The Restricted Mean Survival Time (RMST) serves as a valuable and extensively utilized metric in clinical trials. However, its application becomes intricate when dealing with data affected by length-biased sampling, rendering traditional inference strategies inadequate. To overcome this challenge, we advocate for the adoption of nonparametric techniques. One notably promising approach is the Empirical Likelihood (EL) method, which furnishes robust results without the need for stringent parametric assumptions. In practical scenarios, the underlying sampling distributions often remain elusive, necessitating adjustments in the case of parametric methodologies. The EL method has demonstrated its efficacy in addressing such complexities. Consequently, this paper introduces the EL method for computing RMST in situations involving both length-biased and right-censored data. Additionally, we introduce the concept of adjusted empirical likelihood (AEL) to further enhance the coverage probability, particularly when dealing with smaller sample sizes. To gauge the performance of the EL and AEL methods, we conduct simulations and rigorously compare their results. The findings unequivocally demonstrate that AEL-based confidence intervals consistently provide superior coverage probability when juxtaposed with EL-based intervals. Lastly, we substantiate the practical applicability of our proposed method by employing it in the analysis of a real dataset.

The restricted mean survival time (RMST) in the context of length-biased data is an important addition to clinical studies. The RMST is a widely used measurement for evaluating survival over a specific period, and the area under the survival function is a key component of this metric. However, when the data under study are length-biased, traditional parametric and classical methods for examining the RMST are not applicable. Nonparametric and semi-parametric methods are used to address this issue. We utilize the empirical likelihood (EL) method to investigate RMST.Our proposed EL procedure provides a reliable approach for inferential analysis of RMST in the presence of length-biased data. We have shown that the limiting distribution of the empirical log-likelihood ratio is a chi-square distribution with one degree of freedom. We also demonstrated that the likelihood ratio exhibits weak convergence to a mean-zero Gaussian process, which we used to construct a confidence band.In our simulation section, we compared the confidence intervals obtained from the normal approximation (NA) and EL methods. We showed that the EL method has a better coverage probability than the NA method. Additionally, we provided a real data application using bank customers' monthly taxes to illustrate further the effectiveness of our proposed method.

This study generalizes the joint empirical likelihood (JEL) which is named the joint penalized empirical likelihood(JPEL) and presents a comparative analysis of two innovative empirical likelihood methods: the restricted penalized empirical likelihood (RPEL) and the joint penalized empirical likelihood. These methods extend traditional empirical likelihood approaches by integrating criteria based on the minimum variance and unbiasedness of the estimator equations. In RPEL, estimators are obtained under these two criteria, while JPEL facilitates the joint application of the estimator equations used in RPEL, allowing for broader applicability.\\We evaluate the effectiveness of RPEL and RJEL in regression models through simulation studies, and evaluate the performance of RPEL and JPEL, focusing on parameter accuracy, model selection (as measured by the Empirical Bayesian Information Criterion), predictive accuracy (Mean Square Error), and robustness to outliers. Results indicate that RPEL consistently outperforms JPEL across all criteria, with RPEL yielding simpler models and more reliable estimates, particularly as sample sizes increase. These findings suggest that RPEL provides greater stability and interpretability for regression models, making it a superior choice over JPEL for the scenarios tested in this study.

Recently, it has been shown that the density based empirical likelihood concept extends and standardizes these methods, presenting a powerful approach for approximating optimal parametric likelihood ratio test statistics. In this article, we propose a density based empirical likelihood goodness of fit test for the Cauchy distribution. The properties of the test statistic are stated and the critical points are obtained. Power comparisons of the proposed test with some known competing tests are carried out via simulations. Our study shows that the proposed test is superior to the competitors in most of the considered cases and can confidently apply in practice. Finally, a financial data set is presented and analyzed.