The MRL function inference through empirical likelihood in length-biased sampling

In survival analysis or reliability studies, the mean residual life (MRL) function is the other important function to characterize a lifetime alongside the distribution function. In this paper, an empirical likelihood (EL) procedure based on length-biased data is proposed for inference on the MRL function and the asymptotic distribution of the empirical loglikelihood ratio for the MRL function is derived. We use limiting distribution to obtain EL ratio confidence intervals for the MRL function. Moreover, it is shown that the empirical log-likelihood ratio converges weakly to a mean zero Gaussian process. We apply this result to the construction of a Gaussian process approximation based confidence band for the MRL function. Also, a confidence interval for the MRL function is driven by using the normal approximation (NA) method in a length-biased setting. Simulation results are obtained to reveal the better efficiency and accuracy of the empirical likelihood-based confidence intervals in comparison to the proposed normal approximation-based method. A real data application is presented for better illustration.