Bootstrap-calibrated empirical likelihood confidence intervals for the difference between two Gini indexes

The mean residual life (MRL) function for a given random variable T is the expected remaining lifetime of T after a fixed time point t. It is of great interest in survival analysis, reliability, actuarial applications, duration modelling, etc. Liang, Shen, and He [‘Likelihood Ratio Inference for Mean Residual Life of Length-biased Random Variable’, Acta Mathematicae Applicatae Sinica, English Series, 32, 269–282] proposed empirical likelihood (EL) confidence intervals for the MRL based on length-biased right-censored data. However, their -2log(EL ratio) has a scaled chi-squared distribution. To avoid the estimation of the scale parameter in constructing confidence intervals, we propose a new empirical likelihood (NEL) based on i.i.d. representation of Kaplan–Meier weights involved in the estimating equation. We also develop the adjusted new empirical likelihood (ANEL) to improve the coverage probability for small samples. The performance of the NEL and the ANEL compared to the existing EL is demonstrated via simulations: the NEL-based and ANEL-based confidence intervals have better coverage accuracy than the EL-based confidence intervals. Finally, our methods are illustrated with a real data set.

For an AR(1) process with ARCH(1) errors, we propose empirical likelihood tests for testing whether the sequence is strictly stationary but has infinite variance, or the sequence is an ARCH(1) sequence or the sequence is an iid sequence. Moreover, an empirical likelihood based confidence interval for the parameter in the AR part is proposed. All of these results do not require more than a finite second moment of the innovations. This includes the case of t-innovations for any degree of freedom larger than 2, which serves as a prominent model for real data.

SUMMARY Confidence intervals for the relative frequency of responding cells in limiting dilution assays (LDAs) have not been examined closely. Generally, confidence intervals are calculated by using the normal approximation, sometimes preceded by a log transform. We evaluate and compare with analytical approaches: the normal approximation, the log transform, and two binomial methods-the quadratic approximation and a modification of the Clopper-Pearson exact method. We evaluate these confidence interval methods for use with the maximum likelihood and jackknife estimators of the relative frequency. Our results show that confidence interval construction with maximum likelihood point estimates is more accurate than construction with jackknife estimates. When using maximum likelihood estimates, the normal approximation produces acceptable two-sided confidence intervals, with the smallest length, at a = .05. At a = .01, the normal approximation is anticonservative. In all cases, the normal approximation is unable to produce adequate one-sided confidence intervals. The log transform and both binomial confidence interval methods are shown to be generally superior to

We study statistical inference for the derivative of the nonparametric regression function using local linear model based empirical likelihood. We first derive a normal equation for the derivative through the local linear model and use this equation to construct an empirical likelihood for the derivative. We show that the limiting distribution of the empirical likelihood ratio is a scaled χ 1 2 distribution rather than the usual (unscaled) χ 1 2 distribution. We use this limiting distribution to construct pointwise confidence intervals for the derivative. Such empirical likelihood ratio confidence intervals are easier to obtain than the normal approximation based confidence intervals. A small simulation study also suggests that they are more accurate.

In this paper, we consider the empirical likelihood confidence intervals for mean difference under the assumption of missing at random (MAR). We prove that the empirical log-likelihood ratio of mean difference converges to a standard chi-squared distribution. We conduct simulation study to compare the proposed empirical likelihood with the normal approximation method in terms of the coverage accuracies and average lengths of confidence intervals. By Monte Carlo simulations, we verify that the proposed method performs well.

We propose an estimator for the Gini coefficient, based on a ratio of means. We show how bootstrap and empirical likelihood can be combined to construct confidence intervals. Our simulation study shows the estimator proposed is usually less biased than customary estimators. The observed coverages of the empirical likelihood confidence interval proposed are also closer to the nominal value.

The mean residual life (MRL) function is one of the basic parameters of interest in survival analysis. In this paper, we develop three procedures based on modified versions of empirical likelihood (EL) to construct confidence intervals of the MRL function with length-biased data. The asymptotic results corresponding to the procedures have been established. The proposed methods exhibit better finite sample performance over other existing procedures, especially in small sample sizes. Simulations are conducted to compare coverage probabilities and the mean lengths of confidence intervals under different scenarios for the proposed methods and some existing methods. Two real data applications are provided to illustrate the methods of constructing confidence intervals.

For regression analysis of doubly truncated data, we propose two empirical likelihood (EL) inference approaches, called non-smooth EL and non-smooth Jackknife EL (JEL), to make inference about regression parameters based on the generalized estimating equations of existing weighted rank estimators. The limiting distributions of non-smooth log-EL and log-JEL ratios statistics are derived and non-smooth EL, and JEL confidence intervals for any specified component of regression parameters are obtained. We carry out extensive simulation studies to compare the proposed approaches with the random weighting (RW) approach. The simulation results demonstrate that the non-smooth EL and JEL confidence intervals have better performances than the RW confidence intervals based on coverage probability and average length of confidence intervals of regression parameters when the dependent variable is subject to the double truncation. A real data example is provided to illustrate the proposed approaches.

Pretest-posttest studies are commonly used for assessing the effect of a treatment or an intervention. We propose an empirical likelihood based approach to both testing and estimation of the treatment effect in non-randomized pretest-posttest studies where the posttest outcomes are subject to missingness. The proposed empirical likelihood ratio test and the estimation procedure are multiply robust in the sense that multiple working models are allowed for the propensity score of treatment assignment, the missingness probability and the outcome regression, and the validity of the test and the estimation requires only a certain combination of those multiple working models to be correctly specified. An empirical likelihood ratio confidence interval can be constructed for the treatment effect and has better coverage probabilities than confidence intervals based on the Wald statistic. Simulations are conducted to demonstrate the finite-sample performances of the proposed methods.

Cumulative Sum Curves and Their Prediction Limits

For some time, so-called empirical likelihoods have been used heuristically for purposes of nonparametric estimation. Owen showed that empirical likelihood ratio statistics for various parameters $\theta(F)$ of an unknown distribution $F$ have limiting chi-square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods. Our objective in this paper is twofold: first, to link estimating functions or equations and empirical likelihood; second, to develop methods of combining information about parameters. We do this by assuming that information about $F$ and $\theta$ is available in the form of unbiased estimating functions. Empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood. Efficiency results for estimates of both $\theta$ and $F$ are obtained. The methods are illustrated on several problems, and areas for future investigation are noted.

Recent literature sites ethnic origin as a major factor in developing pulmonary function reference values. Extensive studies established reference values for European and African Americans, but not for Hispanic Americans. The Third National Health and Nutrition Examination Survey defines Hispanic as individuals of Spanish speaking cultures. While no group was excluded from the target population, sample size requirements only allowed inclusion of individuals who identified themselves as Mexican Americans. This research constructs nonparametric reference value confidence intervals for Hispanic American pulmonary function. The method is applicable to all ethnicities. We use empirical likelihood confidence intervals to establish normal ranges for reference values. Its major advantage: it is model free, but shares asymptotic properties of model based methods. Statistical comparisons indicate that empirical likelihood interval lengths are comparable to normal theory intervals. Power and efficiency studies agree with previously published theoretical results.

Commentary

The purpose of this study was to investigate the impact of income inequality and parental socioeconomic status on several birth outcomes in Japan. Data were collected on birth outcomes and parental socioeconomic status by questionnaire from Japanese parents nationwide (n = 41,499) and then linked to Gini coefficients at the prefectural level in 2001. In multilevel analysis, z scores of birth weight for gestational age decreased by 0.018 (95% confidence interval (CI): -0.029, -0.006) per 1-standard-deviation (0.018-unit) increase in the Gini coefficient, while gestational age at delivery was not associated with the Gini coefficient. For dichotomous outcomes, mothers living in prefectures with middle and high Gini coefficients were 1.24 (95% CI: 1.05, 1.47) and 1.23 (95% CI: 1.02, 1.48) times more likely, respectively, to deliver a small-for-gestational-age infant than mothers living in more egalitarian prefectures (low Gini coefficients), although preterm births were not significantly associated with income distribution. Parental educational level, but not household income, was significantly associated with the z score of birth weight for gestational age and small-for-gestational-age status. Higher income inequality at the prefectural level and parental educational level, rather than household income, were associated with intrauterine growth but not with shorter gestational age at delivery.

This article develops empirical likelihood methodology for a class of long range dependent processes driven by a stationary Gaussian process. We consider population parameters that are defined by estimating equations in the time domain. It is shown that the standard block empirical likelihood (BEL) method, with a suitable scaling, has a non‐standard limit distribution based on a multiple Wiener–Itô integral. Unlike the short memory time series case, the scaling constant involves unknown population quantities that may be difficult to estimate. Alternative versions of the empirical likelihood method, involving the expansive BEL (EBEL) methods are considered. It is shown that the EBEL renditions do not require an explicit scaling and, therefore, remove this undesirable feature of the standard BEL. However, the limit law involves the long memory parameter, which may be estimated from the data. Results from a moderately large simulation study on finite sample properties of tests and confidence intervals based on different empirical likelihood methods are also reported.