medical cost. Also we show that the log-jackknife empirical likelihood ratio statistics follow standard χ2 distribution with one degree of freedom for mean medical cost. In the second part of the dissertation, we propose an influence function-based empirical likelihood method to construct a confidence region for the vector of regression parameters in mean cost regression models with censored data. The proposed confidence region can be used to obtain a confidence interval for the expected total cost of a patient with given covariates. The new method has sound asymptotic property (Wilks Theorem). In the third part of the dissertation we propose empirical likelihood method based on influence function to construct confidence intervals for quantile medical costs with censored data. We prove that under very general conditions, −2log (empirical likelihood ratio) has an asymptotic standard chi squared distribution with one degree of freedom for quantile medical cost. Simulation studies are conducted to compare coverage probabilities and interval lengths of the proposed confidence intervals with the existing confidence intervals. The proposed methods are observed to have better finite sample performances than existing methods. The new methods are also illustrated through a real example.

當我們在應用迴歸分析法時，往往會遇到兩個或多個自變數間存在著線性 關係的問題，即所謂多重共線性(multicollinearity)； 多重共線性的存 在會使得一般被廣泛運用的最小平方估計式 (least square estimator) 出現不穩定的情形。此估計式之總變異(total variance)會因共線性之程 度愈高而發散，呈現出不穩定的現象，進而影響其預測模式的能力。因此 相繼有學者提出改良共線性模式的方法，以期達到較精確且穩健的預測結 果。脊迴歸分析法(Ridge regression analysis) 便是其中之一；對於有 共線性存在之模式，若使用傳統脊估式，其總變異會較最小平方估計式穩 定。但傳統脊估式為一個偏量估計式(biased estimator)，故本文考慮採 用Jackknife 取一法以求降低脊迴歸估計式之偏量(bias)，此二法併用所 產生之一個新的估計式即本文所謂改良式脊迴歸估計式。本文將應用線性 模式Jackknife 估計式，配合脊迴歸分析法導出改良式脊迴歸估計式。並 另外利用電腦模擬出不同程度之共線性資料以比較分析傳統脊迴歸係數與 改良式脊迴歸係數，此二者於預測模式上之表現。結果顯示：改良式脊迴 歸估計係數對於降低估計偏差方面有顯著之改善，其預測能力亦優於傳統 脊迴歸係數，因此改良式脊迴歸估計式較傳統脊迴歸估計式更加穩定、精 確。迴歸分析是目前應用最廣泛之統計工具，不論是經濟模型、商業方面 以及醫學上之應用等均以求精求準之預測為主要目的，本文提出之改良式 脊迴歸係數，於共線性存在之迴歸模式下兼備了傳統脊迴歸係數穩定估計 式變異以求精，降低估計偏量以求準之優點，因此改良式脊迴歸係數於預 測模式上之貢獻是值得肯定的。

The Empirical Likelihood method (ELM) was introduced by A. B. Owen to test hypotheses in the early 1990s. It's a nonparametric method and uses the data directly to do statistical tests and to compute confidence intervals/regions. Because of its distribution free property and generality, it has been studied extensively and employed widely in statistical topics. There are many classical test statistics such as the Cramer-von Mises (CM) test statistic, the Anderson-Darling test statistic, and the Watson test statistic, to name a few. However, none is universally most powerful. This thesis is dedicated to extending the ELM to several interesting statistical topics in hypothesis tests. First of all, we focus on testing the fit of distributions. Based on the CM test, we propose a novel Jackknife Empirical Likelihood test via estimating equations in testing the goodness-of-fit. The proposed new test allows one to add more relevant constraints so as to improve the power. Also, this idea can be generalized to other classical test statistics. Second, when aiming at testing the error distributions generated from a statistical model (e.g., the regression model), we introduce the Jackknife Empirical Likelihood idea to the regression model, and further compute the confidence regions with the merits of distribution free limiting chi-square property. Third, the ELM based on some weighted score equations are proposed for constructing confidence intervals for the coefficient in the simple bilinear model. The effectiveness of all presented methods are demonstrated by some extensive simulation studies.

The correlation coefficient (CC) is a standard measure of the linear association between two random variables. The CC plays a significant role in many quantitative researches. In a bivariate normal distribution, there are many types of interval estimation for CC, such as z-transformation and maximum likelihood estimation based methods. However, when the underlying bivariate distribution is unknown, the construction of confidence intervals for the CC is still not well-developed. In this thesis, we discuss various interval estimation methods for the CC. We propose a generalized confidence interval and three empirical likelihood-based non-parametric intervals for the CC. We also conduct extensive simulation studies to compare the new intervals with existing intervals in terms of coverage probability and interval length. Finally, two real examples are used to demonstrate the application of the proposed methods.

The receiver operating characteristic (ROC) curve methodology is the statistical methodology for assessment of the accuracy of diagnostics tests or bio-markers. Currently most widely used statistical methods for the inferences of ROC curves are complete-data based parametric, semi-parametric or nonparametric methods. However, these methods cannot be used in diagnostic applications with missing data. In practical situations, missing diagnostic data occur more commonly due to various reasons such as medical tests being too expensive, too time consuming or too invasive. This dissertation aims to develop new nonparametric statistical methods for evaluating the accuracy of diagnostic tests or biomarkers in the presence of missing data. Specifically, novel nonparametric statistical methods will be developed with different types of missing data for (i) the inference of the area under the ROC curve (AUC, which is a summary index for the diagnostic accuracy of the test) and (ii) the joint inference of the sensitivity and the specificity of a continuous-scale diagnostic test. In this dissertation, we will provide a general framework that combines the empirical likelihood and general estimation equations with nuisance parameters for the joint inferences of sensitivity and specificity with missing diagnostic data. The proposed methods will have sound theoretical properties. The theoretical development is challenging because the proposed profile log-empirical likelihood ratio statistics are not the standard sum of independent random variables. The new methods have the power of likelihood based approaches and jackknife method in ROC studies. Therefore, they are expected to be more robust, more accurate and less computationally intensive than existing methods in the evaluation of competing diagnostic tests.

A simulation study consists of several steps such as data collection, coding and model verification, model validation, experimental design, output data analysis, and implementation. Our research concentrates on output data analysis. In this field, many researchers have studied how to construct confidence intervals for the mean u of a stationary stochastic process. However, the estimation of the value of a nonlinear function f(u) has not received a lot of attention in the simulation literature. Towards this goal, a batch-means-based methodology was proposed by Munoz and Glynn (1997). Their approach did not consider consistent estimators for the variance of the point estimator for f(u). This thesis, however, will consider consistent variance estimation techniques to construct confidence intervals for f(u). Specifically, we propose methods based on the combination of the delta method and nonoverlapping batch means (NBM), standardized time series (STS), or a combination of both. Our approaches are tested on moving average, autoregressive, and M/M/1 queueing processes. The results show that the resulting confidence intervals (CIs) perform often better than the CIs based on the method of Munoz and Glynn in terms of coverage, the mean of their CI half-width, and the variance of their CI half-width.

Great progress has been made in sleep medical research in recent years and sleep medicine has thus evolved into a specialized medical field. Sleep apnea syndrome is one of the mostly commonly seen sleep disorders. It is now clear that sleep apnea has adverse effects on the heart and is a risk factor for several cardiovascular diseases. Studies have found that decreased heart rate variability (HRV) is a prognostic factor for cardiovascular disease and it also associated with higher mortality rate. Considering the confounding effect of BMI and sleep apnea severity, this work investigates the correlation between heart rate variability and AHI (apnea-hypopnea index which is used to characterize the severity of sleep apnea) by dividing patients into different BMI subgroups. This work includes 1068 male subjects with complete overnight ECG recordings. The low-frequency (LF), the high-frequency (HF) component and the LF/HF ratio of HRV are computed for the 10 BMI subgroups. The Bootstrap method and the BCa technique for confidence interval estimation are employed to verify the linear association between the HRV measures and the severity of sleep apnea. The experimental results show that statically significant correlation exist between LF/HF ratio and AHI for BMI ¡Ù28 patient groups. Statically significant correlation between LF and AHI also exists for BMI ¡Ù27 patient groups. These results demonstrate that the associations between some of the HRV measures and AHI are clearly BMI dependent.

This dissertation addresses issues that arise in the classic linear instrumental variables (IV) model when some of the underlying assumptions are violated. / Economics

The estimation of low income proportions plays an important role in comparisons of poverty in different countries. In most countries, the stability of the society and the development of economics depend on the estimation of low income proportions. An accurate estimation of a low income proportion has a crucial role for the development of the natural economy and the improvement of people's living standards. In this thesis, the Jackknife empirical likelihood method is employed to construct confidence intervals for a low income proportion when the observed data had missing values. Comprehensive simulation studies are conducted to compare the relative performances of two Jackknife empirical likelihood based confidence intervals for low income proportions in terms of coverage probability. A real data example is used to illustrate the application of the proposed methods.

The use of doubly robust estimators is a key for estimating the population mean response in the presence of incomplete data. Cao et al. (2009) proposed an alternative doubly robust estimator which exhibits strong performance compared to existing estimation methods. In this thesis, we apply the jackknife empirical likelihood, the jackknife empirical likelihood with nuisance parameters, the profile empirical likelihood, and an empirical likelihood method based on the influence function to make an inference for the population mean. We use these methods to construct confidence intervals for the population mean, and compare the coverage probabilities and interval lengths using both the ``usual'' doubly robust estimator and the alternative estimator proposed by Cao et al. (2009). An extensive simulation study is carried out to compare the different methods. Finally, the proposed methods are applied to two real data sets.