232 research outputs found

    Jackknife Emperical Likelihood Method and its Applications

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    In this dissertation, we investigate jackknife empirical likelihood methods motivated by recent statistics research and other related fields. Computational intensity of empirical likelihood can be significantly reduced by using jackknife empirical likelihood methods without losing computational accuracy and stability. We demonstrate that proposed jackknife empirical likelihood methods are able to handle several challenging and open problems in terms of elegant asymptotic properties and accurate simulation result in finite samples. These interesting problems include ROC curves with missing data, the difference of two ROC curves in two dimensional correlated data, a novel inference for the partial AUC and the difference of two quantiles with one or two samples. In addition, empirical likelihood methodology can be successfully applied to the linear transformation model using adjusted estimation equations. The comprehensive simulation studies on coverage probabilities and average lengths for those topics demonstrate the proposed jackknife empirical likelihood methods have a good performance in finite samples under various settings. Moreover, some related and attractive real problems are studied to support our conclusions. In the end, we provide an extensive discussion about some interesting and feasible ideas based on our jackknife EL procedures for future studies

    Novel Nonparametric Methods For ROC Curves

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    The receiver operating characteristic (ROC) curve is a widely used graphical method for evaluating the discriminating power of a diagnostic test or a statistical model in various areas such as epidemiology, industrial quality control and material testing, etc. One important quantitative measure summarizing the ROC curve is the area under the ROC curve (AUC). The accuracy of two diagnostic tests with right censored data can be compared using the difference of two ROC curves and the difference of two AUC\u27s. Moreover, the difference of two volumes under surfaces (VUS) is investigated to compare two treatments for the discrimination of three-category classification data, extending the ROC curve to the ROC surface in the three-dimensional case. A few scientific progresses have been achieved in ROC curves and its related fields over the past decades. In this dissertation, we propose a plug-in empirical likelihood (EL) procedure combining placement values and weighting of inverse probability techniques, to construct stable and precise confidence intervals of the ROC curves, the difference of two ROC curves, the AUC\u27s and the difference of two AUC\u27s with right censoring. We proved that the limiting distribution of the EL ratio is a weighted χ2\chi^2 distribution. Furthermore, we introduce a jackknife empirical likelihood (JEL) procedure to explore the difference of two correlated VUS\u27s with complete data. We proved that the limiting distribution of the proposed JEL ratio is a χ2\chi^2 distribution, i.e., the Wilk\u27s theorem holds. Extensive simulation studies demonstrate that the new methods have better performance than the existing methods in terms of coverage probability of confidence intervals in most cases. Finally, the proposed methods are applied to analyze data sets of Primary Biliary Cirrhosis (PBC), Alzheimer\u27s disease, etc

    New Non-Parametric Methods for Income Distributions

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    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

    Statistical Evaluation of Continuous-Scale Diagnostic Tests with Missing Data

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    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

    Weighted estimation of the dependence function for an extreme-value distribution

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    Bivariate extreme-value distributions have been used in modeling extremes in environmental sciences and risk management. An important issue is estimating the dependence function, such as the Pickands dependence function. Some estimators for the Pickands dependence function have been studied by assuming that the marginals are known. Recently, Genest and Segers [Ann. Statist. 37 (2009) 2990-3022] derived the asymptotic distributions of those proposed estimators with marginal distributions replaced by the empirical distributions. In this article, we propose a class of weighted estimators including those of Genest and Segers (2009) as special cases. We propose a jackknife empirical likelihood method for constructing confidence intervals for the Pickands dependence function, which avoids estimating the complicated asymptotic variance. A simulation study demonstrates the effectiveness of our proposed jackknife empirical likelihood method.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ409 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

    Collaborative research: statistical interference based on jackknife empirical likelihood methods

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    Issued as final reportUnited States. Dept. of Defens

    Jackknife Empirical Likelihood Inference For The Pietra Ratio

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    Pietra ratio (Pietra index), also known as Robin Hood index, Schutz coefficient (Ricci-Schutz index) or half the relative mean deviation, is a good measure of statistical heterogeneity in the context of positive-valued data sets. In this thesis, two novel methods namely adjusted jackknife empirical likelihood and extended jackknife empirical likelihood are developed from the jackknife empirical likelihood method to obtain interval estimation of the Pietra ratio of a population. The performance of the two novel methods are compared with the jackknife empirical likelihood method, the normal approximation method and two bootstrap methods (the percentile bootstrap method and the bias corrected and accelerated bootstrap method). Simulation results indicate that under both symmetric and skewed distributions, especially when the sample is small, the extended jackknife empirical likelihood method gives the best performance among the six methods in terms of the coverage probabilities and interval lengths of the confidence interval of Pietra ratio; when the sample size is over 20, the adjusted jackknife empirical likelihood method performs better than the other methods, except the extended jackknife empirical likelihood method. Furthermore, several real data sets are used to illustrate the proposed methods

    Empirical Likelihood Confidence Intervals for the Population Mean Based on Incomplete Data

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    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\u27\u27 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

    A New Jackknife Empirical Likelihood Method for U-Statistics

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    U-statistics generalizes the concept of mean of independent identically distributed (i.i.d.) random variables and is widely utilized in many estimating and testing problems. The standard empirical likelihood (EL) for U-statistics is computationally expensive because of its onlinear constraint. The jackknife empirical likelihood method largely relieves computation burden by circumventing the construction of the nonlinear constraint. In this thesis, we adopt a new jackknife empirical likelihood method to make inference for the general volume under the ROC surface (VUS), which is one typical kind of U-statistics. Monte Carlo simulations are conducted to show that the EL confidence intervals perform well in terms of the coverage probability and average length for various sample sizes
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