Jackknife empirical likelihood for the difference of two volumes under ROC surfaces

The volume under a surface (VUS) is an effective measure for evaluating the discriminating power of a diagnostic test with three ordinal diagnostic groups. In this paper, we investigate the difference of two correlated VUS’s to compare two treatments for discrimination of three-class classification data. A jackknife empirical likelihood (JEL) procedure is employed to avoid the variance estimation in the existing methods. We prove that the limiting distribution of the empirical log-likelihood ratio statistic follows a \(\chi ^2\) distribution. Extensive numerical studies show that the JEL confidence intervals outperform those based on the normal approximation method. The proposed method is also applied to the Alzheimer’s disease data.     

Empirical likelihood for single-index models

The empirical likelihood method is especially useful for constructing confidence intervals or regions of the parameter of interest. This method has been extensively applied to linear regression and generalized linear regression models. In this paper, the empirical likelihood method for single-index regression models is studied. An estimated empirical log-likelihood approach to construct the confidence region of the regression parameter is developed. An adjusted empirical log-likelihood ratio is proved to be asymptotically standard chi-square. A simulation study indicates that compared with a normal approximation-based approach, the proposed method described herein works better in terms of coverage probabilities and areas (lengths) of confidence regions (intervals).     

Smoothed jackknife empirical likelihood method for ROC curve

In this paper we propose a smoothed jackknife empirical likelihood method to construct confidence intervals for the receiver operating characteristic (ROC) curve. By applying the standard empirical likelihood method for a mean to the jackknife sample, the empirical likelihood ratio statistic can be calculated by simply solving a single equation. Therefore, this procedure is easy to implement. Wilks’ theorem for the empirical likelihood ratio statistic is proved and a simulation study is conducted to compare the performance of the proposed method with other methods.     

Uniform Laws of Large Numbers for Triangular Arrays of Function-indexed Processes under Random Entropy Conditions

Mean Glivenko Cantelli Theorems are established for triangular arrays of rowwise independent processes. Methods developed by Pollard (1990) are combined with a truncation method essentially due to Alexander (1987). By this, applicability to partial sum processes in particular is achieved, for which Pollard’s truncation method fails. Nevertheless, the metric entropy condition appearing here is kept as weak as Pollard’s by means of application of Hoffmann-Jørgensen’s inequality, which has not been used so far in this context. The main theorem of the paper contains Pollard’s theorem as well as former results by Giné and Zinn (1984) and proves applicable to so-called random measure processes, certain function-indexed processes including empirical processes, partial-sum processes, the sequential empirical process and certain types of smoothed empirical processes. Statistical applications include nonparametric regression and the estimation of the intensity measure of a spatial Poisson process (Poisson point process).     

Empirical likelihood for the parametric part in partially linear errors-in-function models

Partially linear errors-in-function models were proposed by Liang (2000), but their inferences have not been systematically studied. This article proposes an empirical likelihood method to construct confidence regions of the parametric components. Under mild regularity conditions, the nonparametric version of the Wilk’s theorem is derived. Simulation studies show that the proposed empirical likelihood method provides narrower confidence regions, as well as higher coverage probabilities than those based on the traditional normal approximation method.     

Semi-empirical likelihood ratio confidence intervals for the difference of two sample means

We all know that we can use the likelihood ratio statistic to test hypotheses and construct confidence intervals in full parametric models. Recently, Owen (1988,Biometrika,75, 237–249; 1990,Ann. Statist.,18, 90–120) has introduced the empirical likelihood method in nonparametric models. In this paper, we combine these two likelihoods together and use the likelihood ratio to construct confidence intervals in a semiparametric problem, in which one model is parametric, and the other is nonparametric. A version of Wilks's theorem is developed.     

条件密度置信区间的覆盖精度

本文用经验似然方法讨论了条件密度的置信区间的构造. 通过对覆盖概率的Edgeworth展开得到了经验似然置信区间的覆盖精度, 同时证明了条件密度的经验似然置信区间的Bartlett可修正性     

Empirical likelihood for linear models with missing responses

The purpose of this article is to use an empirical likelihood method to study the construction of confidence intervals and regions for the parameters of interest in linear regression models with missing response data. A class of empirical likelihood ratios for the parameters of interest are defined such that any of our class of ratios is asymptotically chi-squared. Our approach is to directly calibrate the empirical log-likelihood ratio, and does not need multiplication by an adjustment factor for the original ratio. Also, a class of estimators for the parameters of interest is constructed, and the asymptotic distributions of the proposed estimators are obtained. Our results can be used directly to construct confidence intervals and regions for the parameters of interest. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths/areas of confidence intervals/regions. An example of a real data set is used for illustrating our methods.     

Coverage Accuracy of Confidence Intervals in Nonparametric Regression

Point-wise confidence intervals for a nonparametric regression function with random design points are considered. The confidence intervals are those based on the traditional normal approximation and the empirical likelihood. Their coverage accuracy is assessed by developing the Edgeworth expansions for the coverage probabilities. It is shown that the empirical likelihood confidence intervals are Bartlett correctable.     

EMPIRICAL LIKELIHOOD RATIO CONFIDENCE INTERVALS FOR VARIOUS DIFFERENCES OF TWO POPULATIONS

Recently the empirical likelihood has been shown to be very useful in nonparametric models. Qin combined the empirical likelihood thought and the parametric likelihood method to construct confidence intervals for the difference of two population means in a semiparametric model. In this paper, we use the empirical likelihood thought to construct confidence intervals for some differences of two populations in a nonparametric model. A version of Wilks' theorem is developed.     

Empirical bounds for ruin probabilities

We consider the classical model for an insurance business where the claims occur according to a Poisson process and where the distribution for the cost of each claim fulfills Cramér's tail-condition. Under these conditions Lundberg's constant R is of fundamental importance for ruin calculations.We derive estimates of R, based on an observation of the insurance business and investigate the statistical properties of those estimates. We further derive bounds and confidence intervals for ruin probabilities.     

相依序列密度函数的经验似然推断

本文利用了强平稳$m-$相依序列的特殊性质,讨论了$m-$相依序列密度函数的经验似然推断, 给出了似然比统计量的极限分布,可构造参数的经验似然置信区间. 并且通过模拟计算来说明有限样本下应用经验似然方法的合理性.     

Empirical likelihood for the contrast of two hazard functions with right censoring

In this paper, we consider the standard two-sample framework with right censoring. We construct useful confidence intervals for the ratio or difference of two hazard functions using smoothed empirical likelihood (EL) methods. The empirical log-likelihood ratio is derived and its asymptotic distribution is a standard chi-squared distribution. Bootstrap confidence bands are also proposed. Simulation studies show that the proposed EL confidence intervals have outperformed normal approximation methods in terms of coverage probability. It is concluded that the empirical likelihood methods provide better inference results.     

Empirical likelihood for single-index models with responses missing at random

The missing response problem in single-index models is studied, and a bias-correction method to infer the index coefficients is developed. Two weighted empirical log-likelihood ratios with asymptotic chisquare are derived, and the corresponding empirical likelihood confidence regions for the index coefficients are constructed. In addition, the estimators of the index coefficients and the link function are defined, and their asymptotic normalities are proved. A simulation study is conducted to compare the empirical likelihood and the normal approximation based method in terms of coverage probabilities and average lengths of confidence intervals. A real example illustrates our methods.     

Self-crossing Points of a Line Segment Process

This paper is devoted to planar stationary line segment processes. The segments are assumed to be independent, identically distributed, and independent of the locations (reference points). We consider a point process formed by self-crossing points between the line segments. Its asymptotic variance is explicitly expressed for Poisson segment processes. The main result of the paper is the central limit theorem for the number of intersection points in expanding rectangular sampling window. It holds not only for Poisson processes of reference points but also for stationary point processes satisfying certain conditions on absolute regularity (β-mixing) coefficients. The proof is based on the central limit theorem for β-mixing random fields. Approximate confidence intervals for the intensity of intersections can be constructed.     

Rigidity around Poisson submanifolds

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].     

Obstructions to the equivalence of Poisson structures near a symplectic leaf of semisimple and compact type

We describe cohomological obstructions to the equivalence of Poisson structures around a symplectic leaf of semisimple and compact type. The result is based on Conn’s linearization theorem and the theory of Poisson coupling.     

广义Lorenz 曲线的非参数统计推断

本文讨论了广义Lorenz 曲线的经验似然统计推断. 在简单随机抽样、分层随机抽样和整群随机抽样下, 本文分别定义了广义Lorenz 坐标的pro le 经验似然比统计量, 得出这些经验似然比的极限分布为带系数的自由度为1 的χ² 分布. 对于整个Lorenz 曲线, 基于经验似然方法类似地得出相应的极限过程. 根据所得的经验似然理论, 本文给出了bootstrap 经验似然置信区间构造方法, 并通过数据模拟, 对新给出的广义Lorenz 坐标的bootstrap 经验似然置信区间与渐近正态置信区间以及bootstrap 置信区间等进行了对比研究. 对整个Lorenz 曲线, 基于经验似然方法对其置信域也进行了模拟研究. 最后我们将所推荐的置信区间应用到实例中.     

Confidence band for expectation dependence with applications

Motivated by the applications of the concept of expectation dependence in economics and finance, we propose a method to construct uniform confidence band for expectation dependence. It is derived based on Hoeffding’s inequality. Our proposed confidence band can be explicitly expressed and thus it is very easy to implement. Our method has applications to demand for a risky asset and first-order risk aversion problems. Simulations suggest our proposed confidence interval can control the coverage probabilities very well, and the average lengths are very short. Two empirical applications are presented to illustrate the usefulness of the constructed confidence band of expectation dependence.     

Computational aspects of statistical intervals based on two Type-II censored samples

In this paper, we propose an efficient branch and bound procedure to compute exact nonparametric statistical intervals based on two Type-II right censored data sets. The procedure is based on some recurrence relations for the distribution and density functions of progressively Type-II censored order statistics which can be applied to compute the coverage probabilities. We illustrate the method for both confidence and prediction intervals of a given level.