Lorenz curves are extensively used in economics to analyze income inequality metrics. In this thesis, we discuss confidence interval estimation methods for generalized Lorenz curve. We first obtain normal approximation (NA) and empirical likelihood (EL) based confidence intervals for generalized Lorenz curves. Then we perform simulation studies to compare coverage probabilities and lengths of the proposed EL-based confidence interval with the NA-based confidence interval for generalized Lorenz curve. Simulation results show that the EL-based confidence intervals have better coverage probabilities and shorter lengths than the NA-based intervals at 100p-th percentiles when p is greater than 0.50. Finally, two real examples on income are used to evaluate the applicability of these methods: the first example is the 2001 income data from the Panel Study of Income Dynamics (PSID) and the second example makes use of households’ median income for the USA by counties for the years 1999 and 2006

Empirical likelihood (EL) is a recently developed nonparametric method of statistical inference. Owen’s 2001 book contains many important results for EL with uncensored data. However, fewer results are available for EL with right-censored data. In this dissertation, we first investigate a right-censored-data extension of Qin and Lawless (1994). They studied EL with uncensored data when the number of estimating equations is larger than the number of parameters (over-determined case). We obtain results similar to theirs for the maximum EL estimator and the EL ratio test, for the over-determined case, with right-censored data. We employ hazard-type constraints which are better able to handle right-censored data. Then we investigate EL with right-censored data and a k-sample mixed hazard-type constraint. We show that the EL ratio test statistic has a limiting chi-square distribution when k = 2. We also study the relationship between the constrained Kaplan-Meier estimator and the corresponding Nelson-Aalen estimator. We try to prove that they are asymptotically equivalent under certain conditions. Finally we present simulation studies and examples showing how to apply our theory and methodology with real data.

La literatura empírica ha encontrado evidencia de una tendencia hacia el envejecimiento de la población en América Latina. Este documento analiza el impacto de los cambios demográficos sobre la pobreza utilizando las proyecciones demográficas de la Organización de las Naciones Unidas junto con distintos escenarios en la estructura educativa. La metodología utilizada en este trabajo es la de microsimulaciones econométricas. Su principal innovación consiste en proponer el método de máxima verosimilitud empírica como estrategia de simulación de ponderadores. Bajo todos los supuestos del modelo de simulación, los resultados sugieren que si la dinámica poblacional se mantiene los niveles de pobreza se verán reducidos. Sin embargo el efecto cuantitativo es muy débil, dejando un amplio margen para la planificación de políticas económicas orientadas a la reducción de la pobreza.

Learning from data, especially ‘Big Data’, is becoming increasingly popular under names such as Data Mining, Data Science, Machine Learning, Statistical Learning and High Dimensional Data Analysis. In this dissertation we propose a new related field, which we call ‘United Nonparametric Data Science’ - applied statistics with “just in time” theory. It integrates the practice of traditional and novel statistical methods for nonparametric exploratory data modeling, and it is applicable to teaching introductory statistics courses that are closer to modern frontiers of scientific research. Our framework includes small data analysis (combining traditional and modern nonparametric statistical inference), big and high dimensional data analysis (by statistical modeling methods that extend our unified framework for small data analysis). The first part of the dissertation (Chapters 2 and 3) has been oriented by the goal of developing a new theoretical foundation to unify many cultures of statistical science and statistical learning methods using mid-distribution function, custom made orthonormal score function, comparison density, copula density, LP moments and comoments. It is also examined how this elegant theory yields solution to many important applied problems. In the second part (Chapter 4) we extend the traditional empirical likelihood (EL), a versatile tool for nonparametric inference, in the high dimensional context. We introduce a modified version of the EL method that is computationally simpler and applicable to a large class of “large p small n” problems, allowing p to grow faster than n. This is an important step in generalizing the EL in high dimensions beyond the p ≤ n threshold where the standard EL and its existing variants fail. We also present detailed theoretical study of the proposed method.

This thesis investigates the statistical properties of Kernel Smoothed Empirical Likelihood (KSEL, e.g. Smith, 1997 and 2004) estimator and various associated inference procedures in weakly dependent data. New tests for structural stability are proposed and analysed. Asymptotic analyses and Monte Carlo experiments are applied to assess these new tests, theoretically and empirically. Chapter 1 reviews and discusses some estimation and inferential properties of Empirical Likelihood (EL, Owen, 1988) for identically and independently distributed data and compares it with Generalised EL (GEL), GMM and other estimators. KSEL is extensively treated, by specialising kernel-smoothed GEL in the working paper of Smith (2004), some of whose results and proofs are extended and refined in Chapter 2. Asymptotic properties of some tests in Smith (2004) are also analysed under local alternatives. These special treatments on KSEL lay the foundation for analyses in Chapters 3 and 4, which would not otherwise follow straightforwardly. In Chapters 3 and 4, subsample KSEL estimators are proposed to assist the development of KSEL structural stability tests to diagnose for a given breakpoint and for an unknown breakpoint, respectively, based on relevant work using GMM (e.g. Hall and Sen, 1999; Andrews and Fair, 1988; Andrews and Ploberger, 1994). It is also original in these two chapters that moment functions are allowed to be kernel-smoothed after or before the sample split, and it is rigorously proved that these two smoothing orders are asymptotically equivalent. The overall null hypothesis of structural stability is decomposed according to the identifying and overidentifying restrictions, as Hall and Sen (1999) advocate in GMM, leading to a more practical and precise structural stability diagnosis procedure. In this framework, these KSEL structural stability tests are also proved via asymptotic analysis to be capable of identifying different sources of instability, arising from parameter value change or violation of overidentifying restrictions. The analyses show that these KSEL tests follow the same limit distributions as their counterparts using GMM. To examine the finite-sample performance of KSEL structural stability tests in comparison to GMM's, Monte Carlo simulations are conducted in Chapter 5 using a simple linear model considered by Hall and Sen (1999). This chapter details some relevant computational algorithms and permits different smoothing order, kernel type and prewhitening options. In general, simulation evidence seems to suggest that compared to GMM's tests, these newly proposed KSEL tests often perform comparably. However, in some cases, the sizes of these can be slightly larger, and the false null hypotheses are rejected with much higher frequencies. Thus, these KSEL based tests are valid theoretical and practical alternatives to GMM's.

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Medical cost has received increasing interest recently in Biostatistics and public health. Statistical analysis and inference of life time medical cost have been challenging by the fact that the survival times are censored on some study subjects and their subsequent cost are unknown. Huang (2002) proposed the calibration regression model which is a semiparametric regression tool to study the medical cost associated with covariates. In this thesis, an inference procedure is investigated using empirical likelihood ratio method. The unadjusted and adjusted empirical likelihood confidence regions are constructed for the regression parameters. We compare the proposed empirical likelihood methods with normal approximation based method. Simulation results show that the proposed empirical likelihood ratio method outperforms the normal approximation based method in terms of coverage probability. In particular, the adjusted empirical likelihood is the best one which overcomes the under coverage problem.

Low income proportion (LIP), Lorenz curve (LC) and generalized Lorenz curve (GLC) are important indexes in describing the inequality of income distribution. They have been widely used for measuring social stability by governments around the world. The accuracy of estimating those indexes is essential to quantify the economics of a country. Established statistical inferential methods for these indexes are based on an asymptotic normal distribution, which may have poor performance when the real income data is skewed or has outliers. Recent applications of nonparametric methods, though, allow researchers to utilize techniques without giving data the parametric distribution assumption. For example, existing research proposes the plug-in empirical likelihood (EL)-based inferences for LIP, LC and GLC. However, this method becomes computationally intensive and mathematically complex because of the presence of nonlinear constraints in the underlying optimization problem. Meanwhile, the limiting distribution of the log empirical likelihood ratio is a scaled Chi-square distribution. The estimation of the scale constant will affect the overall performance of the plug-in EL method. To improve the efficiency of the existing inferential methods, this dissertation first proposes kernel estimators for LIP, LC and GLC, respectively. Then the cross-validation method is proposed to choose bandwidth for the kernel estimators. These kernel estimators are proved to have asymptotic normality. The smoothed jackknife empirical likelihood (SJEL) for LIP, LC and GLC are defined. Then the log-jackknife empirical likelihood ratio statistics are proved to follow the standard Chi-square distribution. Extensive simulation studies are conducted to evaluate the kernel estimators in terms of Mean Square Error and Asymptotic Relative Efficiency. Next, the SJEL-based confidence intervals and the smoothed bootstrap-based confidence intervals are proposed. The coverage probability and interval length for the proposed confidence intervals are calculated and compared with the normal approximation-based intervals. The proposed kernel estimators are found to be competitive estimators, and the proposed inferential methods are observed to have better finite-sample performance. All inferential methods are illustrated through real examples.

Empirical likelihood, which was pioneered by Thomas and Grunkemeier (1975) and Owen (1988), is a powerful nonparametric method of statistical inference that has been widely used in the statistical literature. In this thesis, we investigate the merits of empirical likelihood for various problems arising in ratio estimation. First, motivated by the smooth empirical likelihood (SEL) approach proposed by Zhou & Jing (2003), we develop empirical likelihood estimators for diagnostic test likelihood ratios (DLRs), and derive the asymptotic distributions for suitable likelihood ratio statistics under certain regularity conditions. To skirt the bandwidth selection problem that arises in smooth estimation, we propose an empirical likelihood estimator for the same DLRs that is based on non-smooth estimating equations (NEL). Via simulation studies, we compare the statistical properties of these empirical likelihood estimators (SEL, NEL) to certain natural competitors, and identify situations in which SEL and NEL provide superior estimation capabilities. Next, we focus on deriving an empirical likelihood estimator of a baseline cumulative hazard ratio with respect to covariate adjustments under two nonproportional hazard model assumptions. Under typical regularity conditions, we show that suitable empirical likelihood ratio statistics each converge in distribution to a 2 random variable. Through simulation studies, we investigate the advantages of this empirical likelihood approach compared to use of the usual normal approximation. Two examples from previously published clinical studies illustrate the use of the empirical likelihood methods we have described. Empirical likelihood has obvious appeal in deriving point and interval estimators for time-to-event data. However, when we use this method and its asymptotic critical value to construct simultaneous confidence bands for survival or cumulative hazard functions, it typically necessitates very large sample sizes to achieve reliable coverage accuracy. We propose using a bootstrap method to recalibrate the critical value of the sampling distribution of the sample log-likelihood ratios. Via simulation studies, we compare our EL-based bootstrap estimator for the survival function with EL-HW and EL-EP bands proposed by Hollander et al. (1997) and apply this method to obtain a simultaneous confidence band for the cumulative hazard ratios in the two clinical studies that we mentioned above. While copulas have been a popular statistical tool for modeling dependent data in recent years, selecting a parametric copula is a nontrivial task that may lead to model misspecification because different copula families involve different correlation structures. This observation motivates us to use empirical likelihood to estimate a copula nonparametrically. With this EL-based estimator of a copula, we derive a goodness-of-fit test for assessing a specific parametric copula model. By means of simulations, we demonstrate the merits of our EL-based testing procedure. We demonstrate this method using the data from Wieand et al. (1989). In the final chapter of the thesis, we provide a brief introduction to several areas for future research involving the empirical likelihood approach.