Jackknife Empirical Likelihood Method Research Articles

A new portmanteau test for predictive regression models with possible embedded endogeneity

In the widely used predictive regression model, any possible serial correlation in innovations leads to estimation bias and statistical inference distortions. Hence, it is important to pretest the existence of such serial correlation. Nevertheless, in the presence of embedded endogeneity, which is a common problem in the predictive regression setting, traditional serial correlation tests such as Box–Pierce (BP) and Ljung–Box (LB) tests are found to perform poorly. Motivated by this, we develop a new portmanteau test in this article as a pretest for serial correlation in predictive regression under possible embedded endogeneity. This test is based on the sample splitting idea and the jackknife empirical likelihood method. The asymptotic distribution of the proposed test has been derived, and the Monte Carlo simulations confirm good finite sample performances. As an illustration, we apply our proposed test in pretesting the serial correlation in predictive regression, where financial variables are used to predict the excess return of S&P 500.

Bayesian empirical likelihood inference for the mean absolute deviation

The mean absolute deviation (MAD) is a direct measure of the dispersion of a random variable about its mean. In this paper, the empirical likelihood (EL) and the adjusted EL methods for the MAD are proposed. The Bayesian empirical likelihood, the Bayesian adjusted empirical likelihood, the Bayesian jackknife empirical likelihood and the Bayesian adjusted jackknife empirical likelihood methods are used to construct credible intervals for the MAD. Simulation results show that the proposed EL method performs better than the JEL in Zhao et al. [Jackknife empirical likelihood inference for the mean absolute deviation. Comput Stat Data Anal. 2015;91:92–101], and the proper prior information improves coverage rates of confidence/credible intervals. Two real datasets are used to illustrate the new procedures.

Jackknife empirical likelihood confidence intervals for the categorical Gini correlation

The categorical Gini correlation, ρg, was proposed by Dang et al. (2021) to measure the dependence between a categorical variable, Y, and a numerical variable, X. It has been shown that ρg has more appealing properties than current existing dependence measurements. In this paper, we develop the jackknife empirical likelihood (JEL) method for ρg. Confidence intervals for the Gini correlation are constructed without estimating the asymptotic variance. Adjusted and weighted JEL are explored to improve the performance of the standard JEL. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals. The proposed methods are illustrated in an application on two real datasets.

Jackknife empirical likelihood for the lower-mean ratio

Measuring economic inequality is an important topic to explore the social system. The Gini index and Pietra ratio are used by many people but are limited to reflecting the sampling distribution. In this paper, we study the interval estimates with another measure called the lower mean ratio u. By using jackknife empirical likelihood (JEL), adjusted jackknife empirical likelihood (AJEL), mean jackknife empirical likelihood (MJEL), mean adjusted jackknife empirical likelihood (MAJEL), and adjusted mean jackknife empirical likelihood methods, we propose the interval estimator for u. In the simulation study, we make a comparison of these methods in terms of coverage probability and average confidence interval length. The simulation results indicate that MAJEL performs the best among these methods for small sample sizes of the skewed distribution. For a small sample size of normal distribution, both JEL and MJEL show better performance than the other methods but MJEL is relatively time-consuming. Finally, two real data sets are analysed to illustrate the proposed methods.

Two-sample nonparametric test for proportional reversed hazards

In the last few decades, several works have been undertaken in the context of proportional reversed hazard rates (PRHR) but any specific statistical methodology for the PRHR hypothesis is absent in the literature. This paper proposes a two-sample nonparametric test based on two independent samples to verify the PRHR assumption. Based on a consistent U-statistic three statistical methodologies have been developed exploiting U-statistics theory, jackknife empirical likelihood and adjusted jackknife empirical likelihood method. A simulation study has been performed to assess the merit of the proposed test procedure. Finally, the proposed procedure is applied to a data set in the context of brain injury-related biomarkers and a data set related to Duchenne muscular dystrophy.

Bayesian jackknife empirical likelihood for the error variance in linear regression models

Variance estimation is fundamental in the statistical inference. Due to the nonlinearity of the variance estimator, Lin et al. [Jackknife empirical likelihood for the error variance in linear models. J Nonparametr Stat. 2017;29:151–166.] proposed the jackknife empirical likelihood method for the error variance in a linear regression model. However, people may have some prior information about the error variance. In this article, we propose the Bayesian jackknife empirical likelihood (BJEL) for the error variance in a linear regression model. The validity of the proposed method is verified, and the asymptotic normal properties for the BJEL are also established. A simulation study shows that the new approach for the small sample performs better than its frequentist counterpart. Two real data sets are also used to illustrate the proposed methods.

Transformed jackknife empirical likelihood for probability weighted moments

Probability weighted moments (PWMs) are a generalization of the usual moments of a probability distribution. In this paper, the jackknife empirical likelihood (JEL), the adjusted JEL (AJEL), the transformed JEL, which combines the merits of jackknife and transformed empirical likelihoods (TJEL), the transformed adjusted JEL (TAJEL), the mean jackknife empirical likelihood (MJEL), the mean adjusted jackknife empirical likelihood (MAJEL), and the adjusted mean jackknife empirical likelihood (AMJEL) methods, are considered to construct confidence intervals for probability weighted moments. Simulation results under various distributions show that MAJEL method always gives the best performance in terms of the coverage probability and average length among these methods, and TJEL shows better performance than AJEL and MJEL for small sample sizes, while MJEL is relatively time-consuming. The tests based on the proposed methods for PWMs are also developed. Real datasets are used to illustrate the proposed procedures.

Empirical likelihood test for the equality of several high-dimensional covariance matrices

The testing covariance equality is of importance in many areas of statistical analysis, such as microarray analysis and quality control. Conventional tests for the finite-dimensional covariance do not apply to high-dimensional data in general, and tests for the high-dimensional covariance in the literature usually depend on some special structure of the matrix and whether the dimension diverges. In this paper, we propose a jack-knife empirical likelihood method to test the equality of covariance matrices. The asymptotic distribution of the new test is regardless of the divergent or fixed dimension. Simulation studies show that the new test has a very stable size with respect to the dimension and it is also more powerful than the test proposed by Schott (2007) and studied by Srivastava and Yanagihara (2010). Furthermore, we illustrate the method using a breast cancer dataset.

Jackknife empirical likelihood for the error variance in linear errors-in-variables models with missing data

Measurement errors and missing data are often arise in practice. Under this circumstance, we focus on using jackknife empirical likelihood (JEL) and adjust jackknife empirical likelihood (AJEL) methods to construct confidence intervals for the error variance in linear models. Based on residuals of the models, the biased-corrected inverse probability weighted estimator of the error variance is introduced. Furthermore, we propose the jackknife estimator, jackknife and adjust jackknife empirical log-likelihood ratios of the error variance and establish their asymptotic distributions. Simulation studies in terms of coverage probability and average length of confidence intervals are conducted to evaluate the proposed method. A real data set is used to illustrate the proposed JEL and AJEL methods.

Jackknife empirical likelihood for the mean difference of two zero-inflated skewed populations

In constructing a confidence interval for the mean difference of two independent populations, we may encounter the problem of having a low coverage probability when there are many zeros in the data, and the non-zero values are highly positively skewed. The violation of the normality assumption makes parametric methods inefficient in such cases. In this paper, jackknife empirical likelihood (JEL) and adjusted jackknife empirical likelihood (AJEL) methods are proposed to construct a nonparametric confidence interval for the mean difference of two independent zero-inflated skewed populations. The JEL and AJEL confidence intervals are compared with the confidence intervals by normal approximation and empirical likelihood proposed by Zhou and Zhou (2005). Simulation studies are performed to assess the new methods. Two real-life datasets are also used as an illustration of the proposed methodologies.

Reduce the computation in jackknife empirical likelihood for comparing two correlated Gini indices

ABSTRACTThe Gini index has been widely used as a measure of income (or wealth) inequality in social sciences. To construct a confidence interval for the difference of two Gini indices from the paired samples, Wang and Zhao [‘Jackknife Empirical Likelihood for Comparing Two Gini Indices’, The Canadian Journal of Statistics, 44(1), 102–119] used a profile jackknife empirical likelihood. However, the computing cost with the profile empirical likelihood could be very expensive. In this paper, we propose an alternative approach of the jackknife empirical likelihood method to reduce the computational cost. We also investigate the adjusted jackknife empirical likelihood and the bootstrap-calibrated jackknife empirical likelihood to improve coverage accuracy for small samples. Simulations show that the proposed methods perform better than Wang and Zhao's methods in terms of coverage accuracy and computational time. Real data applications demonstrate that the proposed methods work very well in practice.

A test for equality of two distributions via integrating characteristic functions

In this paper, we investigate the problem of testing the equality of two distributions by integrating the squared norm of the difference between two corresponding empirical characteristic functions. This results in a linear combination of three different U-statistics. So the original testing problem is reduced to testing whether this linear combination is zero or not. We apply the Jackknife Empirical Likelihood (JEL) method to the new hypothesis testing problem. It has been proved that, under multivariate case, the log JEL statistic after scaling tends to a chi-square distribution with degree of freedom 1. Simulation studies are presented to assess the finite-sample performance of our method.

Statistical inference for the difference of two Lorenz curves

ABSTRACTTesting the Lorenz dominance is of importance in economic and social sciences. In this article, we propose new tools to do inferences for the difference of two Lorenz curves. The asymptotic normality of the proposed smoothed nonparametric estimator is proved. We also propose a smoothed jackknife empirical likelihood (JEL) method which avoids to estimate the complicate asymptotic variance. It is proved that the proposed JEL ratio statistics converge to the standard chi-square distribution. Simulation studies and real data analysis are also conducted, and show encouraging finite-sample performance.

Jackknife Empirical Likelihood Methods

Since Jing et al. [1] propose the jackknife empirical likelihood method for U-statistics, the method has been developed and applied to inference problems in statistical theory and other areas such as biostatistics, medical statistics and insurance. In this short review paper, we give an introduction on one sample and two-sample jackknife empirical methods and smoothed jackknife empirical likelihood methods and present a brief literature review on applications of these methods.

Jackknife empirical likelihood methods for Gini correlations and their equality testing

The Gini correlation plays an important role in measuring dependence of random variables with heavy tailed distributions, whose properties are a mixture of Pearson’s and Spearman’s correlations. Due to the structure of this dependence measure, there are two Gini correlations between each pair of random variables, which are not equal in general. Both the Gini correlation and the equality of the two Gini correlations play important roles in Economics. In the literature, there are limited papers focusing on the inference of the Gini correlations and their equality testing. In this paper, we develop the jackknife empirical likelihood (JEL) approach for the single Gini correlation, for testing the equality of the two Gini correlations, and for the Gini correlations’ differences of two independent samples. The standard limiting chi-square distributions of those jackknife empirical likelihood ratio statistics are established and used to construct confidence intervals, rejection regions, and to calculate p-values of the tests. Simulation studies show that our methods are competitive to existing methods in terms of coverage accuracy and shortness of confidence intervals, as well as in terms of power of the tests. The proposed methods are illustrated in an application on a real data set from UCI Machine Learning Repository.

Jackknife empirical likelihood method for multiply robust estimation with missing data

A novel jackknife empirical likelihood method for constructing confidence intervals for multiply robust estimators is proposed in the context of missing data. Under mild regularity conditions, the proposed jackknife empirical likelihood ratio has been shown to converge to a standard chi-square distribution. A simulation study supports the findings and shows the benefits of the proposed method. The latter has also been applied to 2016 National Health Interview Survey data.

Nonparametric Interval Estimators for the Coefficient of Variation.

The coefficient of variation (CV) is a widely used scaleless measure of variability in many disciplines. However the inference for the CV is limited to parametric methods or standard bootstrap. In this paper we propose two nonparametric methods aiming to construct confidence intervals for the coefficient of variation. The first one is to apply the empirical likelihood after transforming the original data. The second one is a modified jackknife empirical likelihood method. We also propose bootstrap procedures for calibrating the test statistics. Results from our simulation studies suggest that the proposed methods, particularly the empirical likelihood method with bootstrap calibration, are comparable to existing methods for normal data and yield better coverage probabilities for nonnormal data. We illustrate our methods by applying them to two real-life datasets.

Jackknife empirical likelihood-based inferences for Lorenz curve with kernel smoothing

ABSTRACTThe Lorenz curve describes the wealth proportion for an income-ordered population. In this paper, we introduce a kernel smoothing estimator for the Lorenz curve and propose a smoothed jackknife empirical likelihood method for constructing confidence intervals of Lorenz ordinates. Extensive simulation studies are conducted to evaluate finite sample performances of the proposed methods. A real dataset of Georgia professor’s income is used to illustrate the proposed methods.

Likelihood Inference on Semiparametric Models: Average Derivative and Treatment Effect

Over the past few decades, much progress has been made in semiparametric modelling and estimation methods for econometric analysis. This paper is concerned with inference (i.e. confidence intervals and hypothesis testing) in semiparametric models. In contrast to the conventional approach based on t-ratios, we advocate likelihood-based inference. In particular, we study two widely applied semiparametric problems, weighted average derivatives and treatment effects, and propose semiparametric empirical likelihood and jackknife empirical likelihood methods. We derive the limiting behaviour of these empirical likelihood statistics and investigate their finite sample performance through Monte Carlo simulation. Furthermore, we extend the (delete-1) jackknife empirical likelihood toward the delete-d version with growing d and establish general asymptotic theory. This extension is crucial to deal with non-smooth objects, such as quantiles and quantile average derivatives or treatment effects, due to the well-known inconsistency phenomena of the jackknife under non-smoothness.

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.