Jackknife Empirical Likelihood Research Articles

Smoothed jackknife empirical likelihood for the one-sample difference of quantiles

The one-sample quantile difference measure, which includes the interquartile range (IQR) of a given distribution, plays an important role in statistical sciences and econometrics. A jackknife empirical likelihood (JEL) method for the quantile difference is proposed using a novel smoothed nonparametric estimating equation. The asymptotic chi-square distribution for the JEL is proved and an algorithm for computing confidence intervals (CIs) is presented. Extensive simulation results demonstrate that JEL CIs have better coverage probability and interval length compared with CIs generated by classical empirical likelihood and normal approximation methods in most cases. The US Census Bureau’s Current Population Survey data set is used to illustrate the trends in household income inequality.

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.

Statistical Inference for a Relative Risk Measure

ABSTRACTFor monitoring systemic risk from regulators’ point of view, this article proposes a relative risk measure, which is sensitive to the market comovement. The asymptotic normality of a nonparametric estimator and its smoothed version is established when the observations are independent. To effectively construct an interval without complicated asymptotic variance estimation, a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation is provided with a Wilks type of result in case of independent observations. When data follow from AR-GARCH models, the relative risk measure with respect to the errors becomes useful and so we propose a corresponding nonparametric estimator. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.

Order-restricted inference for means with missing values.

Missing values appear very often in many applications, but the problem of missing values has not received much attention in testing order-restricted alternatives. Under the missing at random (MAR) assumption, we impute the missing values nonparametrically using kernel regression. For data with imputation, the classical likelihood ratio test designed for testing the order-restricted means is no longer applicable since the likelihood does not exist. This article proposes a novel method for constructing test statistics for assessing means with an increasing order or a decreasing order based on jackknife empirical likelihood (JEL) ratio. It is shown that the JEL ratio statistic evaluated under the null hypothesis converges to a chi-bar-square distribution, whose weights depend on missing probabilities and nonparametric imputation. Simulation study shows that the proposed test performs well under various missing scenarios and is robust for normally and nonnormally distributed data. The proposed method is applied to an Alzheimer's disease neuroimaging initiative data set for finding a biomarker for the diagnosis of the Alzheimer's disease.

Jackknife empirical likelihood for the error variance in linear models

ABSTRACTVariance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.

Uniform interval estimation for an AR(1) process with AR errors

Jonathan Hill1, Deyuan Li2∗ and Liang Peng3 Abstract. An empirical likelihood method was proposed in Hill and Peng (2014) to construct a unified interval estimation for the coefficient in an AR(1) model, regardless of whether the sequence was stationary or near integrated. The error term, however, was assumed independent, and this method fails when the errors are dependent. Testing for a unit root in an AR(1) model has been studied in the literature for dependent errors, but existing methods cannot be used to test for a near unit root. In this paper, assuming the errors are governed by an AR(p) process, we exploit the efficient empirical likelihood method to give a unified interval for the coefficient by taking the structure of errors into account. Furthermore, a jackknife empirical likelihood method is proposed to reduce the computation of the empirical likelihood method when the order in the AR errors is not small. A simulation study is conducted to examine the finite sample behavior of the proposed methods.

Jackknife empirical likelihood confidence regions for the evaluation of continuous-scale diagnostic tests with verification bias

Recently, Wang and Qin proposed various bias-corrected empirical likelihood confidence regions for any two of the three parameters, sensitivity, specificity, and cut-off value, with the remaining parameter fixed at a given value in the evaluation of a continuous-scale diagnostic test with verification bias. In order to apply those methods, quantiles of the limiting weighted chi-squared distributions of the empirical log-likelihood ratio statistics should be estimated. In order to facilitate application and reduce computation burden, in this paper, jackknife empirical likelihood-based methods are proposed for any pairs of sensitivity, specificity and cut-off value, and asymptotic results can be derived accordingly. The proposed methods can be easily implemented to construct confidence regions for the evaluation of continuous-scale diagnostic tests with verification bias. Simulation studies are conducted to evaluate the finite sample performance and robustness of the proposed jackknife empirical likelihood-based confidence regions in terms of coverage probabilities. Finally, a real case analysis is provided to illustrate the application of new methods.

Smoothed jackknife empirical likelihood for the difference of two quantiles

In this paper, we propose a smoothed estimating equation for the difference of quantiles with two samples. Using the jackknife pseudo-sample technique for the estimating equation, we propose the jackknife empirical likelihood (JEL) ratio and establish the Wilk’s theorem. Due to avoiding estimating link variables, the simulation studies demonstrate that JEL method has computational efficiency compared with traditional normal approximation method. We carry out a simulation study in terms of coverage probability and average length of the proposed confidence intervals. A real data set is used to illustrate the JEL procedure.

Jackknife empirical likelihood test for mean residual life functions

ABSTRACTMean residual life (MRL) function is an important function in survival analysis which describes the expected remaining life given survival to a certain age. In this article, we propose a non parametric method based on jackknife empirical likelihood through a U-statistic to test the equality of two mean residual functions. The asymptotic distribution of the test statistic has been derived. Simulations are conducted to illustrate the performance of the proposed test under different distributional assumptions and compare with some existing method. The proposed method is then applied to two real datasets.

A note on the two-sample mean problem based on jackknife empirical likelihood

ABSTRACTIn this article, we employ the jackknife empirical likelihood (JEL) method to construct the confidence regions for the difference of the means of two d-dimensional samples. Compared with traditional EL for the two-sample mean problem, JEL is extremely simpler to use in practice and is more effective in computing. Based on the JEL ratio test, a version of Wilks’ theorem is developed. Furthermore, to improve the coverage accuracy of confidence regions, a Bartlett correction is applied. The effectiveness of the proposed method is demonstrated by a simulation study and a real data analysis.

New non-parametric inferences for low-income proportions

Low-income proportion is an important index in describing the inequality of an income distribution. It has been widely used by governments in measuring social stability around the world. Established inferential methods for this index are based on the empirical estimator of the index. It may have poor finite sample performances when the real income data are skewed or has outliers. In this paper, based on a smooth estimator for the low-income proportion, we propose a smoothed jackknife empirical likelihood approach for inferences of the low-income proportion. Wilks theorem is obtained for the proposed jackknife empirical likelihood ratio statistic. Various confidence intervals based on the smooth estimator are constructed. Extensive simulation studies are conducted to compare the finite sample performances of the proposed intervals with some existing intervals. Finally, the proposed methods are illustrated by a public income dataset of the professors in University System of Georgia.

A consistent jackknife empirical likelihood test for distribution functions

In this paper, a jackknife empirical likelihood based approach is developed to test whether the underlying distribution is equal to a specified one. The limiting distribution of the proposed testing statistic is derived under some mild conditions. It turns out that the proposed test is consistent and easy to be implemented. Some simulation studies are conducted to evaluate the finite sample behaviors by comparing the proposed method with the existing one. A real data example is also analyzed to illustrate the proposed test approach.

Jackknife empirical likelihood for comparing two Gini indices

AbstractThe focus of this paper is to derive the jackknife empirical likelihood for the difference of two Gini indices. For independent data we propose a novel U‐statistic, which allows direct utilization of the jackknife empirical likelihood without involving a nuisance parameter. For paired data we established Wilks’ theorem for the profile likelihood after maximization over the nuisance parameter. Simulation studies show that our method is comparable to existing empirical likelihood methods in terms of coverage accuracy, but obtains much shorter intervals. The proposed methods are illustrated via analyzing a real data set.The Canadian Journal of Statistics44: 102–119; 2016 © 2016 Statistical Society of Canada

Jackknife empirical likelihood confidence interval for the Gini index

Jackknife empirical likelihood for the Gini index is derived. Adjusted jackknife empirical likelihood and bootstrap calibration are further investigated. The resulting interval estimators are comparable to existing empirical likelihood methods in terms of coverage accuracy, but yield much shorter intervals.

Jackknife empirical likelihood inferences for the population mean with ranked set samples

Without requiring any easily violated assumptions needed by existing rank-based nonparametric methods for ranked set samples, we propose a nonparametric approach for interval estimation and hypothesis testing for the population mean and the difference between two population means with balanced and unbalanced ranked set samples using jackknife empirical likelihood.

Jackknife empirical likelihood test for high-dimensional regression coefficients

A novel way to test coefficients in high-dimensional linear regression model is presented. Under the ‘large p small n’ situation, the traditional methods, like F-test and t-test, are unsuitable or undefined. The proposed jackknife empirical likelihood test has an asymptotic chi-square distribution and the conditions are much weaker than those in the existing methods. Moreover, an extension of the proposed method can test part of the regression coefficients, which is practical in considering the significance for a subset of covariates. Simulations show that the proposed test has a good control of the type-I error, and is more powerful than Zhong and Chen (2011)’s method in most cases. The proposed test is employed to analyze a rheumatoid arthritis data to find the association between rheumatoid arthritis and the SNPs on the chromosomes 6.

On high-dimensional two sample mean testing statistics: a comparative study with a data adaptive choice of coefficient vector

The key issues involved in two sample tests in high dimensional problems arise due to large dimension of the mean vector for a relatively small sample size. Recently, Wang et al. (Stat Sin 23:667---690, 2013) proposed a jackknife empirical likelihood test that works under weak assumptions on the dimension of variables (p), and showed that the test statistic has a chi-square limit regardless of whether p is finite or diverges. The sufficient condition required for this statistic is still restrictive. In this paper we significantly relax the sufficient condition for the asymptotic chi-square limit with models allowing flexible dependence structures and derive simpler alternative statistics for testing the equality of two high dimensional means. The proposed statistics have a chi-squared distribution or the maximum of two independent chi-square statistics as their limiting distributions, and the asymptotic results hold for either finite or divergent p. We also propose a data-adaptive method to select the coefficient vector, and compare the various methods in simulation studies. The proposed choice of coefficient vector substantially increases power in the simulation.

Jackknife empirical likelihood for linear transformation models with right censoring

A class of linear transformation models with censored data was proposed as a generalization of Cox models in survival analysis. This paper develops inference procedure for regression parameters based on jackknife empirical likelihood approach. We can show that the limiting variance is not necessary to estimate and the Wilk’s theorem can be obtained. The jackknife empirical likelihood method benefits from the simpleness in optimization using jackknife pseudo-value. In our simulation studies, the proposed method is compared with the traditional empirical likelihood and normal approximation methods in terms of coverage probability and computational cost.

A test for equality of two distributions via jackknife empirical likelihood and characteristic functions

The two-sample problem: testing the equality of two distributions is investigated. A jackknife empirical likelihood (JEL) test is proposed through incorporating characteristic functions, which reduces to a two-sample U-statistic. When the dimension of data is fixed, the nonparametric Wilks’s theorem for the proposed JEL ratio statistics is established. When the dimension diverges with the sample size at a moderate rate, p=o(n1/3), it is proved that under some mild conditions the normalized JEL ratio statistic has a standard normal limit. Moreover, when the dimension exceeds the sample size, p>n, an alternative version of JEL test is proposed. It is verified that under the null hypothesis this alternative version of JEL test has an asymptotical chi-squared distribution with two degrees of freedom. Some numerical results via simulation study and an analysis of a microarray dataset are presented and both confirm theoretical results empirically.

Interval estimation for a measure of tail dependence

Systemic risk concerns extreme co-movement of several financial variables, which involves characterizing tail dependence. The coefficient of tail dependence was proposed by Ledford and Tawn (1996, 1997) to distinguish asymptotic independence and asymptotic dependence. Recently a new measure based on the conditional Kendall’s tau was proposed by Asimit et al. (2015) to measure the tail dependence and to distinguish asymptotic independence and asymptotic dependence. For effectively constructing a confidence interval for this new measure, this paper proposes a smooth jackknife empirical likelihood method, which does not need to estimate any additional quantities such as asymptotic variance. A simulation study shows that the proposed method has a good finite sample performance.