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 log-likelihood 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.