Local likelihood of quantile difference under left-truncated, right-censored and dependent assumptions

We, in this paper, focus on the inference of conditional quantile difference (CQD) for left-truncated and right-censored model. Based on local conditional likelihood function of the observed data, local likelihood ratio function and smoothed local log-likelihood ratio (log-SLL) of the CQD are constructed, and the maximum local likelihood estimator of the CQD is further defined from the log-SLL. When the observations are assumed to be a sequence of stationary alpha-mixing random variables, we establish asymptotic normality of the defined estimator, and prove the Wilks' theorem of adjusted log-SLL. Besides, we define another estimator of the CQD based on product-limit estimator of conditional distribution function and give its asymptotic normality. Also, simulation study and real data analysis are conducted to investigate the finite sample behaviour of the proposed methods.